Porous Medium Equation with A Drift: Free boundary Regularity

TL;DR

This paper investigates the regularity of free boundaries in porous medium equations with drift, establishing $C^{1,eta}$ regularity under directional monotonicity, and introduces new non-degeneracy estimates.

Contribution

It provides the first local non-degeneracy estimates for porous medium equations with drift, leading to $C^{1,eta}$ free boundary regularity under directional monotonicity.

Findings

Proves $C^{1,eta}$ regularity of free boundaries with drift.

Establishes new local non-degeneracy estimates for the equation.

Extends regularity results to cases with drift, including zero drift.

Abstract

We study regularity properties of the free boundary for solutions of the porous medium equation with the presence of drift. We show the $C^{1,\alpha}$ regularity of the free boundary, when the solution is directionally monotone in space variable in a local neighborhood. The main challenge lies in establishing a local non-degeneracy estimate (Theorem 1.3 and Proposition 1.5), which appears new even for the zero drift case.

Equations

$${\varrho }_t=\Delta {\varrho }^m+\nabla \cdot (\varrho \vec{b})inQ:={\mathbb{R}}^d\times (0,\infty ),$$

$${\varrho }_t-\nabla \cdot ((\nabla u+\vec{b})\varrho )=0,$$

$$u=\frac{m}{m-1}{\varrho }^{m-1}.$$

$$V=-(\nabla u+\vec{b})\cdot \vec{n}=|\nabla u|-\vec{b}\cdot \vec{n}on(x,t)\in \Gamma :=\partial \{u>0\},$$

$$\vec{b}\in C_{x,t}^{3,1}(Q)\text{ and }{\varrho }_0\in L^1({\mathbb{R}}^d)\cap L^{\infty }({\mathbb{R}}^d).$$

$$u_t=(m-1)u\Delta u+|\nabla u{|}^2+\nabla u\cdot \vec{b}+(m-1)u\nabla \cdot \vec{b}$$

$$X(t):=X(x_0,t_0;t)is\; the\; unique\; solution\; of\; the\; ODE\{\begin{array}{rl}& {\partial }_{t}X(t)=-\vec{b}(X(t),t_0+t),t\in \mathbb{R},\\ & X(0)=x_0.\end{array}$$

$$u(x,t_0-s)=0\text{ if }|x-X(-s)|\le C_{\ast }{s}^{\beta },$$

$$u(x,t_0+s)>0\text{ if }|x-X(s)|\le C_{\ast }{s}^{\beta }.$$

$$u_0(x)\ge \gamma (d(x,{\Omega }_0^C){)}^{2-\varsigma }for\; some\gamma ,\varsigma >0,$$

$$u_{\epsilon ,A}(x,t):=\frac{1+A\epsilon }{(1+\epsilon {)}^2}u((1+\epsilon )x,(1+A\epsilon )t+B)for\; anyA,B,\epsilon >0,$$

$$x\cdot \nabla u+(At+B)u_t\ge 0\text{ on }\Gamma .$$

$$|\nabla u|=V=\frac{u_t}{|\nabla u|}\ge \frac{1}{(At+B)}\nu \cdot (\frac{x}{|x|})on\; \Gamma ,$$

$$W_{\theta ,\mu }:=\left\{y\in {\mathbb{R}}^d:\left|\frac{y}{|y|}-\mu \right|\le 2\sin \frac{\theta }{2}\right\}with\; axis\mu \in {\mathcal{S}}^{d-1}and\theta \in (0,\pi /2].$$

$$\underset{\epsilon \to 0^{+}}{\mathrm{liminf}}\frac{u(x+\epsilon \mu ,t)}{\epsilon }\ge {\kappa }_{\ast }for(x,t)\in \Gamma \cap Q_1.$$

$$u_t\le A(\mu \cdot \nabla u+u+1)in{Q}_1\text{ for some }A>0.$$

$${\nabla }_{\mu }u(x,t)\ge c_1\text{ in {u>0}∩Qδ.}$$

$$\Omega (u):=\{u>0\},{\Omega }_t(u):=\{u(\cdot ,t)>0\}$$

$${\Gamma }_t(u):=\partial {\Omega }_t,\Gamma (u):=\bigcup _{t\in (0,\infty )}({\Gamma }_t\times \{t\}).$$

$$⟨\nu ,\mu ⟩:=\arccos \left(\frac{\nu \cdot \mu }{|\nu ||\mu |}\right)\in [0,\pi ].$$

$$W_{\theta ,\mu }:=\{p\in {\mathbb{R}}^d:⟨p,\mu ⟩\le \theta \},{\hat{W}}_{\theta ,\nu }:=\{p\in {\mathbb{R}}^{d+1}:⟨p,\nu ⟩\le \theta \}.$$

$$\varrho \in C([0,T],L^1({\mathbb{R}}^d)),\varrho \vec{b}\in L^2([0,T]\times {\mathbb{R}}^d)\text{ and }{\varrho }^m\in L^2(0,T,{\dot{H}}^1({\mathbb{R}}^d))$$

$${\int }_0^T{\int }_{{\mathbb{R}}^d}\varrho {\varphi }_{t}dxdt\ge (\text{resp. }\le ){\int }_{{\mathbb{R}}^d}{\varrho }_0(x)\varphi (0,x)dx+{\int }_0^T{\int }_{{\mathbb{R}}^d}(\nabla {\varrho }^m+\varrho \vec{b})\nabla \varphi dxdt,$$

$$\psi (x,t):=\frac{1}{k}\exp (-\Vert \nabla \cdot \vec{b}{\Vert }_{\infty }t)$$

$${\varrho }_k(x,t)\ge \psi (x,t)>0.$$

$${\psi }_t-\Delta {\psi }^m-\nabla \cdot (\vec{b}\psi )\ge 0\text{ in }U\times [0,T]$$

$${\int }_0^T{\int }_{{\mathbb{R}}^d}\psi {\varphi }_{t}dxdt\le {\int }_{{\mathbb{R}}^d}\psi (0,x)\varphi (0,x)dx+{\int }_0^T{\int }_{{\mathbb{R}}^d}(\nabla {\psi }^m+\psi \vec{b})\nabla \varphi dxdt.$$

$$\Delta u>-\frac{\sigma }{\tau }-\sigma in{\mathbb{R}}^d\times [\tau ,\infty )$$

$$p_t$$

$$+\nabla p\cdot \vec{b}+2\Sigma u_{ij}{b}_j^i+\nabla u\cdot \Delta \vec{b}+(m-1)\left(p\nabla \cdot \vec{b}+2\nabla u\cdot \nabla (\nabla \cdot \vec{b})+u\Delta (\nabla \cdot \vec{b})\right).$$

$$\left|(m-1)p\nabla \cdot \vec{b}+2\Sigma u_{ij}{b}_j^i\right|$$

Full text

1. Introduction

Let us consider the drift-diffusion equation

[TABLE]

with a smooth vector field $\vec{b}:Q\to\mathbb{R}^{d}$, a non-negative initial data $\varrho(\cdot,0)=\varrho_{0}$ and $m>1$. The nonlinear diffusion term in (1.1) represents an anti-congestion effect ([3, 10, 14, 24]).

Our interest is on the regularity of the free boundary $\partial\{\varrho>0\}$, which is present at all times if starting with a compactly supported initial data. We are motivated by the intriguing fact that the free boundary regularity is open even for the travelling wave solutions in two space dimensions, with a smooth and laminar drift $\vec{b}(x_{1},x_{2})=(\sin x_{2},0)$ (see [22]). Our analysis provides a starting point of the discussion in a general framework, but the full answer to this question remains open (see Theorem 1.6 and the discussion below). The presence of the drift generates several significant challenges that are new to the problem, as we will discuss below.

To illustrate the regularizing mechanism of the interface, let us write (1.1) in the form of continuity equation,

[TABLE]

where

[TABLE]

Hence formally the normal velocity for the free boundary can be written as

[TABLE]

where $\vec{n}=\vec{n}_{x,t}$ is the outward normal vector at given boundary points. Given that $\varrho$ solves a diffusion equation, it would be natural to expect that the free boundary is regularized by the pressure gradient $|\nabla u|$ if $\vec{b}$ is smooth, as long as $u$ stays non-degenerate near the free boundary and topological singularities are ruled out. In general neither can be guaranteed even with zero drift. Below we discuss our main results and new challenges in the context of literature. We will always assume that

[TABLE]

Literature

Let us first discuss the case $\vec{b}=0$, in which case our problem (1.1) corresponds to the well-known Porous Medium Equation $(PME)$. In this case a vast amount of literature is available: we refer to the book [23]. What follows is a briefly discussion of several prominent results that are relevant to our results. Aronson and Benilan [2] showd the semi-convexity estimate $\Delta u>-\infty\;\text{ for }t>0$ which played a fundamental role in the regularity theory of $(PME)$. In general there can be a waiting time for degenerate initial data, where the free boundary does not move and regularization is delayed. When the initial data $u_{0}=u(\cdot,0)$ has super-quadratic growth at the free boundary, Caffarelli and Friedman [6] showed that there is no waiting time and the support of solution strictly expands in time. There an expansion rate of the support was obtained, by showing that its free boundary can be represented as $t=S(x)$ where $S$ is Hölder continuous. To discuss further regularity results, it is natural to require some geometric properties of the solution to rule out topological singularities such as merging of two fingers. The $C^{1,\alpha}$ regularity of the free boundary is established by Caffarelli and Wolanski [8], under the assumption of non-degeneracy and Lipschitz continuity of solutions. Their assumptions are shown to hold after a finite time $T_{0}>0$ by Caffarelli, Vazquez and Wolanski [7], where $T_{0}$ is the first time the support of solution expands to contain its initial convex hull. More recently, Kienzler explored the stability of solutions that are close to the flat traveling wave fronts to $(PME)$ [16]. Later Kienzler, Koch and Vazquez [17] improved this result and showed that solutions that are locally close to the traveling waves are smooth: see further discussion on their result in comparison to ours below Theorem 1.3.

When $\vec{b}\neq 0$, few results are available on the free boundary regularity of (1.1). With the exception of the particular choice $\vec{b}=x$, in general there appears to be no change of coordinates that eliminates the drift dependence in (1.1). Numerical experiments in [21] present the interesting possibility that an initially planar solution with smooth drift could develop corners without topological changes. However the non-degeneracy of pressure or the free boundary regularity is unknown even for traveling wave solutions in $\mathbb{R}^{2}$ (see [22]). By comparison, well-posedness and regularity theory for the solutions of (1.1) has been much better understood. Existence and uniqueness results are shown in [4] and [13] for weak solutions and in [20] for viscosity solutions. Asymptotic convergence to equilibrium of (1.1) is shown in [9] using energy dissipation when $\vec{b}$ is the gradient of a convex potential. Recently [19, 15] proved Hölder continuity of solutions for uniformly bounded, but possibly non-smooth drifts.

Discussion of main results and difficulties

For our analysis, we will consider the pressure variable (1.2) and the equation it satisfies:

[TABLE]

in $Q=\mathbb{R}^{d}\times(0,\infty)$.

We first show the semi-convexity (Aronsson-Benilan) estimate through a simple but novel barrier argument on $\Delta u$. This is where we use the $C^{3}_{x}$ norm of $\vec{b}$.

Theorem 1.1**.**

[Theorem 3.1]* Let $\rho$ solve (1.1) in $Q$ with (1.4), and let $u$ be the corresponding pressure variable given by (1.2). Then for some $\sigma>0$, $\Delta u>-\frac{\sigma}{t}-\sigma$ in the sense of distribution for all $t>0$.*

Next we discuss a weak non-degeneracy property in the event of zero initial waiting time. With zero drift this corresponds to the strict expansion property of the positive set, see section 14 [23]. In our case this property needs to be understood in terms of the streamlines, defined as

[TABLE]

While the streamlines are a natural coordinate for us to measure the strict expansion of the positive set over time, it does not cope well with the diffusion term in the equation. The most delicate scenario occurs with degenerate pressure, where the time range we need to observe is much larger than the space range. To deal with such case we need to carefully localize $\vec{b}$.

Theorem 1.2**.**

[Theorem 4.4] * Let $u$ be as given in Theorem 1.1, and fix $(x_{0},t_{0})\in\Gamma:=\partial\{u>0\}\cap\{t>0\}$. Then either of the following holds:*

• (Type one) $X(-s):=X({x_{0},t_{0}};-s)\in\Gamma$ for $s\in[0,t_{0}]$;
• (Type two) there exist $C_{*},\beta>1$ and $h>0$ such that for $s\in(0,h)$

[TABLE]

Moreover, if $u_{0}$ satisfies the near-boundary growth estimate

[TABLE]

then any point on $\Gamma$ is of type two.

The growth condition in (1.7) is optimal, since there is a stationary solution to (1.1) with a corner on its free boundary and with quadratic growth (see Theorem 7.3).

Next we proceed to show the non-degeneracy property of $u$, as it is essential for the regularity of its free boundary. This step presents the most challenging and novel part of our analysis. To illustrate the difficulties, let us briefly go over the main components of the celebrated arguments in [7], which provides non-degeneracy of solutions for (PME) for times $t>T_{0}$. One key ingredient in their analysis was the scale invariance of the equation under the transformation

[TABLE]

In [7] $u_{\varepsilon,A}$ was compared to $u$ to obtain the space-time directional monotonicity

[TABLE]

Applying (1.3) with $\vec{b}=0$ we then have

[TABLE]

where the first equality is from (1.3), the second equality is due to the level set formulation of the normal velocity, and the last inequality is due to (1.8) and the fact that $\nabla u$ is parallel to the negative normal $-\nu$ on the free boundary. Thus the non-degeneracy follows if we know that the free boundary is a Lipschitz graph with respect to the radial direction. This was shown in [7] for $t>T_{0}$ by the celebrated moving planes arguments, and thus we can conclude.

For nonzero drift, neither scaling invariance nor the moving planes method is available due to the inhomogeneity in $\vec{b}$. In fact it is not reasonable to expect consistent free boundary behavior for large times, except possibly when $\vec{b}$ is a potential vector field. Still, it is reasonable to expect that, without topological singularites and waiting time, the diffusive nature of the equation (1.5) regularizes the free boundary. With this in mind we show a local non-degeneracy result under the assumption of directional monotonicity and zero waiting time.

Let us define the spatial cone of directions

[TABLE]

We say $u$ is monotone with respect to $W_{\theta,\mu}$ if $u(\cdot,t)$ is non-decreasing along directions in $W_{\theta,\mu}$. We also denote $Q_{r}:=\{|x|\leq r\}\times(-r,r)$.

Theorem 1.3**.**

[Local Non-degeneracy, Corollary 5.7] * Let $\varrho$ be a weak solution to (1.1) in $Q_{2}$, where $\Gamma$ is of type two, and let $u$ be the pressure. Suppose in $Q_{2}$, $\Delta u>-\infty$ and $u$ is monotone with respect to $W_{\theta,\mu}$ for some $\theta$ and $\mu$. Then there exists $\kappa_{*}>0$ such that*

[TABLE]

For the proof we adopt a local barrier argument introduced in [11] in the context of the Hele-Shaw flow. Heuristically speaking the barrier argument illustrates the fact that the nondegeneracy property of positive level sets propagates to the free boundary as the positive set expands out in diffusive free boundary problems.

As mentioned above, in the zero drift case [17] considered solutions that are locally close to a planar traveling wave solution. Their assumption in particular endows a discrete small-scale flatness and non-degeneracy. It was shown there that over time the flatness improves in its scale to yield the smoothness of the solutions. It was conjectured there whether a cone monotonicity assumption could replace proximity to the planar travelling waves. While we do not pursue improvement of flatness in scale, our result yields a positive partial answer to this question.

Building on the above non-degeneracy result, we proceed to study the free boundary regularity. To prevent sudden changes in the evolution caused by changes in the far-away region, we assume that, in the weak sense,

[TABLE]

Theorem 1.4**.**

[Theorem 6.1]* Let $u$ be given as in Theorem 1.3. If in addition (1.10) holds, then $u$ is Lipschitz continuous and $\Gamma$ is $C^{1,\alpha}$ in $Q_{1/2}$.*

The proof of above theorem is given in Section 6. The novel ingredient in this section is the following result, which propagates the non-degeneracy of the solution at the free boundary to nearby positive level sets.

Proposition 1.5**.**

[Propagation of non-degeneracy, Proposition 6.3] Under the assumption of Theorem 1.4, there exist $\delta<\frac{1}{2}$ and $c_{1}>0$ such that

[TABLE]

From here, the proof of Theorem 1.4 largely follows the iterative argument given in [8], which compares in different scales the solution with its shifted version. For nonzero drifts (1.5) changes under coordinate shifts, and thus a notable modification is necessary in the iteration procedure. See Remark 6.9.

Now we address the traveling wave solutions discussed earlier in the introduciton.

Theorem 1.6**.**

*Let $\alpha:\mathbb{R}\to\mathbb{R}$ be a smooth and bounded function. Let $u$ solve (1.5) in $Q=\mathbb{R}^{2}\times(0,\infty)$ with $\vec{b}=(\alpha(x_{2}),0)$ and the initial data $u_{0}(x)=u(x,0)=(x_{1})_{+}$, under linear growth condition at infinity.

Then $\Gamma$ is locally uniformly $C^{1,\alpha}$ in $Q$.*

In [22] the existence of traveling wave solutions are shown with the above choice of $\vec{b}$. We consider the initially planar solution that was used in [21] to approximate the traveling waves. Our argument yields an exponentially decaying lower bound on the nondegeracy of $u$. While it rules out the possibility of finite time singularity for the approximate solutions, the free boundary regularity of travelling wave solutions remains open.

Lastly we present some examples which illustrate new types of free boundary singularities generated by drifts.

Theorem 1.7**.**

[Theorem 7.3 and  7.4]. There is $\vec{b}\in C_{x}^{3}(\mathbb{R}^{d})$ such that (1.5) has a stationary profile with a corner on its free boundary. There is a continuous spatial vector field $\vec{b}$ such that an initially smooth solution to (1.5) develops singularity on the free boundary in finite time.

Acknowledgements. Both authors are partially supported by NSF grant DMS-1566578. We would like to thank Jean-Michel Roquejoffre and Yao Yao for helpful discussions.

2. Preliminaries

$\circ$** Notations.**

• •

Throughout the paper we denote $\sigma$ as various universal constants, by which we mean constants that only depend on $m$, $d$, $\|\vec{b}\|_{C^{3,1}_{x,t}}$ and $\|\varrho_{0}\|_{L^{1}}+\|\varrho_{0}\|_{L^{\infty}}$.

• •

We use $C$ to represent constants which might depend on universal constants and other constants that are given in the assumptions of corresponding theorems.

• •

For a continuous, non-negative function $u:\mathbb{R}^{d}\times(0,\infty)\to\mathbb{R}$, we denote

[TABLE]

and

[TABLE]

When it is clear from the context we will omit the dependence on $u$.

• •

$B(x,r):=\{x\in\mathbb{R}^{d}:|x|\leq r\}$, $B_{r}:=B(0,r)$, $Q=\mathbb{R}^{d}\times[0,\infty)$ and $Q_{r}:=B_{r}\times(-r,r)$.

• •

$\nabla:=\nabla_{x}$, and $\hat{\nabla}:=(\nabla,\partial_{t})$. We also denote $f_{i}:=\partial_{x_{i}}f$, $f_{ij}:=\partial^{2}_{x_{i}x_{j}}f$.

• •

For $\nu,\mu\in\mathbb{R}^{d}\setminus\{0\}$, the angle between them are denoted by

[TABLE]

For $\mu\in\mathbb{R}^{d}$, $\nu\in\mathbb{R}^{d+1}$ and $\theta\in[0,\pi/2]$, we define the space and space-time cones by

[TABLE]

$\circ$** Notions of solutions and their smooth approximations.**

Next we recall the notion of weak solutions and their properties, including their smooth approximations that will be used in this paper.

Definition 2.1**.**

Let $\varrho_{0}$ be a non-negative function in $L^{\infty}(\mathbb{R}^{d})\cap L^{1}(\mathbb{R}^{d})$. We say that a non-negative and bounded function $\varrho:\mathbb{R}^{d}\times[0,T]\to[0,\infty)$ is a subsolution (resp. supersolution) to (1.1) with initial data $\varrho_{0}$ if

[TABLE]

and

[TABLE]

for all non-negative $\phi\in C_{c}^{\infty}(\mathbb{R}^{d}\times[0,T))$.

We say $\varrho$ is a weak solution to (1.1) if it is both sub- and supersolution of (1.1). We also say that $u:=\frac{m}{m-1}\varrho^{m-1}$ is a solution (resp. super/sub solution) to (1.5) if $\varrho$ is a weak solution (resp. super/sub solution) to (1.1).

The well-posedness result of general degenerate parabolic type equations is established in [1] - [4]. [12, 13] proved the Hölder regularity of solutions.

Theorem 2.1** (Theorem 1.7, [1]).**

Let $\varrho_{0}$ be as given in Definition 2.1. When $\vec{b}\in C^{3}(Q)$, then there exists a weak solution $\varrho$ to (1.1) with initial data $\varrho_{0}$. Moreover $\varrho$ is uniformly bounded for all $t\geq 0$.

Theorem 2.2** (Theorem 1, [13]).**

Suppose $\varrho$ is a non-negative, bounded weak solution to (1.1) in $Q_{1}$. Then $\varrho$ is Hölder continuous in $Q_{\frac{1}{2}}$.

Theorem 2.3** (Theorem 2.2, [1]).**

Suppose $U$ is an open subset of $\mathbb{R}^{d}$ and $\vec{b}\in C^{1,0}_{x,t}$. Let $\bar{\varrho},\underline{\varrho}$ be respectively a subsolution and a supersolution of (1.1) in $U\times\mathbb{R}^{+}$ such that $\bar{\varrho}\leq\underline{\varrho}$ a.e. in the parabolic boundary of $U\times\mathbb{R}^{+}$. Then $\bar{\varrho}\leq\underline{\varrho}$ in $U\times\mathbb{R}^{+}$.

Remark 2.4*.*

Following from Theorem 2.3, we have comparison principle for (1.5): suppose $\bar{u},\underline{u}$ are respectively a subsolution and a supersolution of (1.5) in $U\times\mathbb{R}^{+}$ such that $\bar{u}\leq\underline{u}$ a.e. on the parabolic boundary of $U\times\mathbb{R}^{+}$. Then $\bar{u}\leq\underline{u}$ in $U\times\mathbb{R}^{+}$.

In our analysis it is often convenient to work with classical solutions of (1.1), which is made possible by the following result. We will rely on this approximation lemma in Theorem 3.1 and in Section 5.

Lemma 2.5** (Section 9.3 [23]).**

Let $U$ be either $B_{1}$ or $\mathbb{R}^{d}$, and consider $\varrho_{0}\in L^{1}(U)\cap L^{\infty}(U)\cap C(U)$. Let $\varrho$ be a weak solution of (1.1) in $U\times[0,1]$ that is in $C(\overline{U}\times[0,1])$ with initial data $\varrho_{0}$. Then there exists a sequence of strictly positive, classical solutions $\varrho_{k}$ of (1.1) such that $\varrho_{k}\to\varrho$ locally uniformly in $U\times(0,1]$ as $k\to\infty$.

Proof.

Let us consider $U=B_{1}$. Consider $\varrho_{0,k}=\varrho_{0}+\frac{1}{k}$ and let $\varrho_{k}$ be the weak solution to (1.1) in $U$ with initial data $\varrho_{0,k}$ and Dirichlet boundary condition $\varrho_{k}=\varrho+\frac{1}{k}$ on $\partial U\times(0,1]$. Note that

[TABLE]

is a subsolution to (1.1) in $U\times(0,1]$ with $\psi\leq\frac{1}{k}$ on the parabolic boundary. Thus from the comparison principle it follows that

[TABLE]

Since $\varrho_{k}$ is uniformly bounded away from zero in $U\times[0,T]$, (1.1) is uniformly parabolic. In view of the standard parabolic theory, it follows that $\varrho_{k}$ is smooth in $U\times(0,T]$. The proof for locally uniform convergence of $\varrho_{k}$ to $\varrho$ is parallel to that of Lemma 9.5 in [23]. ∎

To end this section, we state the following technical lemma which is used for comparison.

Lemma 2.6**.**

Set $U:=B_{1}$ or $\mathbb{R}^{d}$. Let $\psi$ be a non-negative continuous function defined in $U\times[0,T]$ such that

• (a)

$\psi$* is smooth in its positive set and in the set we have $\psi_{t}-\Delta\psi^{m}-\nabla\cdot(\vec{b}\,\psi)\geq 0$,*

• (b)

$\psi^{\alpha}$* is Lipschitz continuous for some $\alpha\in(0,m)$,*

• (c)

$\Gamma(\psi)$* has Hausdorff dimension $d-1$.*

Then

[TABLE]

in the weak sense i.e. for all non-negative $\phi\in C_{c}^{\infty}(U\times[0,T))$

[TABLE]

We postpone the proof to the appendix.

3. Regularity of the pressure

In this section we establish two basic properties for the pressure variable $u$ that we will frequently use in the rest of the paper. We begin by obtaining the fundamental estimate.

Theorem 3.1**.**

Let $u$ be a solution of (1.5) in $Q=\mathbb{R}^{d}\times[0,\infty)$ with non-negative initial data $u_{0}$ such that $u_{0}^{\frac{1}{m-1}}\in L^{1}(\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{d})$. Then there exists a universal constant $\sigma$ such that

[TABLE]

*in the sense of distribution. *

Proof.

By Lemma 2.5, it is enough to consider positive smooth solutions with positive smooth initial data. If (3.1) holds for the approximated smooth solutions, from the locally uniform convergence of the approximation we can conclude.

Assume that $u$ is positive and smooth, and consider $p:=\Delta u$. By differentiating (1.5) twice, we get

[TABLE]

By Young’s inequality,

[TABLE]

Thus we obtain

[TABLE]

Viewing $u$ as a known function, we may write the above quasilinear parabolic operator of $p$ as $\mathcal{L}_{0}(p)$ and so we have $\mathcal{L}_{0}(p)\geq 0$. Below will construct a barrier for this operator to obtain a lower bound for $p$.

Suppose that $\Delta u(\cdot,0)\geq-\frac{1}{\tau}$ for some $\tau>0$. By Theorem 1.7 [1], $u$ is uniformly bounded by a universal constant and we denote it as $\sigma_{0}$. Let $w:=-\frac{\sigma_{1}}{t+\tau}+u-\sigma_{2}$ for some $\sigma_{1}\geq 1,\sigma_{2}\geq\sigma_{0}$ to be determined later. Then $p\geq w$ at $t=0$.

Direct computation yields

[TABLE]

[TABLE]

Now we use the equation (1.5) to obtain

[TABLE]

where the last inequality holds if we choose $\sigma_{1}:=d$ and $\sigma_{2}:=\sigma_{0}+(4d\sigma)^{1/2}$. Hence $\mathcal{L}_{0}(w)\leq 0\leq\mathcal{L}_{0}(p)$, and from the comparison principle for $\mathcal{L}_{0}$ we conclude that

[TABLE]

After taking $\tau\to 0$, we obtain that (3.1) holds for smooth solutions. We can conclude by Lemma 2.5.

∎

Remark 3.2*.*

Using the same barrier in the proof of the lemma, it can be seen that if $\Delta u_{0}\geq-C_{0}$ in the sense of distribution, then $\Delta u\geq-\frac{\sigma_{1}}{t+(1/C_{0})}-\sigma_{2}$ in the distribution sense for all time.

Next we prove a useful property: the consistency of positivity set of a solution along streamlines over time. The proof is parallel to the proof of Lemma 3.5 [18] where they used a barrier argument. Recall that we denote ${\Omega}_{t}=\{u(\cdot,t)>0\}$.

Lemma 3.3**.**

Let $u$ solves (1.5) with $\Delta u>-\infty$ in $Q_{2}$. Then for $X(x,t;s)$ given in (1.6) and for $c_{0}:=\frac{1}{2(1+\|\vec{b}\|_{\infty})}$ the following is true.

[TABLE]

If $u$ solves (1.5) in $Q=\mathbb{R}^{d}\times[0,\infty)$ with initial data $u_{0}$ given as in Theorem 3.1, then

[TABLE]

Proof.

In view of Theorem 3.1, the second statement follows easily from the first one. To prove the first statement, it is suffices to show that for all $x\in\Omega_{t}$ and $s\in(0,c_{0}]$, if $X(x,t;s)\in B_{1}$ then $u(X(x,t;s))>0$.

If $x\in B_{\frac{3}{2}}^{c}$, by the choice of $c_{0}$,

[TABLE]

Thus we take $x\in\Omega_{t}\cap B_{\frac{3}{2}}$ and then $X(x,t;s)$ is inside the domain $B_{2}$ for all $s\in(0,c_{0}]$. By Theorem 2.2, $u$ is continuous in $Q_{2}$. Then we can suppose for contradiction that there exists $s_{0}\in(0,c_{0}]$ such that

[TABLE]

Suppose $\Delta u\geq-C_{0}$ in $Q_{2}$. Note that (1.5) is uniformly parabolic in any compact subset of $\{u>0\}$, due to the continuity of $u$. Therefore by the standard parabolic theory, $u$ is smooth in $\Omega\cap Q_{2}$. It follows from (1.5) that for all $s\in(0,s_{0})$,

[TABLE]

where $C:=(m-1)(C_{0}+\|\nabla\cdot\vec{b}\|_{\infty})$. This yields

[TABLE]

which, after taking $s\to s_{0}<1$, contradicts with the assumption that $u(X(x,t;s_{0}),t+s_{0})=0$.

∎

4. Regularity of the Free Boundary

In this section we study finer properties on expansion of the positive set $\{u>0\}$ along the streamlines associated with the drift $\vec{b}$. We largely follow the ideas in [6] applied to the zero drift case, and obtain corresponding statements (Lemma 4.1 and 4.2) for our problem.

Lemma 4.1**.**

Let $u$ be given as in Theorem 3.1. For any $t_{0}\geq\eta_{0}>0$ there exist $\tau_{0},c_{0}$ depending only on $\eta_{0}$ and universal constants such that the following holds. For any $R>0$ and $\tau\in(0,{\tau_{0}})$, if

[TABLE]

then

[TABLE]

Proof.

For simplicity, suppose $x_{0}=0,t_{0}=0$, and consider the rescaled function

[TABLE]

Then $\tilde{u}$ satisfies

[TABLE]

Theorem 3.1 yields

[TABLE]

Set $\varepsilon:=C_{0}\tau_{0}$ so that $\Delta\tilde{u}=\tau\Delta u\geq-\varepsilon$. From our assumption, it follows that

[TABLE]

Using this and that $\tilde{u}+\varepsilon|x|^{2}/(2d)$ is subharmonic, we find for $x\in B(\tilde{X}(1),\frac{1}{2}),$

[TABLE]

Now consider

[TABLE]

Then $\Delta v\geq-\varepsilon$. Moreover, observe that $v$ is the weak solution of

[TABLE]

We used Definition 2.1 as the notion of weak solutions, where $\vec{b}$ is replaced by $\vec{b}^{\prime}(x+\tilde{X},t)-\vec{b}^{\prime}(\tilde{X},t)$. Since the operator $\mathcal{L}$ is locally uniformly parabolic in its positive set, $v$ is smooth in the set due to the standard parabolic theory. From the above equation, $v$ satisfies the following in the classical sense in its positive set

[TABLE]

where the first inequality is due to the fact that $|\nabla\,\vec{b}^{\prime}|\leq\tau\sigma$ and the second inequality follows from Young’s inequality. Because $v$ is continuous and non-negative, the above estimate also holds weakly in the whole domain.

Since $\varepsilon=C_{0}\tau\geq\tau$, we obtain

[TABLE]

and thus by Gronwall

[TABLE]

Using (4.5), we conclude that for all $(x,t)\in B_{\frac{1}{2}}\times(0,1)$ and some $\sigma\geq 1$,

[TABLE]

if $\varepsilon$ is sufficiently small.

To conclude we proceed with a barrier argument applied to the operator $\mathcal{L}$. Define

[TABLE]

and we aim at showing ${\mathcal{L}}(\varphi)\geq 0$ weakly. Using Lemma 2.6 to $(\frac{m-1}{m}\varphi)^{\frac{1}{m-1}}$, the corresponding density variable of $\varphi$, and the Lipschitz continuity of $\varphi$, we find that to show $\varphi$ is a supersolution of ${\mathcal{L}}$, it suffices to prove ${\mathcal{L}}(\varphi)\geq 0$ in the positive set of $\varphi$.

Notice

[TABLE]

so direct computations yield that if

[TABLE]

then $\mathcal{L}(\varphi)\geq 0$ for $\frac{1}{3}-\frac{t}{6}<|x|<\frac{1}{2}$ in the classical sense. The inequality (4.8) is valid for $t\in(0,1)$ provided that we take $0<\varepsilon<<\lambda<<1$. With this choice of $\varepsilon,\lambda$, we get $\mathcal{L}(\varphi)\geq 0$ in $|x|<\frac{1}{2}$ weakly. By the assumption ${v}(x,0)=0$ in $B_{\frac{1}{2}}$ and thus ${v}\leq\varphi$ on $|x|\leq\frac{1}{2},t=0$. On the lateral boundary $|x|=\frac{1}{2},t\in(0,1)$, by (4.7) if $c_{0},\varepsilon$ are small enough depending on universal constants we have

[TABLE]

Hence by comparison principle for the operator $\mathcal{L}$ (see Remark 2.4) in $B_{\frac{1}{2}}\times(0,1)$ we have ${v}\leq\varphi.$ In particular

[TABLE]

for $|x|<\frac{1}{6}$ and we proved the lemma.

∎

Remark 4.2*.*

One can check that the conclusion of the lemma also holds in a local setting: If $u$ solves (1.5) with $\Delta u\geq-C_{0}$ in $Q_{1}$ for some $C_{0}$. Then there exist $\tau_{0},c_{0},\sigma$ such that (4.1) implies (4.2) for any $R\in(0,\sigma)$ and $\tau\in(0,{\tau_{0}})$. Here $\tau_{0},c_{0}$ depend only on $C_{0}$ and universal constants, and $\sigma$ is universal. This local version of the lemma will be used in Lemma 6.2.

Lemma 4.3**.**

Let $u$ be as in Theorem 3.1. For any $t_{0}\geq\eta_{0}>0$ and $c_{1}>0$, there exist $\lambda,c_{2},\tau_{0}>0$ depending on $c_{1}$, $\eta_{0}$ and universal constants such that the following holds. For any $R>0$ and $0<\tau\leq\tau_{0}$, if

[TABLE]

then

[TABLE]

Proof.

Let $C_{0}$ be as in (4.4), and set $(x_{0},t_{0})=(0,0)$ by shifting coordinates. We consider the corresponding density variable $\varrho(x,t):=(\frac{m-1}{m}u(x,t))^{\frac{1}{m-1}}$ and its rescaled version

[TABLE]

Let $\vec{b}^{\prime}$, $\tilde{X}$ be as in (4.3) and let $\varepsilon=C_{0}\tau$ as in the proof of Lemma 4.1. Then $\tilde{\varrho}$ solves the re-scaled density equation

[TABLE]

The fundamental estimate on $u$ implies that $\Delta\tilde{\varrho}^{m}\geq-\varepsilon\tilde{\varrho}$ in the sense of distribution.

Let us define ${\xi}(x,t):=\tilde{\varrho}(x+\tilde{X},t)$ and $Y(t):=\int_{B_{1}}{\xi}^{m}(x,t)dx$. Below we study properties on the growth rate of $Y$ using properties of $\tilde{\varrho}$, namely we derive (4.12) and (4.13). We then use these estimates to argue by a contradiction to prove our main statement.

First let us show that $Y(\lambda)$ stays sufficiently positive if $\varepsilon\lambda$ is small. Since $\tilde{X}(0)=0$, our assumption yields that

[TABLE]

Due to (4.6) and ${v}(x,t)=\frac{m}{m-1}{\xi}^{m-1}(x,t)$, for $\varepsilon$ small enough

[TABLE]

Consequently

[TABLE]

for $t\in(0,\lambda]$ if $\varepsilon\lambda<<_{\sigma}1$.

Next we obtain an upper bound for the growth of $Y$ over time.

Claim: For some universal constants $\sigma_{1},\sigma_{2}$ and $\gamma$,

[TABLE]

Proof of the Claim. As in [6], we introduce the Green’s function in a unit ball so that $G$ solves

[TABLE]

Let us only discuss the dimension $d\geq 3$, where $G$ is defined as

[TABLE]

We want to differentiate $\int_{B(\tilde{X},1)}G(x-\tilde{X})\tilde{\varrho}(x,t)dx$ with respect to $t$. Since $G(x-\tilde{X})=0$ on $\partial B(\tilde{X},1)$,

[TABLE]

Since ${\nabla}\vec{b}^{\prime}\geq-\sigma\varepsilon I_{d}$,

[TABLE]

As for $A_{2}$, applying (4.14), we obtain

[TABLE]

Using (4.17), (4.18), we find for some universal $\sigma>0$

[TABLE]

Hence we derive

[TABLE]

which simplifies to

[TABLE]

Now following the proof of Lemma 2.3 [6], using (4.19) and the integrability property of $G$, we can obtain the upper bound $\int_{{{B_{1}}}}G{\xi}\,dx$ to conclude. We omit the computation since it is parallel to [6].

$\Box$.

Going back to the proof of Lemma 4.3, let us suppose that our statement is false, which means $u(X(\lambda\tau),\lambda\tau)<c_{2}\frac{R^{2}}{\tau}$ for any choice of $\lambda,c_{2},\tau_{0}$, where $X(t):=X(0,0;t)$. Later we will pick the constants satisfying

[TABLE]

In terms of ${\xi}=\tilde{\varrho}(\cdot+\tilde{X},\cdot)$, we have

[TABLE]

Since $\varepsilon\lambda<<1$, by (4.11) again, we obtain

[TABLE]

If follows from (4.13) that for all $t\in(0,\lambda]$ and some $\sigma=\sigma(\sigma_{2})$,

[TABLE]

Recall (4.12), and we have

[TABLE]

Hence we get for $t\in(0,\lambda]$ and some universal $\sigma>0$,

[TABLE]

Writing $Z(t):=\int_{0}^{t}Y(s)ds$, in view of (4.12) we obtain $Z(\frac{\lambda}{2})\geq c_{3}\lambda$ with

[TABLE]

Solving the ODE problem

[TABLE]

shows that

[TABLE]

where

[TABLE]

Since $\sigma\varepsilon<<1$,

[TABLE]

It is obvious that $f$ is monotone increasing in $t$. Notice the right-hand side of (4.21) goes to $+\infty$ as

[TABLE]

which is impossible provided that $f^{-1}((c_{3}\lambda)^{1-m})\leq\frac{\lambda}{2}$. However if $\lambda\geq C(c_{3},\sigma)$ and $\varepsilon\lambda<<1$, we indeed have

[TABLE]

which leads to a contradiction.

We proved that ${\xi}^{m}(0,\lambda)\leq\sigma(m)\,c_{2}^{\frac{m}{m-1}}.$ Since $c_{3}$ only depends on $c_{1},\sigma$, the choices of $\lambda,c_{2},\varepsilon$ only depend on $c_{1},\sigma$. We conclude the lemma with $\tau_{0}=\varepsilon/C_{0}$, $\lambda$ satisfying (4.22), and $c_{2},\varepsilon$ satisfying (4.20) and $\varepsilon\lambda<<1$.

∎

For any $(x_{0},t_{0})\in\Gamma$, we use the notation

[TABLE]

Theorem 4.4**.**

For a given point $(x_{0},t_{0})\in\Gamma$ with $t_{0}\geq\eta_{0}>0$, the following is true:

(1) Either (a) ${\Upsilon}(x_{0},t_{0})\subset\Gamma$ or (b) ${\Upsilon}(x_{0},t_{0})\cap\Gamma=\emptyset$.

(2) If (b) holds, then there exist positive constants $C_{*},\beta,h$ such that for all $s\in(0,h)$

[TABLE]

Here $\beta$ only depends on $\eta_{0}$ and universal constants. If (b) holds for $(x_{0},t_{0})\in\Gamma$, we say $(x_{0},t_{0})$ is “of the second type” free boundary point.

Sketch of the proof

The proof is parallel to those for Theorems 3.1-3.2 [6], based on the Lemmas 4.1 and 4.3. Let us only sketch the proof for part (1) below.

If the assertion of (1) is not true, then we can find $t_{0}>t_{1}>t_{2}>0$ such that $t_{0}-t_{1}>>t_{1}-t_{2}$ and

[TABLE]

Consequently $u(\cdot,t_{2})=0$ in $B(x_{2},R)$ for some $R>0$. Since $x_{1}=X(x_{2},t_{2};t_{1}-t_{2})$, by Lemma 4.1,

[TABLE]

Since $t_{0}-t_{1}>>(t_{1}-t_{2})$, Lemma 4.3 yields $u(x_{0},t_{0})=u(X(x_{1},t_{1};t_{0}-t_{1}),t_{0})>0$, which is a contradiction.

$\Box$

When the initial data grows faster than quadratically near its free boundary, it is possible to characterize the constants $C_{*},h$ in above theorem in terms of time variable. By a compactness argument, iteratively using Theorem 4.4 and arguing as in the remark on Theorem 3.2 in [6], we have the following theorem.

Theorem 4.5**.**

Suppose (1.7). Then any point $x_{0}\in\Gamma_{t_{0}}$ with $t_{0}\leq T$ is of the second type and the constants $C_{*},h$ in Theorem 4.4 (2) only depend on $\gamma,\varsigma,t_{0}$ and universal constants.

5. Monotonicity Implies Non-degeneracy

In this section we discuss non-degeneracy property of solutions in local settings. We start with the following theorem.

Theorem 5.1**.**

Let $u$ solve (1.5) in $Q_{2}$ with $\Delta u\geq-C_{0}$. Suppose that $\Gamma$ is of type two in $Q_{2}$, and that

[TABLE]

Then there exist constants $C,\varepsilon_{0}>0$ such that we have

[TABLE]

Remark 5.2*.*

The constants $C,\varepsilon_{0}$ in Theorem 5.1 only depend on

[TABLE]

where $C_{*},h,\beta$ are constants given in Theorem 4.4. In the global setting, an estimate of $C_{0}$ can be found in Theorem 3.1.

Let us also mention that Theorem 2.2 allows us to consider continuous local solutions.

The central ingredient of the proof is a barrier argument motivated from [11] in the context of Hele-Shaw flow. The barrier argument illustrates the fact that in diffusive free boundary problems the nice regularity properties of $u$ propagate from positive level sets to the free boundary as the positive set expands out. This argument in our setting corresponds to the proof of (5.35). Compared to the Hele-Shaw flow which is driven by a harmonic function, our solutions features a nonlinear diffusion that degenerates near the free boundary and thus it requires more careful arguments. On the other hand, we will benefit from the weak formulation of the problem using the density formula (see $\mathcal{G}$ below.)

For $u$ as given above we consider

[TABLE]

Then $v$ is a weak solution of $\mathcal{L}_{2}(\cdot)=0$, where the operator $\mathcal{L}_{2}$ is given by

[TABLE]

Since the operator $\mathcal{L}_{2}$ is the same as in (1.5) with $\vec{b}$ replaced by $\vec{b}(x+X(t),t)-\vec{b}(X(t),t))$, the notion of sub- and supersolution is given in Definition 2.1.

Below we construct of a supersolution for the operator $\mathcal{L}_{2}$ for the aforementioned barrier argument, using a inf-convolution construction introduced first by [5]. Since the supersolution to be constructed is a rescaled inf-convolution of $v$ (see (5.8)), comparison of the two functions gives a space-time monotonicity of $v$, yielding the theorem. To this end, we will use both smooth approximations of $u$ and the density version of the equation $\mathcal{L}_{2}$.

We begin with some basic properties of the inf-convolution of smooth functions.

Let $\psi,h\in C^{\infty}(B_{2})$ with $0<\psi<\frac{1}{2}$ and $h\geq 0$. Define

[TABLE]

which is Lipschitz continuous. The proofs of the following two lemmas are in the appendix.

Lemma 5.3**.**

Let $h$ and $f$ be as given in (5.5). Furthermore, suppose $\Delta h\geq-C$ for some $C\in\mathbb{R}$ and $\|\nabla\psi\|_{\infty}\leq 1$. Then there are dimensional constants $\sigma_{1}>0$ and $\sigma_{2}\geq 3$ such that if $\psi$ satisfies

[TABLE]

we have

[TABLE]

where $y(\cdot)$ satisfies that $f(\cdot)=h(y(\cdot))$ a.e. in $B_{1}$.

Lemma 5.4**.**

Let $h,f$ be as given in (5.5). Then for a.e. $x\in B_{1}$ we have

[TABLE]

Now for a weak solution $u$ to (1.5) in $Q_{2}$, let $\{u_{k}\}_{k}$ be its smooth approximations as given in Lemma 2.5. In particular $u_{k}$ is positive in $Q_{2}$ for each $k$. Set $v_{k}(x,t):=u_{k}(x+X(t),t)$ and introduce the corresponding density variable of $v_{k}$ as

[TABLE]

We define the density version of the operator ${\mathcal{L}}_{2}$ as ${\mathcal{G}}(\xi):={\mathcal{L}}_{2}(v)$ where $\xi=(\frac{m-1}{m}v)^{\frac{1}{m-1}}$ i.e.

[TABLE]

and thus ${\mathcal{G}}(\xi_{k})=0$.

Let $\varphi:\mathbb{R}^{d}\to(0,\infty)$ be a smooth function and $\sigma_{1},\sigma_{2}$ be from Lemma 5.3. For some constants $\alpha,A_{0},M_{0}\geq 1$ to be determined, we define

[TABLE]

[TABLE]

and

[TABLE]

where

[TABLE]

Then $w_{k}$ is Lipschitz continuous, and

[TABLE]

Thus to show that $w_{k}$ is a supersolution for ${\mathcal{L}}_{2}$, it suffices to show that $\eta_{k}$ is a supersolution for ${\mathcal{G}}$.

We will apply Lemmas 5.3, 5.4 with

[TABLE]

Based on these lemmas we estimate the density equation $\mathcal{G}(\eta_{k})$ in the weak sense, to go around the potential lack of smoothness for inf-convolutions, to conclude.

We will choose the constants $A_{0}=A_{0}(M_{0})$ and $\alpha=\alpha(M_{0})$ in Proposition 5.5, the constants $M_{0},r$ and the function $\varphi$ in the proof of Theorem 5.1.

Proposition 5.5**.**

Let $u_{k},w_{k}$ be defined from above, and suppose that $u_{k}$ satisfies $\Delta u_{k}\geq-C_{0}$ in $Q_{2}$. Fix any $M_{0}\geq 1$ and consider $\varphi:B_{2}\to\mathbb{R}$ such that

[TABLE]

Then there exist positive constants $A_{0},\alpha,\tau$ depending only on $M_{0}$ and universal constants such that for all $\varepsilon<\frac{1}{M_{0}}$ the function $w_{k}$ given in (5.7) is a weak supersolution of

[TABLE]

Proof.

Let $\xi_{k},\eta_{k}$ be from (5.6), (5.9) respectively. As discussed before to prove the statement, it suffices to show that ${\mathcal{G}}(\eta_{k})\geq 0$ weakly in $B_{r}\times(0,\tau)$.

Below we estimate each term in ${\mathcal{G}}(\eta_{k})$ in $B_{r}\times(0,\tau)$ using $\xi_{k}$. We begin with some preliminary estimates on $\eta_{k}$.

Since $u_{k}$ is smooth and positive, $\xi_{k}$ is also smooth and positive. From the definition of the inf-convolution, it follows that $\eta_{k}$ is Lipschitz continuous. Since $\Delta u_{k}\geq-C_{0}$, direct computation yields that

[TABLE]

Let us set the constants

[TABLE]

for some $\sigma_{3}\geq\sigma_{2}$ to be determined, and

[TABLE]

By definition of ${\eta_{k}}$, there is $z(x)$ satisfying

[TABLE]

such that

[TABLE]

where we use the notation $g(t):=e^{A_{1}\varepsilon t}$.

It follows from the definition of ${p_{\varepsilon}}(t)$ in (5.11) that

[TABLE]

and

[TABLE]

We now proceed to estimating each terms in $\mathcal{G}(\eta_{k})$, starting with $\partial_{t}{\eta_{k}}$. All estimates in the domain $B_{r}(0)\times(0,\tau)$. In the rest of the proof, for simplicity, $X(t):=X(0,0,;t)$, $p_{\varepsilon}$, $\eta_{k}$ denotes the values of them at $(x,t)$, and $\xi_{k},\partial_{t}\xi_{k},\nabla\xi_{k},\Delta\xi_{k}$ denotes the values of them evaluated at point $(z(x),p_{\varepsilon}(t))$.

In [18], $\partial_{t}{\eta_{k}}$ is computed in the viscosity sense. Since our $\eta_{k}$ is Lipschitz continuous, the same computation carries out almost everywhere in $B_{r}\times(0,\tau)$. We have

[TABLE]

Applying (5.10), (5.17) and the assumption that $\varphi\geq\frac{r}{M_{0}}$, (5.19) implies

[TABLE]

From the assumptions on $\varphi$, $\|R_{\varepsilon}\|_{\infty}\leq rM_{0}\varepsilon$, $\|\nabla R_{\varepsilon}\|_{\infty}\leq M_{0}\varepsilon$. We now apply Lemma 5.3 with $h=\xi^{m}_{k}(\cdot+r\varepsilon\mu,p_{\varepsilon})$ and $\psi=R_{\varepsilon}(\cdot,t)$. From (5.12) and (5.13), the following holds in the sense of distribution:

[TABLE]

Next we consider the terms coming from the drift. Due to Lemma 5.4,

[TABLE]

since $\varepsilon<\frac{1}{M_{0}}$, we have $|\nabla{\eta_{k}}-g\nabla\xi_{k}|\leq\sigma M_{0}\varepsilon|\nabla\eta_{k}|.$ This implies that for $t\leq\tau$,

[TABLE]

Next using the regularity of $\vec{b}$ and $|x|\leq r$, we have

[TABLE]

and, by (5.16),

[TABLE]

Then (5.22)-(5.24) imply

[TABLE]

Parallel computations yield

[TABLE]

Combining the estimates (5.20), (5.21), (5.25) and (5.26), we have

[TABLE]

Since ${\mathcal{G}}(\xi_{k})=0$ we obtain

[TABLE]

where

[TABLE]

Finally we proceed from $\widetilde{{\mathcal{G}}}(\eta_{k})$ to ${\mathcal{G}}(\eta_{k})$:

[TABLE]

Let us estimate $V_{0}$:

[TABLE]

Similarly,

[TABLE]

Thus it follows from (5.28) that, if $C_{1}\geq\sigma$, $C_{2}\geq\sigma$,

[TABLE]

In view of (5.27), $C_{1},C_{2}\geq\sigma$ if $\sigma_{3}$ in (5.14) is chosen to be large enough depending only on universal constants. Hence with this choice of $\sigma_{3}$ we have proved that ${\mathcal{G}}(\eta_{k})\geq 0$ in the sense of distribution in $B_{r}\times(0,\tau)$. From the Lipschitz continuity of $\eta_{k}$ we conclude that ${\mathcal{G}}(\eta_{k})\geq 0$ weakly in $B_{r}\times(0,\tau)$.

Lastly it is not hard to see that the choices of $A_{0},\alpha,\tau$ are independent of $r$ and $k$. ∎

Corollary 5.6**.**

Let $u$ be from Theorem 5.1, and let $v$ and $w$ be given by (5.3) and (5.8) respectively. Suppose that the assumptions in Proposition 5.5 are satisfied. Then for any open set $U\subseteq B_{r}$, if $w\geq v$ on the parabolic boundary of $U\times(0,\tau)$, then

[TABLE]

Proof.

Let $\{u_{k}\}_{k}$ be the smooth approximations of $u$ and $u_{k}\geq u$. Let $v_{k}(x,t)=u_{k}(x+X(t),t)$ and $w_{k}$ be from (5.7). It follows from the proposition that ${\mathcal{L}}_{2}(w_{k})\geq 0$ weakly in $B_{r}\times(0,\tau)$. We have $w_{k}\geq w$ due to the fact that $u_{k}\geq u$. Then by the assumption, $w_{k}\geq v$ on the parabolic boundary of $U\times(0,\tau)$. By comparison principle for ${\mathcal{L}}_{2}$, we get $w_{k}\geq v$ in $U\times(0,\tau)$. Due to Lemma 2.5, $u_{k}$ converges locally uniformly to $u$, and so $w_{k}$ converges locally uniformly to $w$. We conclude by sending $k\to\infty$. ∎

Now we are able to prove Theorem 5.1.

Proof of Theorem 5.1. Let $\sigma_{1}$ be given in Lemma 5.3, and let $\Phi$ be the unique solution of

[TABLE]

Here $A_{d,\theta}$ is chosen sufficiently large so that

[TABLE]

Then for some $M_{0}(\theta,d)\geq 1$

[TABLE]

With this $M_{0}$, let ${A_{0}},\alpha,\tau$ be as given in Proposition 5.5.

Fix any $(\hat{x},\hat{t})\in Q_{1}\cap\Gamma$ and let $C^{*},h,\beta$ be from Theorem 4.4 and $\tau$ be from (5.15). We will show that the support of the solution strictly expands relatively to the streamlines at $(\hat{x},\hat{t})$.

Let $\delta=\delta(\theta,C_{0})>0$, which will be chosen as a constant satisfying (5.41) and (5.43). Define

[TABLE]

and

[TABLE]

Due to Theorem 4.4,

[TABLE]

After translation, we assume $(X(\hat{x},\hat{t};-t_{\delta}),\hat{t}-t_{\delta})$ to be the origin. Using the notation $X(t)=X(0,0;t)$, we have

[TABLE]

Let $v$ be as given in (5.3), and then $\mathcal{L}_{2}(v)=0$ weakly in $Q_{\frac{1}{2}}$, where ${\mathcal{L}}_{2}$ is given in (5.4). It follows from (5.33) that

[TABLE]

For $P:=-\frac{{r_{\delta}}}{5}{{\mu}}$, set $\varphi(x):={r_{\delta}}\Phi(\frac{x-P}{{r_{\delta}}})$.

Let $w$ be defined as in (5.8) with the above $\varphi$ and $r=r_{\delta}$:

[TABLE]

Next denote the cylindrical domain

[TABLE]

where $r_{\theta}:=\frac{{r_{\delta}}}{10}\sin\theta$. We claim that

[TABLE]

Roughly speaking, (5.35) states that the nondegeneracy property of $u$ propagates from the positive set to the free boundary, as the positive set expands out relative to the streamlines.

The proof of (5.35) will be given below. We first discuss its consequences.

Using (5.15) and (5.30),

[TABLE]

From this, it follows that for all $|x|\leq\frac{r_{\delta}\varepsilon}{5}\leq\frac{r_{\delta}}{10}$,

[TABLE]

In the inclusion, we used that $\alpha t_{\delta}\leq\frac{1}{5}$. Then using (5.35) and the definition of $w$, we get for $|x|\leq\frac{r_{\delta}\varepsilon}{5}$,

[TABLE]

From (5.11) it follows that $p_{\varepsilon}(t_{\delta})=t_{\delta}+c\varepsilon$ for some $c=c(t_{\delta},\sigma)$ which is independent of $\varepsilon$. Thus

[TABLE]

Recall that $(X(t_{\delta}),t_{\delta})=(\hat{x},\hat{t})\in\Gamma(u)$ and $X(t_{\delta}+c\varepsilon)=X(X(t_{\delta}),t_{\delta};c\varepsilon)$. We proved

[TABLE]

which implies

[TABLE]

Now we proceed to prove our claim.

Proof of (5.35). Here we apply Corollary 5.6 with the choice of $U:=B(P,\frac{r_{\delta}}{2})\setminus B(P,{r_{\theta}})$. To this end, it suffices to show that $w\geq v$ on the parabolic boundary of $\Sigma$.

First observe that from (5.34),

[TABLE]

Since $v(0,t_{\delta})=u(X(t_{\delta}),t_{\delta})=0$ and due to Lemma 3.3,

[TABLE]

Due to the cone monotonicity assumption (5.1),

[TABLE]

Hence to show (5.35), it remains to show that $w\geq v$ on $\partial B(P,\frac{r_{\delta}}{2})\times[0,t_{\delta}]$.

By definition of $\varphi$, we have $\varphi(x)=\frac{{r_{\delta}}}{2}\sin\theta$ on $\partial B(P,{{r_{\delta}}/2})$. From (5.14), we know $A_{0}\geq\sigma_{2}M_{0}$. It follows that for $x\in\partial B(P,{r_{\delta}}/2)$,

[TABLE]

In view of (5.1), we have

[TABLE]

Thus it remains to show that

[TABLE]

Take any $(x,t)\in\partial B(P,\frac{{r_{\delta}}}{2})\times[0,t_{\delta}]$, and denote

[TABLE]

With this notation we can rewrite $V_{1}(x,t)$ as

[TABLE]

By (5.18) and (5.31), we know

[TABLE]

Then

[TABLE]

Note that, for some universal $\sigma$,

[TABLE]

Therefore, (5.40) and (5.39) imply that

[TABLE]

where the last inequality holds if

[TABLE]

Combining above estimate with (5.38), it follows that

[TABLE]

Due to (3.2), for ${C}:=(m-1)(C_{0}+\|\nabla\cdot\vec{b}\|_{\infty})$,

[TABLE]

In view of (5.36), we derive

[TABLE]

Using (5.39) again shows

[TABLE]

Now after fixing $\delta=\delta(\theta,C_{0})>0$ such that (5.41) and (5.43) hold, we can conclude with (5.37) and then the claim (5.35).

$\Box$

In view of the velocity law (1.3), non-degeneracy follows once we know that the positive set of the solution is strictly expanding relatively to the streamlines. In the following theorem, we are going to show that indeed the solution $u$ grows linearly near the free boundary.

Corollary 5.7**.**

Under the conditions of Theorem 5.1, there exist $\varepsilon_{0},\kappa_{*}>0$ depending only on constants in (5.2) such that, for all $\varepsilon\in(0,\varepsilon_{0})$,

[TABLE]

Proof.

Let $c_{0}$ be from Lemma 4.1 and $C$ be from Theorem 5.1. Define $\kappa:=\frac{c_{0}\,\sin^{2}\theta}{4C}$. We first claim that for all $\varepsilon>0$ sufficiently small

[TABLE]

We argue by contradiction. Suppose that the above claim is false. Then for any $\varepsilon_{0}>0$ there exist $\varepsilon\in(0,\varepsilon_{0}]$ and $(\hat{x},\hat{t})\in\Gamma\cap Q_{1}$ such that (5.45) fails.

Set $t_{1}:=\hat{t}-C\varepsilon$ and consider the map $X(\cdot,t_{1};C\varepsilon):\mathbb{R}^{d}\to\mathbb{R}^{d},$ which is an isomorphism when $\varepsilon_{0}$ is small enough. Since the positive set of $u$ is strictly expanding relatively to the streamlines, we have

[TABLE]

Using the cone monotonicity condition (5.1) and the fact that $u(\hat{x},\hat{t})=0$, it follows that $(\hat{x}+\mathbb{R}^{+}\mu)\cap\Gamma_{t_{1}}\neq\emptyset$. Therefore there exists $(x_{1},t_{1})\in\Gamma$ such that

[TABLE]

Due to (5.1) again, we have

[TABLE]

In view of Theorem 5.1, for all $\varepsilon$ sufficiently small

[TABLE]

Therefore, combining with the fact that

[TABLE]

we obtain $C_{1}\geq 1$.

Next define

[TABLE]

Due to (1.6),

[TABLE]

Thus

[TABLE]

Using this, (5.46) and the fact that $C_{1}\geq 1$, if $\varepsilon\leq\varepsilon_{0}$ is sufficiently small compared to $C$, it follows that

[TABLE]

which yields

[TABLE]

Note that $t_{1}+C\varepsilon=\hat{t}$ and $X(x_{2},t_{1};C\varepsilon)=\hat{x}$ from definition. Therefore the failure of (5.45) implies that

[TABLE]

In the last equality, we used that $\kappa=\frac{c_{0}\,\sin^{2}\theta}{4C}$.

With (5.47)-(5.48), we are able to apply Lemma 4.1 to get

[TABLE]

which contradicts with the assumption that $(\hat{x},\hat{t})\in\Gamma$. We proved (5.45). It can be seen from the proof that $\varepsilon_{0}$ only depends on constants in (5.2).

Now we show (5.44). Let us take $\gamma\in(0,1)$ to be small enough depending only on $\theta$ such that $B(\mu,\gamma)\subseteq W_{\theta,\mu}$, which implies that for any $\varepsilon\in(0,1)$,

[TABLE]

Fix any $(x,t)\in\Gamma\cap Q_{1}$, and set $\kappa_{*}:=\kappa\gamma$. By (5.45), there exists $\varepsilon_{0}>0$ such that

[TABLE]

Therefore we can find $y\in{B(x,\gamma\varepsilon)}$ that $u(y,t)\geq\kappa_{*}\varepsilon$. It follows from (5.49) that $x+\varepsilon\mu\in y+W_{\theta,\mu}$. Due to (5.1), we conclude with

[TABLE]

∎

6. Flatness Implies Smoothness

In this section we prove the following theorem.

Theorem 6.1**.**

Let $u$ be as given in Theorem 5.1. If (1.10) holds in $Q_{1}$, then $u$ is Lipschitz continuous and $\Gamma\cap Q_{1/2}$ is a $d$-dimensional $C^{1,\alpha}$ surface for some $\alpha\in(0,1)$.

The cone monotonicity and (1.10) provide sufficient monotonicity properties for the solution to rule out topological singularities and to localize the regularization phenomena driven by the diffusion in the interior of the domain. We follow the outline for the zero drift built on [8] and [7], while we elaborate on the differences. Most notable difference is in establishing Proposition 6.3.

Lemma 6.2**.**

Under the conditions of Theorem 6.1, $u$ is Lipschitz continuous in $Q_{1}$, and $\Gamma\cap\,Q_{{1}/{2}}$ is a $d$-dimensional Lipschitz continuous surface.

Proof.

First let us prove that $u$ is Lipschitz continuous in $Q_{1}$. Since $u$ satisfies a parabolic equation locally uniformly in its positive set, $u$ is smooth in $\{u>0\}$. From the equation and $\Delta u\geq-C_{0}$, we obtain

[TABLE]

where $\sigma$ is universal. Above estimate combined with condition (1.10) yields

[TABLE]

which turns into a bound on $|\nabla u|$ in $\{u>0\}$. From (1.10), we also get a bound on $|u_{t}|$. Notice the bounds are independent of the ellipticity constants of the equation satisfied by $u$. Indeed we have,

[TABLE]

for some $C$ only depending on $A,C_{0}$ and universal constants. Since $u$ is continuous and nonnegative, it is not hard to see that the same estimate holds weakly in $Q_{1}$.

Next we turn to the Lipschitz continuity of $\Gamma$, using the cone monotonicity and Lipschitz continuity of $u$. The spatial cone monotonicity of $u$ implies that for each $t\in(-1,1)$, $\Gamma_{t}$ is a Lipschitz continuous graph in $\mathbb{R}^{d}$. Thus it remains to show that for each $\tau\in(-1,1)$, $\Gamma_{t+\tau}\cap B_{\frac{1}{2}}$ is in a $C\tau$ neighbourhood of $\Gamma_{t}\cap B_{1}$ for some $C>0$. To this end it is enough to show the following: for $(x,t)\in\Gamma\cap Q_{\frac{1}{2}}$ and for $\tau>0$ sufficiently small, we have

[TABLE]

To show (6.3) let us fix $(x,t)\in\Gamma\cap Q_{\frac{1}{2}}$. Observe that from Lemma 3.3 there exists $C>0$ such that, if $\tau>0$ is small,

[TABLE]

Thus it remains to show the second inequality in (6.3). Let $C_{1}>0$ be a sufficiently large constant to be chosen later. From the cone monotonicity

[TABLE]

where $y:=x-C_{1}\tau\mu$ and $R:=C_{1}\sin\theta\,\tau$. By the Lipschitz continuity of $u$,

[TABLE]

where $C$ depends on $Lip(u)$ and $\|\vec{b}\|_{\infty}$. Thus, for $c_{0}$ given in Lemma 4.1,

[TABLE]

where the last inequality holds if $C_{1}$ is large enough compared to $1/c_{0},1/\theta,Lip(u),\|\vec{b}\|_{\infty}$. Remark 4.2 then yields for small $\tau$,

[TABLE]

and therefore (6.3) is proved. ∎

Now we start proving the $C^{1,\alpha}$ regularity of the free boundary. By considering $\tilde{u}(x,t):=2u(x_{0}+\frac{1}{2}x,t_{0}+\frac{1}{2}t)$ for any $(x_{0},t_{0})\in Q_{\frac{1}{2}}\cap\Gamma$, we can assume $(0,0)\in\Gamma$. And to prove the rest of Theorem 6.1, it suffices to show that $\Gamma$ is $C^{1,\alpha}$ at point $(0,0)$.

The following proposition propagates the free boundary non-degeneracy in Corollary 5.7 to the nearby level sets.

Proposition 6.3**.**

Under the conditions of Theorem 6.1, there exist constants $0<\delta_{1}<\frac{1}{2}$ and $c_{1}>0$ such that

[TABLE]

Proof.

Fix a sufficiently small $\delta>0$ to be determined and pick $(\hat{x},\hat{t})\in\{u>0\}\cap Q_{\delta}$. Let $h:={d}(\hat{x},\Gamma_{\hat{t}})<\delta$. From Lemma 6.1, $\Gamma(u)$ is space-time Lipschitz continuous, and actually it can be written as the graph of $x_{\nu}=F_{u}(x^{\perp},t)$ where $x_{\nu}:=x\cdot\nu$ and $x^{\perp}\in\{x\cdot\nu=0\}$. Let us denote the space-time Lipschitz constant of $F_{u}$ as $C$, and choose $C_{2}:=C+1$. Then

[TABLE]

Denote $(y,s)$ such that $s=\hat{t}-h,y\in\Gamma_{s}$ and ${d}(\hat{x},y)={d}(\hat{x},\Gamma_{s})\leq(C_{2}-1)h$. Thus $B(y,h)\subseteq B(\hat{x},{C_{2}}h)$. Also by Lipschitz continuity of $\Gamma_{s}$ in space, $\partial B(y,h)\cap\{u>0\}$ is of strictly positive measure $Ch$ with $C$ independent of $h$.

By the fundamental theorem of calculus and (5.44),

[TABLE]

for some ${\kappa}>0$ only depending on $\kappa_{*}$ and $C_{2}$.

Let us define

[TABLE]

Fix $\gamma\in(0,\frac{1}{2})$ to be a small constant only depending on ${\kappa}$ such that

[TABLE]

Therefore there exists a point

[TABLE]

such that

[TABLE]

We will apply Harnack inequality to $\phi:=\nabla_{\mu}u$, using the fact that it solves a locally uniform parabolic equation in the positive set of $u$.

Let us consider $\{u>0\}$ and then $u$ is $C^{1}$ inside the open region. Differentiating (1.5) in $\{u>0\}$, we can check that $\phi$ satisfies the following parabolic equation

[TABLE]

in $\Omega^{r}$, where

[TABLE]

Since $u$ is Lipschitz continuous and $\vec{b}$ is smooth, $f$ is uniformly bounded. Then the new function

[TABLE]

satisfies

[TABLE]

Next let us define

[TABLE]

where $\Omega^{r}$ is as given in (6.4). We have

[TABLE]

For any $(x,t)\in\Sigma_{2}^{h}$ which is $\frac{\gamma h}{2}$ away from $\Gamma$, by the cone monotonicity and (5.44) we have

[TABLE]

Thus $w(x,t):=\tilde{\phi}(xh+\hat{x},th+s)$ satisfies

[TABLE]

Also we denote

[TABLE]

Notice that $\Sigma_{1},\Sigma_{2}$ are domains with Lipschitz boundary with Lipschitz constant depending only on $C,\sigma$. Writing $\Sigma_{i}(t)=\{x\,|\,(x,t)\in\Sigma_{i}\}$ for $i=1,2$, we have

[TABLE]

Since $\frac{u}{h}\geq\frac{\kappa_{*}\gamma}{2}>0$ in $\Sigma_{1}$ due to (6.6), the operator in (6.7) is uniformly parabolic in $\Sigma_{2}$. Let us apply the Harnack inequality to $w$ in $\Sigma_{1}$ and write it in terms of $\phi$, to obtain

[TABLE]

for some constant $C=C(\theta,\kappa_{*},C_{2})>0$, which is larger than $\frac{{\kappa}}{2C}$ due to (6.5).

Since $\hat{t}-s=h\leq\delta$, further assuming $\delta$ to be small enough, we can get $\phi(\hat{x},\hat{t})\geq\frac{{\kappa}}{4C}>0.$ Finally we conclude that $\nabla_{\mu}u\geq\frac{{\kappa}}{4C}>0$ in ${\Omega\cap Q_{\delta}}$.

∎

Next we show the strict monotonicity of $u$ along the streamlines.

Lemma 6.4**.**

Let $u$ be given as in Proposition 6.3. Then there exist $\delta_{2}\in(0,\delta_{1})$ and $c_{2}>0$ such that, for $v(x,t):=u(x+X(t),t)$ with $X(t)=X(0,0;t)$, we have

[TABLE]

Proof.

By definition, $v$ solves $\mathcal{L}_{2}(v)=0$, where $\mathcal{L}_{2}$ is as given in (5.4). By the equation, we have

[TABLE]

where the second inequality comes from the fact that $v\leq C\delta$ due to (6.2), and the third inequality follows from Proposition 6.3.

Since $c_{1}$ is independent of $\delta$, the last quantity is positive if $\delta=\delta_{2}$ is small enough compared to $C_{0},c_{1},$ the Lipchitz constant of $u$ and universal constants. We thus conclude. ∎

Now we are ready to follow the celebrated iteration procedure given in [8]. Their argument describes the enlargement of cone of monotonicity as we zoom in near a free boundary point. More precise discussions are below.

Our reference point is $(0,0)\in\Gamma$, and let $v$ be from Lemma 6.4. For $\delta\in(0,\delta_{2})$, define

[TABLE]

Then $X_{\delta}$ is the streamline generated by $\vec{b}_{\delta}$ starting at $(0,0)$. We have that $v_{\delta}$ is a solution to ${\mathcal{L}}_{2}(\cdot)=0$ with $\vec{b},X$ replaced by $\vec{b}_{\delta},X_{\delta}$. From Lemmas 6.2 - 6.4, we have for some $L>0$ independent of $\delta$ (depending on constants in (5.2)) such that

[TABLE]

Denoting $\sigma$ as the $C^{2}$ norm of $\vec{b}$, we have

[TABLE]

Let $\widehat{W}_{\theta,{\nu}}$ be given as in (2.1). We say $v$ has the cone of monotonicity $\widehat{W}_{\theta,\nu}$ in $Q_{1}$ if

[TABLE]

The following lemma, yielding the initial cone of monotonicity for $v_{\delta}$, can be proven using (6.9)- (6.10) with a parallel proof to Proposition 2.1 of [8]. Let us denote the positive time direction as $e_{d+1}$.

Lemma 6.5**.**

Let $v_{\delta}$ be as given in (6.8). Then there exists $\mathcal{\theta}_{0}>0$ such that

[TABLE]

where ${\mu}_{0}:=\frac{1}{\sqrt{2}}[(\mu,0)+e_{d+1}]$ and $L$ is as given in (6.9).

Now we begin our iteration procedure. Fix some $J(L)\in(0,1)$ to be chosen later, define

[TABLE]

Then $v_{k}$ satisfies

[TABLE]

where $\vec{b}_{k}(x,t):=\vec{b}_{\delta}(J^{k}x,J^{k}t),X_{k}(t):=\frac{1}{J^{k}}X_{\delta}(J^{k}t)$.

Due to (6.9) - (6.10) the following holds in $Q_{1}$:

• $(A_{k})$

$0\leq v_{k}\leq{L},$ $\Delta v_{k}\geq-{L}\delta,$  $|\nabla v_{k}|+|\partial_{t}v_{k}|\leq{L}$;

• $(B_{k})$

$\nabla_{{{\mu}}}v_{k},$ $\partial_{t}v_{k}\geq\frac{1}{{L}}$;

• $(C_{k})$

$\|\vec{b}_{k}\|_{\infty}\leq\sigma,$ $\|{\nabla}\vec{b}_{k}\|_{\infty}+\|\partial_{t}\vec{b}_{k}\|_{\infty}\leq\sigma\delta{J^{k}},$ $\|D^{2}\vec{b}_{k}\|_{\infty}+\|\nabla\partial_{t}\vec{b}_{k}\|_{\infty}\leq\sigma\delta^{2}{J^{2k}}$.

The main step in the proof of Theorem 6.1 is to show the following property inductively.

• $(D_{k})$

there exist $s\in(0,1)$ and ${\mu}_{k}\in\mathbb{R}^{d+1}$ such that for $\theta_{k}:=\frac{\pi}{2}-s^{k}(\frac{\pi}{2}-\theta_{0})$,

[TABLE]

Once establishing ($D_{k}$), it shows that the cone of monotonicity $\widehat{W}_{\theta_{k},{\mu}_{k}}$ for $v_{k}$ has strictly increasing $\theta_{k}$, converging to $\pi/2$ as $k\to\infty$. The rate of its increasing angles leads to the $C^{1,\alpha}$ regularity of the free boundary.

In [8], (6.13) is stated with the weaker requirement $\hat{\nabla}_{p}v_{k}\geq 0$. However for us the competition between diffusion and drift requires a stronger inductive property: see Remark 6.9. This extra observation follows from the enlargement of cones as well as the non-degeneracy of the solution.

We will proceed with several lemmas that leads to the enlargement of cones in Proposition 6.10. The proofs of the lemmas will be postponed until after the proof of the Proposition.

First we show that some improvements on monotonicity can be obtained on the set $\{v_{k}=\varepsilon\}$.

Lemma 6.6**.**

[Enlargement of Cones] Let $v_{k}$ be as given in (6.11), and suppose that $v_{k}$ satisfies $(D_{k})$. For any $\varepsilon\in(0,1)$, there exist positive constants $r\leq\frac{1}{10},\delta_{0}<\delta_{2},C$ only depending on $\varepsilon,L,\sigma$ such that the following holds:

For any $\gamma\in(0,\varepsilon)$, $\delta\in(0,\delta_{0})$, $p\in\widehat{W}_{\theta_{k},{\mu}_{k}}\cap\,\mathcal{S}^{d}$ and $\tau:=C\varepsilon^{-1}\cos\langle\,p,\hat{\nabla}v_{k}({{\mu}},-2r)\,\rangle$, we have

[TABLE]

Next we show that this improvement can propagate to the zero level set of $v$.

Lemma 6.7**.**

Let $v_{k}$ be as given in Lemma 6.6. Let $\delta(\varepsilon,L),r(\varepsilon,L),\tau(\varepsilon,L)$ be as given in Lemma 6.6. Let $w$ be a supersolution of (6.12), and suppose that $w\geq v_{k}$ in $Q_{1}$ and

[TABLE]

Then, if $\varepsilon$ is small enough (independently of $\delta,r,\tau$),

[TABLE]

Lastly we further improve the monotonicity in a smaller domain of size $r$.

Lemma 6.8**.**

Let $v_{k},w,\tau$ be as in Lemma 6.7. There exists a small $\kappa>0$ depending only on $L$ and universal constants such that the following holds. Consider any smooth function $\phi:\mathbb{R}^{d}\to\mathbb{R}^{+}$ such that $\phi$ is supported in $B_{2r}$ and $\phi,|\nabla\phi|,|D^{2}\phi|\leq\kappa\tau\gamma$. If $v_{k}\leq\varepsilon$ in $Q_{2r}$ then we have

[TABLE]

Remark 6.9*.*

In [8] for the zero drift case, $w$ in the above lemmas is chosen as a translation of $v_{k}$ to derive monotonicity properties of $v_{k}$. Since our equation is not translation invariant, we instead choose $w$ of the form $v_{k}((x,t)+p)+Et$ with $E>0$. To control the extra term $Et$ we rely on the inductive property $(D_{k})$. The order between $v_{k},w$ is still enough to derive the Proposition below.

Now we state the main proposition.

Proposition 6.10**.**

[Improvement of Monotonicity] Let $v_{\delta}$ be as given in (6.8) with $\delta<\delta_{0}$ is as given in Lemma 6.6. Then there exist constants $J,s\in(0,1)$ independent of $k$ such that the following holds. Suppose $(0,0)\in\Gamma$ and (6.9) - (6.10). Then there exist a monotone family of cones $\widehat{W}_{\theta_{k},{\mu}_{k}}$ with $\theta_{k}=\frac{\pi}{2}-s^{k}(\frac{\pi}{2}-\theta_{0})$ such that

[TABLE]

The $C^{1,\alpha}$ regularity of $\Gamma$ at $(0,0)$ is a result of the relation $\theta_{k}=\theta_{k-1}+S(\pi/2-\theta_{k-1})$ which describes quantitatively the enlargement of cone of monotonicity of solutions near the free boundary. Then Theorem 6.1 follows. We omit detailed discussion of this part since it is parallel to Theorem 1 in [8].

Proof.

Fix a small $\varepsilon>0$ such that the conclusion of Lemma 6.7 holds, and let $r,\delta_{0}$ be as given in Lemma 6.6. Then $\varepsilon,\delta_{0},r$ only depend on $L$ and universal constants. Define $\tau$ as in Lemma 6.6. Let $v_{k}$ be as in (6.11), and set $\vec{b}_{k},X_{k}$ as before and we take $J\leq r$ to be determined. It is straightforward that for all $k\geq 0$, $(A_{k})-(C_{k})$ hold. When $k=0$, due to Lemma 6.5, $(D_{0})$ holds for $v=v_{0}$.

Let us suppose that $(D_{k})$ holds for some $k\geq 0$ with ${\mu}_{k},\theta_{k}\geq\theta_{0}$ i.e. the hypothesis of Lemmas 6.6- 6.8 are satisfied. We will show $(D_{k+1})$.

For any $\gamma\in(0,\varepsilon)$ and a unit vector $p\in\widehat{W}_{\theta_{k},{\mu}_{k}}$, define

[TABLE]

Note that $\widetilde{w}\geq v_{k}$ in $Q_{1}$ due to $D_{k}$. Next, (6.12) implies that

[TABLE]

By $(A_{k})-(C_{k})$ and the fact that $|\partial_{t}X_{k}|\leq|\vec{b}_{k}|\leq\sigma$, we have

[TABLE]

Then for $w:=\widetilde{w}+E_{k}(t+2r)$, we have $w\geq v_{k}$ in $Q_{2r}$.

In view of Lemma 6.6, $v_{k}\leq\varepsilon$ in $Q_{r}$ and $w$ satisfies the hypothesis of Lemma 6.7. Let $\tau$ be defined as in Lemma 6.6, and let $\kappa<\kappa_{0}$ be from Lemma 6.8. We select a smooth function $\phi:\mathbb{R}^{d}\to\mathbb{R}^{+}$ such that $\phi$ is supported in $B_{2r}$, and $\phi,|\nabla\phi|,|D^{2}\phi|\leq\kappa\tau\gamma$, and

[TABLE]

Clearly such $\phi$ exists.

It follows from Lemmas 6.6-6.8 that

[TABLE]

By $(B_{k})$ and (6.14), for $c^{\prime}:=\frac{\sigma r^{3}\kappa}{L}$ we have

[TABLE]

This implies that

[TABLE]

Using the definition of $\tau$, we obtain

[TABLE]

where $C_{1}:=c^{\prime}C\varepsilon^{-1}$ only depending on $L,\sigma$ (since $\varepsilon$ is fixed).

It follows from ($A_{k}$) and ($D_{k}$) that

[TABLE]

Taking $\delta$ to be small enough only depending on $L$ and $\sigma$, (6.15) yields

[TABLE]

Thus in $Q_{r}\cap\{v_{k}>0\}$,

[TABLE]

For $p\in\mathcal{S}^{d+1}$, set

[TABLE]

For any $q\in B(p,\rho(p))$ we have $\sin\langle\,p,q\,\rangle\leq\rho(p)$ and thus

[TABLE]

In view of ($A_{k}$) and (6.16), we get

[TABLE]

Since the above holds for all $q\in B(p,\rho(p))$, there exists a larger cone $\widehat{W}_{\theta_{k+1},{\mu}_{k+1}}$ for some ${\mu}_{k+1}\in\mathbb{R}^{d+1}$, $S\in(0,1)$ and $\theta_{k+1}=\theta_{k}+S(\frac{1}{2}\pi-\theta_{k})$ such that

[TABLE]

Here $S$ is independent of $k$, because $\rho(p)$ only depends on the angle between $p$ and $\hat{\nabla}v_{k}({{\mu}},-2r)$. From the iterative definition of $\theta_{k}$, we obtain $\theta_{k}=\frac{\pi}{2}-s^{k}(\frac{\pi}{2}-\theta_{0})$ with $s=1-S$. We refer readers to [8, 5] for more details.

Let $J:=\min\{{C_{1}}/{(4L^{3})},r\}$. Recalling $v_{k+1}(x,t)=\frac{1}{J}v_{k}(Jx,Jt)$, we obtain for all unit $p\in\widehat{W}_{\theta_{k+1},{\mu}_{k+1}}$

[TABLE]

We checked $(D_{k+1})$ and therefore by induction we conclude the proof of the theorem.

∎

Now we give the proofs of Lemmas 6.6-6.8. To simplify notations, we write $v:=v_{k}$, $\vec{b}:=\vec{b}_{k}$ and $X:=X_{k}$ in the following proofs.

Proof of Lemma 6.6. First note that if $r\leq\frac{\varepsilon}{2{L}}$ , then $v\leq\varepsilon\text{ in }Q_{2r}$ from ($A_{k}$) and the fact that $0\in\Gamma_{0}$. Next observe that in $Q_{1}$, $g:=\hat{\nabla}_{p}v$ solves

[TABLE]

[TABLE]

By the condition $(A_{k})(C_{k})$,

[TABLE]

Now we apply Harnack’s inequality to $g$, using (6.18), in $({B_{\frac{7}{8}}}\times[-3r,3r])\cap\{v\geq\frac{1}{2}\varepsilon\}$. As done in Proposition 2.2 in [8], if we restrict to a smaller region $({B_{\frac{3}{4}}}\times(-2r,2r))\cap\{v\geq\varepsilon\}$ for $r$ small enough (depending on $\varepsilon$), there exist $C,C^{\prime}$ (depending on $L,r,\varepsilon$) such that

[TABLE]

By ($D_{k}$), we have $\hat{\nabla}_{p}v({{\mu}},-2r)\geq J^{k}$. Thus we can select $\delta$ small enough such that for some $C>0$

[TABLE]

To show the assertion, we need to show

[TABLE]

which holds by the definition of $\tau$ and (6.19). $\Box$

Proof of Lemma 6.7. Let $f\in C^{1}(B_{1/2})$ be a non-negative function such that

[TABLE]

For $\alpha\in(-2r,2r)$, define

[TABLE]

We claim that $\xi$ is a subsolution in $\Sigma:=(B_{\frac{1}{2}}\times(-2r,-\alpha))\cap\{v\leq\varepsilon\}$ if $\varepsilon$ is small enough, independent of $r$. Let us follow [8] and only point out the differences coming from the drift. We recall the operator ${\mathcal{L}}_{2}$ defined in (5.4) and denote the drift independent part as $\widetilde{\mathcal{L}}$:

[TABLE]

Let $g(s):=\tau\gamma s_{+}$ and thus $g^{\prime}=\tau\gamma\,\chi_{\{s>0\}}$, $g^{\prime\prime}\geq 0$ in the sense of distribution. Below, we write $g=g(v+\varepsilon(t+\alpha)-f)$. Direct computations yield

[TABLE]

Following the computations in Lemma 3.1 of [8] and using $|\nabla v|\geq\frac{1}{L}$, we obtain

[TABLE]

Since $\mathcal{L}_{2}(\xi)=\widetilde{\mathcal{L}}(\xi)-\nabla\xi\cdot(\vec{b}(x+X)-\vec{b}(X))-(m-1)\xi\nabla\cdot\vec{b}$, then

[TABLE]

By $(D_{k})$, we have $\|\vec{b}\|_{\infty}\leq\sigma,\|{\nabla}\vec{b}\|_{\infty}\leq\sigma\delta J^{k}$. Since we assumed $\delta\leq\varepsilon$ and $J<1$, $\|\nabla\vec{b}\|_{\infty}\leq\sigma\varepsilon$. Also for $(x,t)\in\Sigma$, $s:=v+\varepsilon(t+\alpha)-f\leq\varepsilon$ and hence $g(s)\leq\varepsilon g^{\prime}(s)$. We get

[TABLE]

Thus $\mathcal{L}_{2}(\xi)\leq 0$ if $\varepsilon$ is small enough.

The rest of the proof follows from the proof of Proposition 2.3 [8], where we compare $w$ and $\xi$ in $\Sigma$ to conclude that

[TABLE]

for all $\alpha\in(-2r,2r)$.

$\Box$

Proof of Lemma 6.8. Based on $(A_{k})$,$(B_{k})$ and the elliptic regularity estimate applied to $v$, one can argue as in Lemma 3.2 of [8] to conclude that

[TABLE]

where $C_{4}$ depends only on $L$, universal constants and the Lipschitz constant of $\Gamma(v)$. We will use this fact in the computation below.

Define

[TABLE]

Note that $|y-x|\leq{\kappa}\tau\gamma$. Lemma 6.7 implies that $w\geq h$ on the parabolic boundary of

[TABLE]

We claim that $\mathcal{L}_{2}(h)\leq 0$ in $\Sigma$. Write $\tau^{\prime}:=\tau\gamma$. We have

[TABLE]

From (6.22) and the computations in Proposition 2.4 [8]

[TABLE]

where $\widetilde{\mathcal{L}}$ is given by (6.20) and $C$ depends only on $m,L,C_{4},\sigma$. Thus

[TABLE]

Now apply ($C_{k}$) and since $\delta\leq\varepsilon$, we have $\left\|D\vec{b}\right\|_{\infty}\leq\sigma\varepsilon,\left\|D^{2}\vec{b}\right\|_{\infty}\leq\sigma\varepsilon^{2}$. Since $|\nabla\phi|\leq\kappa\tau^{\prime}$, we obtain

[TABLE]

if ${\kappa}$ is small enough. By comparison principle applied to $w$ and $h$ in $Q_{2r}$ we can conclude that

[TABLE]

$\Box$

7. Discussion of traveling waves and potential singularities

In this section we discuss evolution of solutions in two space dimensions, in several explicit scenario.

7.1. A discussion on Traveling Waves

For simplicity, we restrict to two space dimensions $d=2$. The drift is chosen as

[TABLE]

When $\alpha$ is periodic and $\max\{\alpha\}<c$, it is shown in [22] that there exist traveling wave solutions of the form $U(x+cte_{1})$ for the corresponding pressure equation (7.2), with the growth condition $\lim_{x_{1}\to\infty}\frac{U(x)}{x_{1}}=c$. While Lipschitz regularity of the solutions are established therein, the free boundary regularity and possibility of a corner remain open.

Our regularity analysis cannot address the traveling waves themselves, but we are able to say that such singularity, if at all, is of asymptotic nature. More precisely we show that dynamic solutions, used in [21] to approximate the travelling waves, stay smooth in any finite time interval.

Theorem 7.1**.**

Let $u$ solve (1.5) in $\mathbb{R}^{2}\times(0,\infty)$, with $\vec{b}$ given in (7.1), with the initial data $u_{0}(x)=(x_{1})_{+}$. Further impose that $\frac{u(x,t)}{x_{1}}\to 1\text{ as }x_{1}\to\infty.$ Then the following holds:

• (a)

$u$* is uniformly Lipschitz continuous in $\mathbb{R}^{2}\times[0,\infty)$.*

• (b)

For any fixed $T>0$, there exists $\tau_{0}(T)>0$ such that for all $t\in[0,T]$ and $\tau\leq\tau_{0}$

[TABLE]

• (c)

$u$* is non-degenerate, and $\Gamma(u)$ is $C^{1,\alpha}$ in $\mathbb{R}^{2}\times[0,T]$.*

Proof.

Let us rewrite (1.5) with our choice of $\vec{b}$:

[TABLE]

Define $\varphi(x,t):=(x_{1}+\sigma_{1}t)_{+}$ with $\sigma_{1}:=\sup|\alpha|+1$. Then $\varphi$ is a supersolution of (7.2) with the same initial data as $u$, and thus $u\leq\varphi$. In particular, for any $\varepsilon>0$

[TABLE]

where we denote the positive $x_{1}$ direction as $e_{1}$.

For $\varepsilon>0$ let $u^{\varepsilon}(x,t):=u(x-\sigma_{1}\varepsilon e_{1},t+\varepsilon)$. From (7.3), it follows that $u^{\varepsilon}(\cdot,0)\leq u_{0}$. Since $u^{\varepsilon}$ also solves (7.2), by comparison principle it follows that $u^{\varepsilon}\leq u$, and thus

[TABLE]

Above inequality with (6.1) yields that $u$ is uniformly Lipschitz continuous in space and time.

Next to show $(b)$, for $\varepsilon>0$ and $\sigma_{2}=\sup|\partial_{x_{2}}\vec{b}|$ we define

[TABLE]

For each $x$, pick $y=y(x,t)$ that realizes the supremum. As in the proof of Lemma 5.4, for a.e. $(x,t)\in\mathbb{R}^{2}\times(0,\infty)$ we have

[TABLE]

Therefore for a.e. $(x,t)\in\mathbb{R}^{2}\times(0,\infty)$,

[TABLE]

where for the second equality above we used the fact that $\vec{b}$ only depends on $x_{2}$. Thus $w$ is a subsolution. Since $w(\cdot,0)\leq u_{0}$, the comparison principle for (7.2) yields $w\leq u$. In particular we have

[TABLE]

which yields (b) with $\tau\leq\tan(\arcsin(e^{-{\sigma_{2}}T}))$. Since (a)-(b) imply (1.10) and that $u$ is cone monotone, Proposition 6.3 and Theorem 6.1 yield $(c)$.

∎

Remark 7.2*.*

Let us consider the travelling wave solution $u(x,t)=U(x+cte_{1})$ of (7.2) with smooth and periodic $\alpha$, studied in [22] . It was shown there that, assuming non-degeneracy, the free boundary $\Gamma_{0}=\partial\{U(x)>0\}$ can be represented by a Lipschitz graph $x_{1}=f(x_{2})$.

Our analysis shows that under the same assumption the graph function $f$ is at least $C^{1,\alpha}$. Indeed $|\nabla U|$ is globally bounded due to Theorem 1 of [22] and thus (1.10) holds for $u$. Now Theorem 6.1 applies to yield the desired regularity of $f$. This improvement suggests that singularity of the free boundary such as corner formulation could happen only when non-degeneracy fails.

The rest of the section discusses examples of singular solutions that are not present in the zero drift problem. First we discuss global-time persistence and aggravation of corners.

Theorem 7.3**.**

There exist solutions $u_{1},u_{2}$ to (1.5) in $Q$ with bounded smooth spatial vector fields and non-negative, Lipschitz initial data such that

$u_{1}$* is stationary and $\Gamma(u_{1})$ has a corner at the origin.*

• 2.

For a finite time, there is a corner of shrinking angles on $\Gamma(u_{2})$.

Proof.

Write $(x,y)$ as the space coordinate. Let

[TABLE]

and then it can be checked directly that

[TABLE]

is a stationary solution to (1.5). Notice $\Gamma_{0}(u_{1})$ is the [math]-level set of $\Phi$ and we claim that if $\Phi$ is degenerate, the interface can be non-smooth.

For example, we can take

[TABLE]

where $g$ is a function on $\mathbb{R}$ that it is only positive in $(0,1)$. Then $\partial\{u_{1}>0\}$ is a square. In particular, $\partial\{u_{1}>0\}$ contains a Lipschitz corner at the origin.

Next we show (2). Take $\vec{b}:=(ax,by)$ (for a moment) and

[TABLE]

where

[TABLE]

Then the $\Gamma_{t}(\varphi)$ contains a corner with vertex at the origin.

Let us show that $\varphi$ is a supersolution to (1.1) for $t\in(0,1/{{\sigma_{1}}})$. Due to Lemma 2.6, we only need to check this for $x>k^{1/2}|y|$.

[TABLE]

Now we fix $a$ and take $b$ such that

[TABLE]

if ${{\sigma_{1}}}\geq 10$ and $t\leq 1/{{\sigma_{1}}}$. Next we further take ${{\sigma_{1}}}$ to be large enough such that, the first part of (7.5) is also non-negative. We conclude that for $t\in(0,1/\sigma_{1})$, $\varphi$ is indeed a supersolution and its support contains a corner with angles shrinking from $2\arctan(k_{0}^{-\frac{1}{2}})$ to $2\arctan(k(t)^{-\frac{1}{2}})$.

Now consider a solution $u_{2}$ with initial data $u_{0}$ such that $u_{0}=\varphi(x,y,0)$ in ${{B_{1}}}$ and $u_{0}\leq\varphi(x,y,0)$. By comparison, $\varphi\geq u_{2}$ for all times and so

[TABLE]

Since $\vec{b}=0$ at the origin, the origin is a one-point streamline. By Lemma 3.3, $0\in\overline{\Omega_{t}(u_{2})}$ for all $t\geq 0$. Thus $\Gamma_{t}(u_{2})$ has a shrinking corner for a short time. Lastly since $u_{2}$ is compactly supported, we can truncate $\vec{b}$ to be bounded which does not affect $u_{2}$ and its support.

∎

Next we consider formation of corners and cusps over time.

Theorem 7.4**.**

There is a solution $u$ to (1.5) in $Q$ with some bounded continuous vector field and non-negative, bounded and Lipshitz initial data $u_{0}$ such that:

$\Gamma_{0}(u)$* is smooth;*

• 2.

$\Gamma_{t}(u)$* contains a corner/a cusp for a range of time.*

Proof.

First we consider $\vec{b}:=-(x+|y|,\,y).$ We will construct a supersolution for this choice of $\vec{b}$. For some ${\sigma_{0}},\sigma_{1},\varepsilon>0$, set $\lambda(t)={\sigma_{0}}\,e^{{\sigma_{1}}t},\,\alpha(t)=\varepsilon t$ and

[TABLE]

When $t=0$, the support of $\varphi$ is a half-plane, while for any $t>0$ there forms a corner on $\Gamma_{t}(\varphi)$.

In the positive set of $\varphi$ $(x>\alpha|y|)$, we have

[TABLE]

Here $\delta_{y}$ is the Dirac mass of variable $y$. Since $\delta_{y}\geq 0$, the above simplifies to

[TABLE]

Select ${\sigma_{1}}=4m,{\sigma_{0}}\leq\frac{1}{2}e^{-4m},\varepsilon\leq 1/4$ and then $\lambda^{\prime}\geq 2(m-1)\lambda+2m\lambda^{2}$. Therefore for $t\in[0,1]$,

[TABLE]

In the last inequality we used that $\lambda\leq 1/2,\varepsilon+\varepsilon t\leq 1/2$.

Thus $\varphi$ is a supersolution in $\mathbb{R}^{2}\times[0,1]$. Now $u_{0}=\varphi(x,y,0)$ in ${B_{1}}$ and $u$ be a solution with initial data $u_{0}$. Then by comparison we conclude that a corner forms on $\Gamma_{t}(u)$ for $t>0$.

Next we show the possibility of the formation of cusps. Consider

[TABLE]

which is continuous but not Lipschitz continuous at $x=0$. In particular in our barrier argument we will use approximations. For some ${\sigma_{2}}$ is large enough, let

[TABLE]

and let $\tau,\varepsilon,\delta>0$ be such that

[TABLE]

Set

[TABLE]

Then as $\varepsilon\to 0$, for $x\geq 0$,

[TABLE]

Directly from the definition, the support of $\varphi$ is smooth when $\alpha>1$, while a cusp appears when $\alpha=1$ i.e. $t>\tau$. Set the domain

[TABLE]

Let us check that $\varphi_{\varepsilon}$ is a supersolution to (1.5) in $\Sigma_{\varepsilon}$. Notice

[TABLE]

By direct computation, in $\Sigma$

[TABLE]

Note we can assume $\alpha\geq 1/2$ and $\nabla\cdot\vec{b}\leq\sigma$ for some universal $\sigma$ in $\Sigma$, and therefore the above

[TABLE]

To have $A_{1}\geq 0$, we only need

[TABLE]

Using $r^{2}\log r$ is negative and decreasing for $r\in[0,\frac{1}{2}]$ and (7.6), we have

[TABLE]

Also note by (7.6), we have $\lambda\leq 1,\alpha\leq 1+\tau^{2}\leq 2$, $4\alpha-2\geq\alpha(2-\delta)$, So

[TABLE]

In all $\Sigma_{i=1}^{4}A_{i}\geq 0$. We proved that $\varphi_{\varepsilon}$ is a supersolution in $\Sigma_{\varepsilon}$, so by Lemma 2.6, it is a supersolution in $B_{\frac{1}{2}}\times[0,2\tau]$.

Now for $h\in(0,1)$, we select $u^{h}_{0,\varepsilon}$ to be smooth with initial data $u^{h}_{0,\varepsilon}=h\,\varphi_{\varepsilon}(\cdot,0)$ in $B_{h}$ and $u^{h}_{0,\varepsilon}=0\text{ in }B^{c}_{2h}$. Let $u^{h}_{\varepsilon}$ solve (1.1) with vector field $\vec{b}$ and initial data $u^{h}_{0,\varepsilon}$. By finite propagation property, we can take $h$ to be small enough such that for all $\varepsilon\in(0,1)$

[TABLE]

By comparison (which is valid since $\vec{b}$ is smooth in $\Sigma_{\varepsilon}$), $u^{h}_{\varepsilon}\leq\varphi_{\varepsilon}$ in $B_{\frac{1}{2}}\times[0,2\tau]$. Now passing $\varepsilon\to 0$ gives a solution $u^{h}$ with initial data $h\varphi(\cdot,0)$ in $B_{h}$ such that $u^{h}\leq\varphi$ for $t\in[0,2\tau]$. As before we conclude by the geometry of $\Omega(\varphi)$ and Lemma 3.3 that a cusp appears for $\tau<t<2\tau$.

∎

Appendix A Proof of Lemma 2.6

Let us only consider the case when $U=\mathbb{R}^{d}$. The case of $U=B_{1}$ follows similarly.

Fix one non-negative $\phi\in C_{c}^{\infty}(\mathbb{R}^{d}\times[0,T))$. Denote

[TABLE]

For any $\varepsilon>0$, take finitely many space time balls $U_{i},i=1,...,n$ such that

for each $i\geq 1$, $|U_{i}|\leq\varepsilon^{d}$ and $U_{i}$ is in the $\varepsilon$-neighbourhood of $\Gamma(\psi)$,

• 2.

$\{U_{i}\}_{i=1,...,n}$ is an open cover of $\Gamma(\psi)\cap\{\phi>0\}.$

Since $\Gamma(\psi)$ is of dimension $d-1$, we can assume

[TABLE]

Take a partition of unity $\{\rho_{i},i=0,...,n\}$ which is subordinate to the open cover $\{U_{i}\}_{i\geq 0}$. Then for $i\geq 1$,

[TABLE]

By the assumption, $\psi$ is a supersolution in the interior of its positive set. And since $\varepsilon$ can be arbitrarily small, to show (2.4) we only need to show

[TABLE]

as $\varepsilon\to 0$.

By property 1 of $U_{i}$ and the regularity assumption on $\psi$, in all $U_{i},i\geq 1$ we have

[TABLE]

Now from (A.1), (A.2) and $\alpha<m$, it follows that

[TABLE]

which indeed converges to [math] as $\varepsilon\to 0$.

Appendix B Sketch of the proof of Lemma 5.3

We follow the idea of Lemma 9 [5] and compute

[TABLE]

Without loss of generality, suppose locally near the origin

[TABLE]

because otherwise $\Delta f(0)=0$. Choosing an appropriate system of coordinates, we can have

[TABLE]

We will evaluate $w$ by above by choosing $\nu(x)=\frac{\nu_{*}(x)}{|\nu_{*}(x)|}$ where

[TABLE]

where $\gamma$ satisfies

[TABLE]

With this choice of $\nu$, we define $y:=x+\psi(x)\nu(x)$ and so $y(0)=\psi(0)e_{n}$. After direct computations (also see [5]), we can write

[TABLE]

such that the first-order term, except the translation $\varphi(0)e_{n}$, satisfies

[TABLE]

Hence $Y_{*}(x)$ is a rigid rotation plus a dilation and we have

[TABLE]

Then

[TABLE]

By the condition on $\psi$ and the computations done in Lemma 9 [5], the first term is non-positive.

Since $h$ is smooth, the second term converges to

[TABLE]

Now using (B.1) and the assumption that $\Delta h\geq-C$ and $\|\nabla\psi\|_{\infty}\leq 1$, we get

[TABLE]

Thus we finished the proof.

Appendix C Proof of Lemma 5.4

Let us suppose $x=0$ and $f(0)=h(y)$ for a unique $y$. We only compute $\partial_{1}f(0)=\partial_{x_{1}}f(0)$. If $\nabla h(y)=0$, it is not hard to see

[TABLE]

Next suppose $\nabla h(y)\neq 0$. We know that $h$ obtains its minimum over $B(0,\psi(0))$ at point $y\in\partial B(0,{\psi(0)})$. Let us assume

[TABLE]

For smooth $h$, it is not hard to see that

[TABLE]

Near point $y$

[TABLE]

To estimate $w((\delta,0,...,0))$, consider the leading terms:

[TABLE]

By a standard argument, under the constrain

[TABLE]

$A(\delta)$ achieves its minimum at

[TABLE]

with value

[TABLE]

Thus

[TABLE]

Notice that $\partial_{1}h(y)=-ky_{1}$. So we find

[TABLE]

This leads to the conclusion.