Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint. *

TL;DR

This paper proves that a tissue growth model with a non-overlapping constraint converges to a Hele-Shaw free boundary problem in the incompressible limit, despite using a singular pressure law.

Contribution

It introduces a non-overlapping constraint into a tissue growth model and shows convergence to the Hele-Shaw problem even with a singular pressure law.

Findings

The model converges to the Hele-Shaw free boundary problem.

The non-overlapping constraint is preserved in the limit.

Singular pressure laws can enforce non-overlapping constraints.

Abstract

A mathematical model for tissue growth is considered. This model describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that such cell population model converges at the incompressible limit towards a Hele-Shaw type free boundary problem. The novelty of this work is to impose a non-overlapping constraint. This constraint is important to be satisfied in many applications. One way to guarantee this non-overlapping constraint is to choose a singular pressure law. The aim of this paper is to prove that, although the pressure law has a singularity, the incompressible limit leads to the same Hele-Shaw free boundary problem.

Equations

$${\partial }_{t}n+\nabla \cdot (nv)=nG(p),\text{mbox}on{\mathbb{R}}^{+}\times {\mathbb{R}}^d,$$

$$v=-\nabla p,p=\Pi (n).$$

$$P(n)=ϵ\frac{n}{1-n}.$$

$${\partial }_t{n}_{ϵ}-\nabla \cdot (n_{ϵ}\nabla p_{ϵ})=n_{ϵ}G(p_{ϵ}),$$

$$p_{ϵ}=P(n_{ϵ})=ϵ\frac{n_{ϵ}}{1-n_{ϵ}}.$$

$${\partial }_t{p}_{ϵ}-(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\Delta p_{ϵ}-|\nabla p_{ϵ}{|}^2=(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G(p_{ϵ}).$$

$$-p_0^2\Delta p_0=p_0^{2}G(p_0).$$

$$\{\begin{array}{rl}& \exists G_m>0,\Vert G{\Vert }_{\infty }\le G_m,\\ & G^{\prime }<0,\text{mbox}and\exists \gamma >0,\underset{[0,P_M]}{\min }|G^{\prime }|=\gamma ,\\ & \exists P_M>0,G(P_M)=0.\end{array}$$

$$\{\begin{array}{rl}& 0\le n_{ϵ}^{ini},p_{ϵ}^{ini}:=ϵ\frac{n_{ϵ}^{ini}}{ϵ+n_{ϵ}^{ini}}\le P_M,\\ & \Vert {\partial }_{x_i}{n}_{ϵ}^{ini}{\Vert }_{L^1({\mathbb{R}}^d)}\le C,i=1,...,d,\\ & \exists n_0^{ini}\in L_{+}^1({\mathbb{R}}^d),\Vert n_{ϵ}^{ini}-n_0^{ini}{\Vert }_{L^1({\mathbb{R}}^d)}\to 0\text{mbox}asϵ\to 0,\\ & \exists K\subset {\mathbb{R}}^d,K\text{mbox}compact,\forall ϵ\in (0,{ϵ}_0),\text{mbox}supp{n}_{ϵ}^{ini}\subset K.\end{array}$$

$$0\le n_0\le 1,0\le p_0\le P_M,$$

$${\partial }_t{n}_0-\Delta p_0=n_{0}G(p_0),\text{mbox}in{\mathcal{D}}^{\prime }(Q_T),$$

$${\partial }_t{n}_0-\nabla \cdot (n_0\nabla p_0)=n_{0}G(p_0),$$

$$(1-n_0)p_0=0,$$

$$p_0^2(\Delta p_0+G(p_0))=0,$$

$${\partial }_t{n}_{ϵ}-\nabla \cdot (n_{ϵ}\nabla p_{ϵ})=n_{ϵ}G(p_{ϵ}).$$

$$\frac{d}{dt}|n_{ϵ}{|}_{-}-\nabla \cdot (|n_{ϵ}{|}_{-}\nabla p_{ϵ})=|n_{ϵ}{|}_{-}G(p_{ϵ}).$$

$$\frac{d}{dt}{\int }_{{\mathbb{R}}^d}|n_{ϵ}{|}_{-}dx\le {\int }_{{\mathbb{R}}^d}|n_{ϵ}{|}_{-}G(p_{ϵ})dx\le G_m{\int }_{{\mathbb{R}}^d}|n_{ϵ}{|}_{-}dx.$$

$${\int }_{{\mathbb{R}}^d}|n_{ϵ}{|}_{-}dx\le e^{G_{m}t}{\int }_{{\mathbb{R}}^d}|n_{ϵ}^{ini}{|}_{-}dx.$$

$${\partial }_t{n}_{ϵ}-\Delta H(n_{ϵ})=n_{ϵ}G(P(n_{ϵ})),$$

$$0\le n_{ϵ}\in L^{\infty }([0,T];L^1\cap L^{\infty }({\mathbb{R}}^d));$$

$$0\le p_{ϵ}\le P_M,0\le n_{ϵ}\le \frac{P_M}{P_M+ϵ}\le 1.$$

$${\partial }_t(n_{ϵ}-m_{ϵ})-\Delta (H(n_{ϵ})-H(m_{ϵ}))\le n_{ϵ}G(P(n_{ϵ}))-m_{ϵ}G(P(m_{ϵ})).$$

$$\Delta f(y)=f^{\prime \prime }(y)|\nabla y{|}^2+f^{\prime }(y)\Delta y,$$

$${\partial }_t|n_{ϵ}-m_{ϵ}{|}_{+}-\Delta |H(n_{ϵ})-H(m_{ϵ}){|}_{+}\le$$

$$+m_{ϵ}(G(P(n_{ϵ}))-G(P(m_{ϵ})))1_{n_{ϵ}-m_{ϵ}>0}.$$

$${\partial }_t|n_{ϵ}-m_{ϵ}{|}_{+}-\Delta |H(n_{ϵ})-H(m_{ϵ}){|}_{+}\le G_m|n_{ϵ}-m_{ϵ}{|}_{+}.$$

$$\frac{d}{dt}{\int }_{{\mathbb{R}}^d}|n_{ϵ}-m_{ϵ}{|}_{+}dx\le G_m{\int }_{{\mathbb{R}}^d}|n_{ϵ}-m_{ϵ}{|}_{+}dx.$$

$${\int }_{{\mathbb{R}}^d}|n_{ϵ}-m_{ϵ}{|}_{+}dx\le e^{G_{m}t}{\int }_{{\mathbb{R}}^d}|n_{ϵ}^{ini}-m_{ϵ}^{ini}{|}_{+}dx.$$

$$0\le p_{ϵ}\le P_M.$$

$$\frac{d}{dt}\Vert n_{ϵ}{\Vert }_{L^1({\mathbb{R}}^d)}\le G_m\Vert n_{ϵ}{\Vert }_{L^1({\mathbb{R}}^d)},$$

$$\Vert n_{ϵ}{\Vert }_{L^1({\mathbb{R}}^d)}\le e^{G_{m}t}\Vert n_{ϵ}^{ini}{\Vert }_{L^1({\mathbb{R}}^d)}.$$

$$\Vert p_{ϵ}{\Vert }_{L^1(R^d)}\le (ϵ+P_M){\int }_{{\mathbb{R}}^d}|n_{ϵ}|dx\le C{e}^{G_{m}t}\Vert n_{ϵ}^{ini}{\Vert }_{L^1({\mathbb{R}}^d)}.$$

$${\partial }_t{\partial }_{x_i}{n}_{ϵ}-\Delta (H^{\prime }(n_{ϵ}){\partial }_{x_i}{n}_{ϵ})={\partial }_{x_i}{n}_{ϵ}G(p_{ϵ})+n_{ϵ}{G}^{\prime }(p_{ϵ}){\partial }_{x_i}{p}_{ϵ}.$$

Full text

Keywords: Nonlinear parabolic equation; Incompressible limit; Free boundary problem; Tissue growth modelling.

AMS Subject Classification: 35K55; 76D27; 92C50.

1 Introduction

Mathematical models are now commonly used in the study of growth of cell tissue. For instance, a wide literature is now available on the study of the tumor growth through mathematical modeling and numerical simulations [2, 3, 14, 18]. In such models, we may distinguish two kinds of description: Either they describe the dynamics of cell population density (see e.g. [6, 8]), or they consider the geometric motion of the tissue through a free boundary problem of Hele-Shaw type (see e.g. [16, 15, 11, 18]). Recently the link between both descriptions has been investigated from a mathematical point of view thanks to an incompressible limit [22].

In this paper, we depart from the simplest cell population model as proposed in [7]. In this model the dynamics of the cell density is driven by pressure forces and cell multiplication. More precisely, let us denote by $n(t,x)$ the cell density depending on time $t\geq 0$ and position $x\in\mathbb{R}^{d}$, and by $p$ the mechanical pressure. The mechanical pressure depends only on the cell density and is given by a state law $p=\Pi(n)$. Cell proliferation is modelled by a pressure-limited growth function denoted $G$. Mechanical pressure generates cells displacement with a velocity whose field $v$ is computed thanks to the Darcy’s law. After normalizing all coefficients, the model reads

[TABLE]

The choice $\Pi(n)=\frac{\gamma}{\gamma-1}n^{\gamma-1}$ has been taken in [22, 23, 24]. This choice allows to recover the well-known porous medium equation for which a lot of nice mathematical properties are now well-established (see e.g. [26]). The incompressible limit is then obtained by letting $\gamma$ going to $+\infty$.

However, this state law does not prevent cells to overlap. In fact, it is not possible with this choice to avoid the cell density to take value above $1$ (which corresponds here to the maximal packing density after normalization). A convenient way to avoid cells overlapping is to consider a pressure law which becomes singular when the cell density approaches $1$. Such type of singularity is encountered, for instance, in the kinetic theory of dense gases where the interaction between molecules is strongly repulsive at very short distance [9]. Similar singular pressure laws have been also considered in [12, 13] to model collective motion, in [4, 5] to model the traffic flow, and in [21] to model crowd motion (see also the review article [19]). Then, in order to fit this non-overlapping constraint, we consider the following simple model of pressure law given by

[TABLE]

Finally, the model under study in this paper reads, for $\epsilon>0$,

[TABLE]

This system is complemented by an initial data denoted $n_{\epsilon}^{ini}$. The aim of this paper is to investigate the incompressible limit of this model, which consists in letting $\epsilon$ going to [math] in the latter system.

At this stage, it is of great importance to observe that from (1.1), we may deduce an equation for the pressure by simply multiplying (1.1) by $P^{\prime}(n_{\epsilon})$ and using the relation $n_{\epsilon}=\frac{p_{\epsilon}}{\epsilon+p_{\epsilon}}$ from (1.2),

[TABLE]

Formally, we deduce from (1.3) that when $\epsilon\to 0$, we expect to have the relation

[TABLE]

Moreover, passing formally to the limit into (1.2), it appears clearly that $(1-n_{0})p_{0}=0$. We deduce from this relation that if we introduce the set $\Omega_{0}(t)=\{p_{0}>0\}$, then we obtain a free boundary problem of Hele-Shaw type: On $\Omega_{0}(t)$, we have $n_{0}=1$ and $-\Delta p_{0}=G(p_{0})$, whereas $p_{0}=0$ on $\mathbb{R}^{d}\setminus\Omega_{0}(t)$. Thus although the pressure law is different, we expect to recover the same free boundary Hele-Shaw model as in [22].

The incompressible limit of the above cell mechanical model for tumor growth with a pressure law given by $\Pi(n)=\frac{\gamma}{\gamma-1}n^{\gamma-1}$ has been investigated in [22] and in [23] when taking into account active motion of cells. In [24], the case with viscosity, where the Darcy’s law is replaced by the Brinkman’s law, is studied. We mention also the recent works [17, 20] where the incompressible limit with more general assumptions on the initial data has been investigated. However, in all these mentionned works the pressure law do not prevent the non-overlapping of cells. Up to our knowledge, this work is the first attempt to extend the previous result with this constraint, i.e. with a singular pressure law as given by (1.2).

The outline of the paper is the following. In the next section we give the statement of the main result in Theorem 2.1, which is the convergence when $\epsilon$ goes to [math] of the mechanical model (1.1)–(1.2) towards the Hele-Shaw free boundary system. The rest of the paper is devoted to the proof of this result. First, in section 3 we establish some a priori estimate allowing to obtain space compactness. Then, section 4 is devoted to the study of the time compactness. Thanks to compactness results, we can pass to the limit $\epsilon\to 0$ in system (1.1)–(1.2) in section 5, up to the extraction of a subsequence. Finally the proof of the complementary relation (1.4) is performed in section 6.

2 Main result

The aim of this paper is to establish the incompressible limit $\epsilon\to 0$ of the cell mechanical model with non-overlapping constraint (1.1)–(1.2). Before stating our main result, we list the set of assumptions that we use on the growth fonction and on the initial data. For the growth function, we assume

[TABLE]

The quantity $P_{M}$, for which the growth stops, is commonly called the homeostatic pressure [25]. This set of assumptions on the growth function is quite similar to the one in [22], except for the bound on the growth term which is needed here due to the singularity in the pressure law.

For the initial data, we assume that there exists $\epsilon_{0}>0$ such that for all $\epsilon\in(0,\epsilon_{0})$,

[TABLE]

Notice that this set of assumptions imply that $n_{\epsilon}^{ini}$ is uniformly bounded in $W^{1,1}(\mathbb{R}^{d})$.

We are now in position to state our main result.

Theorem 2.1

Let $T>0$, $Q_{T}=(0,T)\times\mathbb{R}^{d}$. Let $G$ and $(n^{ini}_{\epsilon})$ satisfy assumptions (2.5) and (2.6) respectively. After extraction of subsequences, both the density $n_{\epsilon}$ and the pressure $p_{\epsilon}$ converge strongly in $L^{1}(Q_{T})$ as $\epsilon\rightarrow 0$ to the limit $n_{0}\in C([0,T];L^{1}(\mathbb{R}^{d}))\cap BV(Q_{T})$ and $p_{0}\in BV(Q_{T})\cap L^{2}([0,T];H^{1}(\mathbb{R}^{d}))$, which satisfy

[TABLE]

and

[TABLE]

Moreover, we have the relation

[TABLE]

and the complementary relation

[TABLE]

This result extends the one in [22] to singular pressure laws with non-overlapping constraint. We notice that we recover the same limit model whose uniqueness has already been stated in [22, Theorem 2.4].

Although our proof follows the idea in [22], several technical difficulties must be overcome due to the singularity of the pressure law. Indeed, we first recall that with the choice $\Pi(n)=\frac{\gamma}{\gamma-1}n^{\gamma-1}$, equation (1.1) may be rewritten as the porous medium equation $\partial_{t}n+\Delta n^{\gamma}=nG(\Pi(n))$. A lot of estimates are known and well established for this equation (see [26]), in particular a semiconvexity estimate is used in [22] which allows to obtain estimate on the time derivative and thus compactness. With our choice of pressure law, (1.1) should be consider as a fast diffusion equation. Thus we have first to state a comparison principle to obtain a priori estimates (see Lemma 3.2). Unlike in [22], we may not use a semiconvexity estimate to obtain estimate on the time derivative. To do so, we use regularizing effects (see section 4). Then the convergence proof has to be adapted for these new estimates.

Finally, we illustrate the comparison between the two pressure laws $P$ and $\Pi$ by a numerical simulation. We display in Figure 1 the density computed thanks to a discretization with an upwind scheme of (1.1). In Figure 1-left, the pressure law is $p=P(n)=\epsilon\frac{n}{1-n}$ as in (1.2) with $\epsilon=0.5$. In Figure 1-right, the pressure law is $p=\Pi(n)=\frac{\gamma}{\gamma-1}n^{\gamma}$ with $\gamma=20$. We take $G(p)=10(10-p)_{+}$ as growth function (which satisfies obviously assumption (2.5) with $P_{M}=10$). The dashed lines in these plots correspond to the constant value $1$. As expected, we observe that the density $n$ is bounded by $1$ in the case of the pressure law $P$ whereas it takes values greater than $1$ for the pressure law $\Pi$. This observation illustrates the fact that the choice of the pressure law $\Pi$ does not prevent from overlapping.

3 A priori estimates

3.1 Nonnegativity principle

The following Lemma establishes the nonnegativity of the density.

Lemma 3.1

Let $(n_{\epsilon},p_{\epsilon})$ be a solution to (1.1) such that $n_{\epsilon}^{ini}\geq 0$ and $\|G\|_{\infty}\leq G_{m}<\infty$. Then, for all $t\geq 0$, $n_{\epsilon}(t)\geq 0$.

Proof.

We have the equation

[TABLE]

We use the Stampaccchia method. We multiply by $\mathbf{1}_{n_{\epsilon}<0}$, then using the notation $|n|_{-}=\max(0,-n)$ for the negative part, we get

[TABLE]

We integrate in space, using assumption (2.5), we deduce

[TABLE]

So, after a time integration

[TABLE]

With the initial condition $n_{\epsilon}^{ini}\geq 0$, we deduce $n_{\epsilon}\geq 0$. ∎

3.2 A priori estimates

In order to use compactness results, we need first to find a priori estimates on the pressure and the density. We first observe that we may rewrite system (1.1) as, by using (1.2),

[TABLE]

with $H(n)=\int_{0}^{n}uP^{\prime}(u)du=P(n)-\epsilon\ln(P(n)+\epsilon)+\epsilon\ln\epsilon$.

Lemma 3.2

Let us assume that (2.5) and (2.6) hold. Let $(n_{\epsilon},p_{\epsilon})$ be a solution to (3.12)–(1.2). Then, for all $T>0$, we have the uniform bounds in $\epsilon\in(0,\epsilon_{0})$,

[TABLE]

More generally, we have the comparison principle: If $n_{\epsilon}$, $m_{\epsilon}$ are respectively subsolution and supersolution to (3.12), with initial data $n_{\epsilon}^{ini}$, $m_{\epsilon}^{ini}$ as in (2.6) and satisfying $n_{\epsilon}^{ini}\leq m_{\epsilon}^{ini}$. Then for all $t>0$, $n_{\epsilon}(t)\leq m_{\epsilon}(t)$.

Finally, we have that $(n_{\epsilon})_{\epsilon}$ is uniformly bounded in $L^{\infty}([0,T],W^{1,1}(\mathbb{R}^{d}))$ and $(p_{\epsilon})_{\epsilon}$ is uniformly bounded in $L^{1}([0,T],W^{1,1}(\mathbb{R}^{d}))$.

Proof.

Comparison principle.

Let $n_{\epsilon}$ be a subsolution and $m_{\epsilon}$ a supersolution of (3.12), we have

[TABLE]

Notice that, since the function $H$ is nondecreasing, the sign of $n_{\epsilon}-m_{\epsilon}$ is the same as the sign of $H(n_{\epsilon})-H(m_{\epsilon})$. Moreover,

[TABLE]

so for $y=H(n_{\epsilon})-H(m_{\epsilon})$ and $f(y)=y_{+}$ is the positive part, the so-called Kato inequality reads $\Delta f(y)\geq f^{\prime}(y)\Delta y$. Thus multiplying the latter equation by $\mathbf{1}_{n_{\epsilon}-m_{\epsilon}>0}$, we obtain

[TABLE]

From assumption (2.5), we have that $G$ is nonincreasing. Thus, since $n\mapsto P(n)$ is increasing, we deduce that the last term of the right hand side is nonpositive. Since $G$ is uniformly bounded we obtain

[TABLE]

After an integration over $\mathbb{R}^{d}$,

[TABLE]

Then, integrating in time, we deduce

[TABLE]

Since we have $n^{ini}_{\epsilon}\leq m^{ini}_{\epsilon}$, we deduce that for all $t>0$, $|n_{\epsilon}-m_{\epsilon}|_{+}(t)=0$.

$L^{\infty}$** bounds.**

We define $n_{M}=\frac{P_{M}}{\epsilon+P_{M}}$, such that $p_{M}=P(n_{M})$, then applying the comparison principle with $m_{\epsilon}=n_{M}$, we deduce, using also the assumption on the initial data (2.6) that for all $0<\epsilon\leq\epsilon_{0}$, $n_{\epsilon}\leq n_{M}.$ Moreover, since [math] is clearly a subsolution to (3.12), we also have by the comparison priniciple $n_{\epsilon}\geq 0$. Since $n_{M}\leq 1$, we have $0\leq n_{\epsilon}\leq n_{M}\leq 1$ which implies

[TABLE]

$L^{1}$** bound of $n,p$.**

By nonnegativity, after a simple integration in space of equation (1.1), we deduce

[TABLE]

where we use (2.5). Integrating in time give the $L^{1}$ bound,

[TABLE]

Then, using $p_{\epsilon}=n_{\epsilon}(\epsilon+p_{\epsilon})$ by (1.2), we get from the bound $p_{\epsilon}\leq P_{M}$, which has been proved above,

[TABLE]

Estimates on the $x$ derivative.

We derive equation (3.12) with respect to $x_{i}$ for $i=1,\ldots,d$,

[TABLE]

Multiplying by sign$(\partial_{x_{i}}n_{\epsilon})$, we get

[TABLE]

We can remark that $\mbox{sign}(\partial_{x_{i}}n_{\epsilon})=\mbox{sign}(\partial_{x_{i}}H(n_{\epsilon}))$, so, by the same token as above, we have

[TABLE]

Moreover, $\mbox{sign}(\partial_{x_{i}}n_{\epsilon})=\mbox{sign}(\partial_{x_{i}}p_{\epsilon})$, thus $\partial_{x_{i}}p_{\epsilon}\mbox{sign}(\partial_{x_{i}}n_{\epsilon})=|\partial_{x_{i}}p_{\epsilon}|$. By assumption (2.5), we know that

[TABLE]

we deduce

[TABLE]

After an integration in time and space,

[TABLE]

This latter inequality provides us with a uniform bound on the space derivative of $n_{\epsilon}$ in $L^{1}$. Then

[TABLE]

We split the integral in two: Either $n_{\epsilon}\leq 1/2$ and then $\frac{\epsilon}{(1-n_{\epsilon})^{2}}\leq C$; or $n_{\epsilon}>1/2$.

[TABLE]

where we have used the estimate (3.14) for the last inequality. Then, integrating in time, we deduce, using again the estimate (3.14)

[TABLE]

It concludes the proof. ∎

3.3 Compact support

The following Lemma proves that assuming that the initial data is compactly supported, then the pressure is compactly supported for any time with a control of the growth of the support.

Lemma 3.3** **(Finite speed of propagation)

Under the same assumptions as in Theorem 2.1, we have that $\mbox{supp }p_{\epsilon}\subset B(0,R(t))$ with $R(t)\leq 2\sqrt{C(T+t)}$, where $B(0,R(t))$ is the ball of center [math] and radius $R(t)$.

Proof.

Using the equation on $p_{\epsilon}$ (1.3),

[TABLE]

Let us introduce for $C>0$,

[TABLE]

with $\theta=\frac{d}{4G_{m}}$. Then $\tilde{p}$ is compactly supported in $B(0,R_{\theta}(t))$ with $R_{\theta}(t)=2\sqrt{C(\theta+t)}.$ We have

[TABLE]

and

[TABLE]

Then, for all $t\in[0,\theta]$,

[TABLE]

In other words, $\tilde{p}$ is a supersolution for the equation for the pressure. Let us show that it implies that $p\leq\tilde{p}$. We define $\tilde{n}=\frac{\tilde{p}}{\epsilon+\tilde{p}}=N(\tilde{p})$. We know that

[TABLE]

Then, on the one hand, multiplying (3.15) with by $N^{\prime}(\tilde{p})$ we get

[TABLE]

On the other hand, from (1.1),

[TABLE]

By the comparison principle (see Lemma 3.2), we have

[TABLE]

Thus, for all $t\in[0,\theta]$,

[TABLE]

and $p_{\epsilon}(t)$ is compactly supported in $B(0,R_{\theta}(t))$ provided we choose $C$ large enough such that $p_{\epsilon}^{ini}(x)\leq\tilde{p}(t=0,x)$, which can be done thanks to our assumption on the initial data (2.6).

Since $p_{\epsilon}$ is uniformly bounded in $L^{\infty}$, we may iterate the process on $[\theta,2\theta]$. After several iterations, we reach the time $T$ and prove the result on $[0,T]$. ∎

3.4 $L^{2}$ estimate for $\nabla p$

In the following Lemma, we state a uniform $L^{2}$ estimate on the gradient of the pressure.

Lemma 3.4** **($L^{2}$ estimate for $\nabla p$)

Under the same assumptions as in Theorem 2.1, we have a uniform bound on $\nabla p_{\epsilon}$ in $L^{2}(Q_{T})$.

Proof.

For a given function $\psi$ we have, multiplying (1.1) by $\psi(n_{\epsilon})$,

[TABLE]

Let $\Psi$ be an antiderivative of $\psi$, we have thanks to an integration by parts

[TABLE]

We choose $\psi$ such as $n_{\epsilon}\nabla n_{\epsilon}\cdot\nabla p_{\epsilon}\psi^{\prime}(n_{\epsilon})=|\nabla p_{\epsilon}|^{2}$, i.e. $n_{\epsilon}\psi^{\prime}(n_{\epsilon})=p^{\prime}(n_{\epsilon})$. After straightforward computations, we find $\psi(n)=\epsilon(\ln(n)-\ln(1-n)+\frac{1}{1-n})$ and $\Psi(n)=\epsilon n(\ln(n)-\ln(1-n))$. It gives

[TABLE]

We integrate in time, using also the expression of $p_{\epsilon}$ in (1.2),

[TABLE]

Then, to have a bound on the $L^{2}$-norm of $\nabla p_{\epsilon}$, it suffices to prove a uniform control on $\int_{\mathbb{R}^{d}}\epsilon n_{\epsilon}|\ln(\frac{p_{\epsilon}}{\epsilon})|dx$. We have

[TABLE]

The second term of the right hand side is small when $\epsilon$ is small thanks to the $L^{1}$ bound on $n_{\epsilon}$, thus it is uniformly bounded. Using the expression of $p_{\epsilon}$ in (1.2), we get

[TABLE]

Then, since $0\leq p_{\epsilon}\leq P_{M}$ and since $x\mapsto x|\ln x|$ is uniformly bounded on $[0,P_{M}]$, we get

[TABLE]

We conclude thanks to Lemma 3.3, which provides a uniform control on the support of $p_{\epsilon}$. ∎

4 Regularizing effect and time compactness

As already noticed in [23], regularizing effects, similar to the ones observed for the heat equation [1, 10], allow to deduce estimates on the time derivatives.

Lemma 4.1

Under the assumptions (2.5) and (2.6), the weak solution $(\rho_{k},p_{k})$ satisfies the estimates

[TABLE]

for a large enough (independent of $\epsilon$) constant $\kappa$.

Proof.

Let us denote $w_{\epsilon}=\Delta p_{\epsilon}+G(p_{\epsilon})$, the equation on the pressure (1.3) reads

[TABLE]

The proof is divided into several steps. We first find a lower bound for $w_{\epsilon}$ by using the comparison principle. Then we deduce estimates on the density and on the pressure.

1st step. Thanks to (4.16), we deduce an equation satisfied by $w_{\epsilon}$. On the one hand, by multiplying (4.16) by $G^{\prime}(p_{\epsilon})$, we deduce, since $G$ is decreasing from (2.5)

[TABLE]

On the other hand, we have

[TABLE]

Thus, with (4.17), we deduce that $w_{\epsilon}=\Delta p_{\epsilon}+G(p_{\epsilon})$ satisfies

[TABLE]

By definition of $w_{\epsilon}$, we have $(\Delta p_{\epsilon})^{2}\geq w_{\epsilon}^{2}-2G(p_{\epsilon})w_{\epsilon}$. Thus we deduce that

[TABLE]

where we have used the notation

[TABLE]

Following an idea of [10] which has been generalized in [23], we introduce the function

[TABLE]

where the function $h$ will be defined later such that $W$ is a subsolution for (4.18). We compute

[TABLE]

Using again equation (4.16), we have

[TABLE]

By definition of $\mathcal{F}(W)$ in (4.19), we deduce with (4.21),

[TABLE]

We may rearrange it into

[TABLE]

Let us choose

[TABLE]

where $\kappa>0$ is chosen large enough (independent of $\epsilon$) such that

[TABLE]

Thanks to this choice, we have

[TABLE]

and

[TABLE]

Finally, we obtain from (4.22)

[TABLE]

where we use the fact that by definition (4.20) we have $W\leq 0$ (recalling also that $G$ is decreasing by assumption (2.5)).

Thus, by the sub- and super-solution technique, we deduce, using also (4.18) that

[TABLE]

2nd step. Using again equation (4.16), we get from (4.24)

[TABLE]

which is the first inequality of Lemma 4.1. Finally, by definition (1.2), we have also $n_{\epsilon}=\frac{p_{\epsilon}}{p_{\epsilon}+\epsilon}$. Thus

[TABLE]

where we use the definition (1.2) for the last identity. We conclude easily the proof. ∎

Thanks to this latter Lemma, we may deduce uniform estimates on the time derivative of $n_{\epsilon}$ and $p_{\epsilon}$.

Lemma 4.2

For any $\tau>0$, we have that $\partial_{t}n_{\epsilon}$ is uniformly bounded in $L^{\infty}([\tau,T];L^{1}(\mathbb{R}^{d}))$ and $\partial_{t}p_{\epsilon}$ is uniformly bounded in $L^{1}([\tau,T]\times\mathbb{R}^{d})$.

Proof.

We use the equality $|\partial_{t}n_{\epsilon}|=\partial_{t}n_{\epsilon}+2|\partial_{t}n_{\epsilon}|_{-}$, where we recall that $|\cdot|_{-}$ denotes the negative part. Thus

[TABLE]

where we have used equation (3.13) to bound the first term and Lemma 4.1 for the second term. By the same token, we have

[TABLE]

We conclude the proof thanks to the estimates on $n_{\epsilon}$ and $p_{\epsilon}$ in $L^{1}\cap L^{\infty}$ obtained in Lemma 3.2. ∎

5 Convergence

This section is devoted to the proof of Theorem 2.1 apart from the complementary relation (2.11) which is postponed to the next section.

Since the sequences $(n_{\epsilon})_{\epsilon}$ and $(p_{\epsilon})_{\epsilon}$ are bounded in $W^{1,1}_{loc}(Q_{T})$, due to Lemma 3.2 and 4.2, we may apply the Helly theorem and recover strong convergence in $L^{1}_{loc}(Q_{T})$, up to an extraction. If we want to extend this local convergence to a global convergence in $L^{1}(Q_{T})$ we need to prove that we can control the mass in an initial strip and in the tail. Indeed, let $\epsilon,\epsilon^{\prime}>0$, $R>0$, $\tau>0$

[TABLE]

Since we have strong convergence of $n_{\epsilon}$ in $L^{1}_{loc}(Q_{T})$,

[TABLE]

Then we have to control the two other terms in the right hand side.

The control of the initial strip comes from the $L^{1}$ estimate of $n$,

[TABLE]

For the control of the tail we consider $\phi\in C^{\infty}(\mathbb{R})$ such that $0\leq\phi\leq 1$, $\phi(x)=0$ for $|x|<R-1$ and $\phi(x)=1$ for $|x|>R$. We define $\phi_{R}(x)=\phi(x/R)$. Then

[TABLE]

where the notation $C$ stand for a generic nonnegative constant. Moreover, using equation (3.12), we deduce

[TABLE]

Then, integrating on $[0,T]$, we get

[TABLE]

By assumption (2.6), since the initial data is uniformly compactly supported, we deduce that the right hand side tends to [math] as $R$ goes to $+\infty$ and ${\epsilon}$ goes to [math]. Then $(n_{\epsilon})_{\epsilon}$ is a Cauchy sequence in $L^{1}(Q_{T})$. It implies its convergence in $L^{1}(Q_{T})$. The convergence of the pressure follows from the same kind of computation. The only difference is for the control of the tail and which is directly given by the estimate

[TABLE]

Therefore, we can extract subsequences and pass to the limit in the equation

[TABLE]

which implies

[TABLE]

This is the relation (2.10). We can also pass to the limit in the uniform estimate of Lemma 3.2 which provides (2.7) and $n_{0},p_{0}\in BV(Q_{T})$.

Limit model.

We first recall that from (3.12), we have

[TABLE]

We get,

[TABLE]

Thus, the term in the Laplacien converges strongly to $p_{0}$ as $\epsilon$ goes to [math]. Then, thanks to the strong convergence of $n_{\epsilon}$ and $p_{\epsilon}$, we deduce that in the sense of distribution $(n_{0},p_{0})$ satisfies (2.8). Moreover, due to the uniform estimate on $\nabla p$ in $L^{2}(Q_{T})$ of Lemma 3.4, we can show, by passing into the limit in a product of a weak-strong convergence, that in the sense of distribution $(n_{0},p_{0})$ satisfies (2.9).

Time continuity.

Let us define $0<t_{1}<t_{2}\leq T$, $\eta>0$. For a given $R>0$, we consider a smooth function $\zeta_{R}$ on $\mathbb{R}^{d}$ such that $0\leq\zeta_{R}\leq 1$, $\zeta_{R}(x)=1$ for $|x|<R-1$ and $\zeta_{R}(x)=0$ for $|x|>R$. We have

[TABLE]

We have

[TABLE]

with $1-\zeta_{R}$ a function which is zero on $B(0,R-1)$. Thus, as for the control of the tail, for $R$ large enough, we have, uniformly for $0<t_{1}<t_{2}\leq T$,

[TABLE]

In addition, we know from Lemma 4.1 (and the $L^{\infty}$ bound on $n_{0}$) that $\partial_{t}n_{0}\geq-\frac{C}{t}$, so $\partial_{t}(n_{0}+C\ln(t))\geq 0$. Then, since $t_{1}<t_{2}$,

[TABLE]

Then, using equation (2.8) and an integration by parts, we obtain

[TABLE]

Then we can choose $(t_{1},t_{2})$ close enough such that

[TABLE]

We conclude that $n_{0}\in C((0,T),L^{1}(\mathbb{R}^{d}))$.

Initial trace

For any test function $0\leq\zeta(x)\leq 1$, we have from (3.12),

[TABLE]

Letting $\epsilon$ going to [math], we obtain with (2.6),

[TABLE]

Letting $t\rightarrow 0$ we can conclude that $n_{0}(0)=n_{0}^{ini}$.

6 Complementary relation

In this section we prove the complementary relation

[TABLE]

In the weak sense, this identity reads, for any test function $\phi$,

[TABLE]

The proof is divided into two steps.

1st step. In this first step we prove the inequality $\geq 0$ in (6.25). We start with the pressure equation (1.3) that we multiply by $\epsilon$

[TABLE]

We multiply by a test function $\phi\in\mathcal{D}((0,T)\times\mathbb{R}^{d})$ and integrate,

[TABLE]

where we use an integration by parts for the last identity. From the estimates in Lemma 3.2, we have

[TABLE]

We deduce that for any test function $\phi\in\mathcal{D}((0,T)\times\mathbb{R}^{d})$,

[TABLE]

Since we have strong convergence of $(p_{\epsilon})_{\epsilon}$ and weak convergence of $(\nabla p_{\epsilon})_{\epsilon}$, we can pass into the limit in the last two term in (6.26),

[TABLE]

Now we are looking for the limit of the first term in (6.26). We have $p_{\epsilon}|\nabla p_{\epsilon}|^{2}=\frac{4}{9}|\nabla p_{\epsilon}^{3/2}|^{2}$. By weak convergence of $\nabla p_{\epsilon}^{3/2}=p_{\epsilon}^{1/2}\nabla p_{\epsilon}$ and with Jensen inequality (since $x\mapsto x^{2}$ is convex),

[TABLE]

Thus, we conclude from (6.26) that

[TABLE]

which is a first inequality for (6.25).

2nd step. Now we want to show the reverse inequality, i.e.

[TABLE]

We know that

[TABLE]

with $q_{\epsilon}=p_{\epsilon}-\epsilon\ln(p_{\epsilon}+\epsilon).$ Thanks to the inequality $\epsilon\ln(\epsilon)\leq\epsilon\ln(p_{\epsilon}+\epsilon)\leq\epsilon\ln(p_{M}+\epsilon)$, and the strong convergence $p_{\epsilon}\to p_{0}$, we know that $q_{\epsilon}\rightarrow p_{0}$ as $\epsilon\to 0$. Because

[TABLE]

we deduce from Lemma 3.2 that $\Delta q_{\epsilon}\in L^{\infty}([0,T];L^{1}(\mathbb{R}^{d}))$. It gives us compactness in space but not in time. Thus, following the idea of [22], we use a regularization process ’à la Steklov’.

Let introduce a time regularizing kernel $\omega_{\eta}\geq 0$ such that $\mbox{supp}(\omega_{\eta})\subset\mathbb{R}_{-}$. Then with the notations $n_{\epsilon,\eta}=\omega_{\eta}*_{t}n_{\epsilon}$, $q_{\epsilon,\eta}=\omega_{\eta}*_{t}q_{\epsilon}$, where the convolution holds only in the time variable,

[TABLE]

We denote $U_{\epsilon}=\Delta q_{\epsilon,\eta}$, then

[TABLE]

Since $n_{\epsilon}$ and $n_{\epsilon}G(p_{\epsilon})$ are uniformly bounded in $W^{1,1}(Q_{T})$ from Lemma 3.2, $(U_{\epsilon})_{\epsilon}$ is bounded in $W^{1,1}(Q_{T})$ and we can extract a converging subsequence, still denoted $(U_{\epsilon})_{\epsilon}$, converging towards $U_{0}$ in $L^{1}_{loc}(\mathbb{R}^{d})$ for $\eta$ fixed. Moreover

[TABLE]

We multiply (6.27) by $P^{\prime}(n_{\epsilon})=\frac{\epsilon}{(1-n_{\epsilon})^{2}}=\frac{1}{\epsilon}(p_{\epsilon}+\epsilon)^{2}$,

[TABLE]

Then, passing to the limit $\epsilon\to 0$, we obtain, thanks to the above remark

[TABLE]

So we are left to prove that for any $\eta>0$, we have

[TABLE]

We compute for a fixed $\eta>0$,

[TABLE]

where $C$ is a constant such that $\partial_{t}n_{\epsilon}(s,x)+\frac{C}{t}\geq 0$.

For the first term we have

[TABLE]

For the second term, we have

[TABLE]

and $\partial_{t}(p_{\epsilon}+\epsilon)^{2}=2(p_{\epsilon}+\epsilon)\partial_{t}p_{\epsilon}\geq-\frac{C^{\prime}}{t}$. Let $0\leq\xi\in\mathcal{C}^{\infty}_{c}(Q)$ and $\tau>0$ the smallest time in its support, we then have for $t\geq\tau$

[TABLE]

So integrating on $(t,s)\subset(\tau,+\infty)$

[TABLE]

Then

[TABLE]

where we use the bound on $\partial_{t}n$ in Lemma 4.2.

For the third term, since $s\geq t>0$, for any test function $\xi$ as above,

[TABLE]

So for all test function $\xi$ as above, and all $\eta>0$,

[TABLE]

Now it remain to pass to the limit $\eta\rightarrow 0$ in the regularization process. Thanks to an integration by parts,

[TABLE]

From the $L^{2}$ estimate on $\nabla p_{0}$ (Lemma 3.4) and the $L^{1}\cap L^{\infty}$ estimate on $p_{0}$ (Lemma 3.2), we deduce that we can pass to the limit $\eta\to 0$ and get

[TABLE]

Finally, from (2.10), we have $p_{0}n_{0}=p_{0}$. It concludes the proof.