On Global-in-$x$ Stability of Blasius Profiles

TL;DR

This paper proves that solutions to the 2D stationary Prandtl system with localized data near Blasius profiles converge globally in $x$ to these profiles, with explicit decay rates in strong norms, using novel energy and inequality techniques.

Contribution

It introduces a new division estimate and a quantity $ ext{ extOmega}$ to establish sharp convergence rates of Prandtl solutions to Blasius profiles in stronger norms.

Findings

Convergence of solutions in $W^{k,p}$ norms with explicit decay rates.

Introduction of a new nonnegative quantity $ ext{ extOmega}$ for Blasius solutions.

Development of a weighted Nash-type inequality for the analysis.

Abstract

We characterize the well known self-similar Blasius profiles, $[\bar{u}, \bar{v}]$, as downstream attractors to solutions $[u,v]$ to the 2D, stationary Prandtl system. It was established in \cite{Serrin} that $\| u - \bar{u}\|_{L^\infty_y} \rightarrow 0$ as $x \rightarrow \infty$. Our result furthers \cite{Serrin} in the case of localized data near Blasius by establishing convergence in stronger norms and by characterizing the decay rates. Central to our analysis is a "division estimate", in turn based on the introduction of a new quantity, $\Omega$, which is globally nonnegative precisely for Blasius solutions. Coupled with an energy cascade and a new weighted Nash-type inequality, these ingredients yield convergence of $u - \bar{u}$ and $v - \bar{v}$ at the essentially the sharpest expected rates in $W^{k,p}$ norms.

Equations

$$u{u}_x+v{u}_y-u_{yy}=0,u_x+v_y=0,(x,y)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$$

$$u{|}_{x=0}=u_0(y),[u,v]{|}_{y=0}=0,u{|}_{y\uparrow \infty }=u_E(x).$$

$$u_0(y)>0\text{ for }y>0,$$

$$u_0^{\prime }(0)>0,$$

$$u_0\in C^{\infty }(y\ge 0),$$

$$u_0^{\prime \prime }(y)\sim y^2\text{ near }y=0,$$

$$|u_0(y)-1|\text{ and }{\partial }_y^k{u}_0(y)\text{ decay exponentially for }k\ge 1.$$

$$\underset{x}{\sup }\underset{y\in (0,y_0)}{\sup }|u,v,u_y,u_{yy},u_x|\lesssim 1,$$

$$u_y(x,0)>0\text{ and }u>0.$$

$$\displaystyle[\bar{u},\bar{v}]=\Big{[}f^{\prime}(\eta),\frac{1}{\sqrt{x+x_{0}}}\{\eta f^{\prime}(\eta)-f(\eta)\}\Big{]},\text{ where }\eta=\frac{y}{\sqrt{x+x_{0}}},$$

$$f{f}^{\prime \prime }+f^{\prime \prime \prime }=0,f^{\prime }(0)=0,f^{\prime }(\infty )=1,\frac{f(\eta )}{\eta }\stackrel{n\to \infty }{\to }1.$$

$$0\le f^{\prime }\le 1,f^{\prime \prime }(\eta )\ge 0,f^{\prime \prime }(0)>0,f^{\prime \prime \prime }(\eta )<0.$$

$$\Vert u-\bar{u}{\Vert }_{L_y^{\infty }}\to 0\text{ as }x\to \infty .$$

$$\begin{array}{rl}& \Vert {\partial }_x^{\alpha }{\partial }_y^{\beta }\{u-\bar{u}\}⟨\eta {⟩}^{\gamma }{\Vert }_{L_y^p}\lesssim C_{\kappa }⟨x{⟩}^{-\frac{1}{2}+\frac{1}{2p}-\alpha -\frac{\beta }{2}+\kappa },\\ & \Vert {\partial }_x^{\alpha }{\partial }_y^{\beta }\{v-\bar{v}\}⟨\eta {⟩}^{\gamma }{\Vert }_{L_y^p}\lesssim C_{\kappa }⟨x{⟩}^{-1+\frac{1}{2p}-\alpha -\frac{\beta }{2}+\kappa }\end{array}$$

$$\psi ={\int }_0^{y}u(x,y^{\prime })\mathrm{d}{y}^{\prime }.$$

$${\partial }_x(u^2)-u{\partial }_{\psi \psi }(u^2)=0.$$

$$\varphi (x,\psi ):=u^2(x,\psi )-{\bar{u}}^2(x,\psi ),\rho (x,\psi ):=u(x,\psi )-\bar{u}(x,\psi ).$$

$${\varphi }_x-u{\varphi }_{\psi \psi }+A\varphi =0,A=-2\frac{{\bar{u}}_{yy}}{\bar{u}(\bar{u}+u)}$$

$$\xi :=\frac{\psi }{\sqrt{x+1}}.$$

$$\psi =\psi (x,y)={\int }_0^y\bar{u}(x,y^{\prime })\mathrm{d}{y}^{\prime }={\int }_0^y{\eta }^{\prime }\mathrm{d}{y}^{\prime }=\frac{1}{\sqrt{x}}{\int }_0^y{y}^{\prime }\mathrm{d}{y}^{\prime }\sim y\eta ={\eta }^2\sqrt{x},$$

$$\sqrt{\xi }=\eta \text{ for }\eta \le 1.$$

$$\Vert \varphi {\Vert }_X:=\sum _{k=0}^{K_0}\Vert \varphi {\Vert }_{X_k},$$

$$\Vert \varphi {\Vert }_{X_k}:=\Vert {\varphi }^{(k)}⟨x{⟩}^{k-{\sigma }_k}{\Vert }_{L_x^{\infty }{L}_{\psi }^2}+\Vert {\varphi }^{(k)}\frac{⟨\sqrt{\psi }⟩}{\sqrt{u}}⟨x{⟩}^{k-{\sigma }_k}{\Vert }_{L_x^{\infty }{L}_{\psi }^2}$$

$$+\Vert \sqrt{u}{\varphi }_{\psi }^{(k)}⟨x{⟩}^{k-{\sigma }_k}{\Vert }_{L_x^2{L}_{\psi }^2}+\Vert {\varphi }_{\psi }^{(k)}\sqrt{\psi }⟨x{⟩}^{k-{\sigma }_k}{\Vert }_{L_x^2{L}_{\psi }^2}.$$

$${\varphi }_x-(\bar{u}+\epsilon \rho ){\varphi }_{\psi \psi }+A\varphi =0,A=-2\frac{{\bar{u}}_{yy}}{\bar{u}(2\bar{u}+\epsilon \rho )}.$$

$${\varphi }_x-\bar{u}{\varphi }_{\psi \psi }+\bar{A}\varphi ,\bar{A}=-\frac{{\bar{u}}_{yy}}{{\bar{u}}^2}.$$

$$\int {\varphi }^2\times \text{positive quantities}\times \Omega ,\text{ where }\Omega =-{\bar{u}}_{yy}+\frac{1}{2}\bar{u}{\bar{u}}_x.$$

$${\varphi }_x-\bar{u}{\varphi }_{\psi \psi }+\bar{A}\varphi -\frac{{\partial }_x\bar{u}}{\bar{u}}\varphi .$$

$$\int {\varphi }^{\text{high}}{\varphi }_{\psi \psi }^{\text{low}}{\varphi }^{\text{high}}\chi (\xi \lesssim 1).$$

$$\frac{{\partial }_x}{2}\int {\varphi }^2+\int \bar{u}|{\varphi }_{\psi }{|}^2\lesssim \epsilon ⟨x{⟩}^{-(1+)}{E}_{K_0}(x).$$

$$\frac{{\partial }_x}{2}\int {\varphi }^2+\int \bar{u}|{\varphi }_{\psi }{|}^2-\frac{1}{2}\int {\bar{u}}_{\psi \psi }|\varphi {|}^2+\int \epsilon \rho {\varphi }_{\psi \psi }\varphi +\int A|\varphi {|}^2=0.$$

$$|\int \epsilon \rho {\varphi }_{\psi \psi }\varphi \Vert \lesssim$$

$$\lesssim$$

$$\frac{{\partial }_x}{2}\int {\varphi }^2\frac{1}{u}⟨\psi ⟩+\int {\varphi }_{\psi }^2⟨\psi ⟩\lesssim \epsilon ⟨x{⟩}^{-(1+)}{E}_{K_0}(x).$$

Full text

1 Introduction

The 2D, stationary, homogeneous Prandtl equations are given by:

[TABLE]

The system is typically supplemented with initial data at $\{x=0\}$ and boundary data at $\{y=0\}$, and $y\uparrow\infty$:

[TABLE]

For simplicity, we will take $u_{E}(x)=1$, but any constant will also work. The $x$ direction is considered a time-like direction, while the $y$-direction is considered a space-like direction, and the equation (1) is considered as an evolution in the $x$ variable. Correspondingly, $u_{0}(y)$ is called the “initial data” and as a general matter of terminology, in this paper the words “global” and “local” refer to the $x$-direction.

The following is a classical result due to Oleinik (see [OS99], P. 21, Theorem 2.1.1):

Theorem 1** ([OS99])**

Assume:

[TABLE]

Then there exists a global solution, $[u,v]$ to (1) satisfying, for some $y_{0},m>0$,

[TABLE]

Given the global existence of a solution to (1), the next point is to describe more precisely the asymptotics of the evolution as $x\rightarrow\infty$. In order to do, let us introduce the self-similar Blasius solutions:

[TABLE]

where $f$ satisfies

[TABLE]

Here, $x_{0}>0$ is a free parameter. The following hold:

[TABLE]

We now recall the following result of Serrin’s:

Theorem 2** ([Ser66])**

Let $u$ be a solution to (1), (2) such that $\partial_{y}u_{0}(y)$ is continuous. Then the following asymptotics hold

[TABLE]

First, let us mention that the results in [Ser66] are more general than the theorem stated above in the sense that $u_{E}(x)$ in (2) is allowed to be have polynomial growth in $x$, whereas in the present paper we are only concerned with constant $u_{E}$ (which corresponds to shear flow).

The purpose of the present work is to further Theorem 2 under the assumption of small, localized perturbations of the Blasius profile.

Theorem 3

Fix any $0<\varepsilon<<1$. Let $u_{0}(y)$ satisfy Oleinik’s conditions as stated in Theorem 1. Suppose $|\partial_{y}^{l}\{u_{0}-1\}\langle y\rangle^{M}|\leq\varepsilon$ for some large $M$. Fix $\gamma<M$, any $\kappa>0$ small but arbitrary. Let $[u,v]$ be the unique solution to (1), (2) with initial data $u_{0}(y)$. Then the following asymptotics are valid:

[TABLE]

The asymptotic information obtained in estimate (LABEL:est.main) is much more precise than (7). In fact, the estimates (LABEL:est.main) are essentially the optimal expected rates in the following sense:

Remark 4

Given localized initial data, $h_{0}(y)$, one expects solutions, $h(x,y)$ to the heat equation, $(\partial_{x}-\partial_{yy})h=0$ to decay at rates $\|\partial_{x}^{\alpha}\partial_{y}^{\beta}h\|_{L^{p}_{y}}\lesssim\langle x\rangle^{-\frac{1}{2}+\frac{1}{2p}-\alpha-\frac{\beta}{2}}$.

Note that we require the small $\kappa>0$ in (LABEL:est.main) to avoid logarithmic singularities at $x=\infty$.

One of the motivations for establishing quantitive estimates of the type (LABEL:est.main) is due to recent advances in the validity theory for steady Navier-Stokes flows, for instance the works of [GI18], [GI18b]. In particular, using the estimates (LABEL:est.main) we can generalize the class of data treated by [GI18]:

Corollary 5

Consider initial data, $u_{0}(y)$, that is a small perturbation of Blasius in the sense of Theorem 3. Then for $x_{0}>>1$, we may take $[u(x_{0},\cdot),v(x_{0},\cdot)]$ as the $\{x=0\}$ data in Theorem 1 of [GI18].

Proof. This follows immediately upon applying the estimates (LABEL:est.main) above in the proof of Lemma 9 of [GI18].

A second motivation for this work is that in order to prove the global validity of steady Prandtl expansions, a work currently underway by the author, one needs a precise understanding of the decay mechanism in the Prandtl equations, which is established in the present work.

Let us now point the reader towards the related work of [DM18], which studies the formation of singularities (in this context called “separation”) for the inhomogeneous version of (1) (with adverse pressure gradient).

1.1 Notation and Main Objects

We now introduce the main notations that will be in use for this paper. First, we introduce the stream function,

[TABLE]

A classical idea ([OS99]) is to write the Prandtl system, (1) in the variables $(x,\psi)$:

[TABLE]

Define the difference unknowns:

[TABLE]

It is shown in [Ser66], equation (11), that $\phi$ satisfies the equation

[TABLE]

Recall the self-similar variable $\eta$ as defined in (5). For simplicity, we will set the parameter $x_{0}=1$. We define a new self-similar variable, which reflects the diffusive scaling in (10) via

[TABLE]

By using that $u\sim\eta$ for $\eta\lesssim 1$ (as established in Theorem 1) and integrating equation (9) via

[TABLE]

we obtain the relation

[TABLE]

The basic object of study throughout the paper will be $\phi$, which satisfies the equation (10), in the variables $(x,\psi)$ and correspondingly the self-similar variable $\xi$.

Let us now give a brief review of the properties of $\bar{u}$ and $u$. First, Oleinik’s global existence result, Theorem 1, gives that $u\sim\eta$ near $\eta\leq 1$. Regarding $\bar{u}$, the main properties are summarized in (6). Of particular note is the concavity of $\bar{u}$, guaranteed by $f^{\prime\prime\prime}<0$. In particular this implies that $A\geq 0$ in (10).

We will now introduce the norms in which we measure the solution $\phi$. First, we simplify notation throughout the paper by putting $\phi^{(k)}:=\partial_{x}^{k}\phi$.

[TABLE]

Above, we let $\sigma_{k}$ be a sequence such that $\sigma_{0}=0$ and $\sigma_{k+1}>\sigma_{k}$. $K_{0}$ will be a fixed, large number. We will denote by $E_{k}(x)$ an arbitrary quantity satisfying $\sup_{x}|E_{k}(x)|\lesssim\sum_{j=0}^{k}\|\phi\|_{X_{j}}$. Similarly, we will denote by $I_{k}(x)$ an arbitrary quantity satisfying $\int_{0}^{\infty}|I_{k}(x)|\,\mathrm{d}x\lesssim\sum_{j=0}^{k}\|\phi\|_{X_{j}}$.

By rescaling, we may arrange that the quadratic terms in (10) have a factor of $\varepsilon$ in front of them:

[TABLE]

1.2 Main Ideas

The main mechanisms can be summarized in four steps listed below. Overall, at each order of $x$ regularity up to $\partial_{x}^{K_{0}}$ for a fixed $K_{0}$ large, there are two estimates that are performed. We call these the “Energy estimate” and the “Division estimate”. This results in the control of the norm $\|\phi\|_{X}$ as shown above.

Step 1: $L^{2}$ level

At the $L^{2}$ level, we may center our discussion around the linearized operator from (13), which reads

[TABLE]

The standard energy estimate gives a bound on $\sup_{x}\|\phi\|_{L^{2}_{\psi}}^{2}+\|\sqrt{\bar{u}}\phi_{\psi}\|_{L^{2}_{x}L^{2}_{\psi}}^{2}$. The $\bar{A}$ term does not play a role at this point, as $\bar{A}\geq 0$ and may thus be ignored.

The second estimate at the $L^{2}$ level is the “Division estimate”, which can be found in Lemma 7. There are two distinguished features of the quantities that are controlled (see the estimate (18)). First, there is a far-field weight of $\langle\psi\rangle$. Second, there is a nonlinear weight $\frac{1}{u}$ which gives additional control near the boundary $\{\psi=0\}$.

The reason we can close this Division estimate is due to the precise structure of Blasius solutions. Indeed, the choice of weight $\frac{\langle\psi\rangle}{u}$ is specially designed so that the interaction with the linearized equation, (14), produces the quantity

[TABLE]

This type of quantity would be out of reach of the norm $X$. However, because $\bar{u}$ is a Blasius solution (not just a generic Prandtl solution), we are able to show that $\Omega(x,y)$ is globally positive. This is the content of Lemma 18.

The reason we need the division estimate is two-fold, corresponding to the two weights. The weight $\langle\psi\rangle$ comes in for Step 4, whereas the boundary weight $\frac{1}{u}$ comes in for Step 3.

Step 2: $H^{k}$ for $1\leq k\leq K_{1}$

We now fix $K_{1}$ so that $1<<K_{1}<<K_{0}$. The tier of derivatives between $1$ and $K_{1}$ we call the “middle tier”. The middle tier is distinguished from the top tier because we are able to expend derivatives. The middle tier is distinguished from the bottom ($L^{2}$) tier because the linearized equation is no longer (14), but rather

[TABLE]

We arrive here by substituting the equation (13) upon differentiating it in $x$. The reason the linearized equation has changed is due to the quasilinearity present in (14). At this stage we repeat the process of Step 1, taking advantage of the further property of Blasius solutions that $\partial_{x}\bar{u}<0$.

Step 3: $H^{k}$ for $K_{1}+1\leq k\leq K_{0}$

We now arrive at the top tier of derivative in the norm $X$. The top tier is distinguished because we do not have derivatives to expend. First of all, we select $K_{0}$ large enough so that the “tame principle” kicks in. For instance, terms like $\partial_{x}^{j}f\times\partial_{x}^{K_{0}-j}g\times\partial_{x}^{K_{0}}h$ are bound to have either $j$ or $K_{0}-j$ to be much smaller than $K_{0}-3$. This is standard in quasilinear problems.

The important part, however, is that the crucial weight of $\frac{1}{u}$ available due to the division estimate, is used to “save derivatives”. This is most easily seen in a term such as (“high” and “low” refer to order of $x$ derivative):

[TABLE]

For a term such as this, we are forced to put $\phi^{\text{low}}_{\psi\psi}$ in an $L^{\infty}$ type norm in order to conserve the high derivatives. To do this, with the weights of $\bar{u}$ distributed as optimally as we are allowed with the $X$ norm, we must invoke the additional $\frac{1}{\sqrt{u}}$ weight available due to the division estimate. This is quantified by proving a localized, optimal weight, uniform estimate on $\phi^{\text{low}}_{\psi\psi}$ (see for instance, (46)).

Step 4: Optimal Decay

Using Steps 1 - 3 we are able to show global existence of $\phi$ in the space $X$. The space $X$ certainly encodes decay information regarding the solution $\phi$ - this is evident by consulting (11). However, if one is comparing to the expected optimal asymptotics of parabolic equations in dimension one, in the sense of Remark 4, then one notices that the space $X$ is a factor of $\langle x\rangle^{\frac{1}{4}}$ weaker than optimal.

The reader should now recall the classical Nash inequality, [N58], which states that $\|\phi\|_{L^{2}_{\psi}}^{2}\lesssim\|\phi_{\psi}\|_{L^{2}_{\psi}}^{\frac{2}{3}}\|\phi\|_{L^{1}_{\psi}}^{\frac{4}{3}}$. Typically, one uses this by saying $\|\phi\|_{L^{1}_{\psi}}$ is conserved (say) and thus one inserts the Nash inequality to the basic energy bound to obtain an ODE of the form $\dot{\eta}+\eta^{3}=0$, for $\eta=\|\phi\|_{L^{2}_{\psi}}^{2}$, which immediately results in $\langle x\rangle^{-\frac{1}{4}}$ decay of $\|\phi\|_{L^{2}_{\psi}}$.

In our case, two difficulties are present in order to carry out this procedure to optimize the decay. First, we only have the degenerate weighted quantity $\|\sqrt{\bar{u}}\phi_{\psi}\|_{L^{2}_{\psi}}$ appearing in the energy. Second, we cannot control $\|\phi\|_{L^{1}}$ by integrating the equation.

To contend with these difficulties, we establish a new Nash-type inequality in Lemma 19 which (1) accounts for the degenerate weight of $\sqrt{\bar{u}}$ and (2) replaces the $L^{1}$ norm by $L^{2}(\sqrt{\langle\psi\rangle})$ (which scales the same way). The type of inequality we are able to establish is piecewise (as is seen from Lemma 19). Remarkably, both upper bounds in estimate (47) yield the same, optimal, decay rate of $\langle x\rangle^{-\frac{1}{4}}$.

2 Baseline Tier: $L^{2}$ Estimates

In this section, we obtain two estimates at the $L^{2}$ level - the energy estimate and the division estimate. The reader is urged to keep in mind the linearized structure which is present at the $L^{2}$ level, equation (14).

Lemma 6** (Energy Estimate)**

Let $\phi$ solve (13). Then for $K_{0}>>1$,

[TABLE]

Proof. We take inner product against $\phi$ to obtain

[TABLE]

We now use that $u\geq 0$ and $\bar{u}_{\psi\psi}\leq 0$. For $\bar{u}_{\psi\psi}$, we use the relation $\bar{u}_{yy}=\bar{u}^{2}\bar{u}_{\psi\psi}+\bar{u}|\bar{u}_{\psi}|^{2}$ to conclude that $\bar{u}_{\psi\psi}\leq 0$ if $\bar{u}_{yy}\leq 0$, which holds by properties of the Blasius profile. We also use that $A\geq 0$, which again holds by the concavity of the Blasius profile. Finally, we estimate

[TABLE]

where we have used inequalities (41.11) and (44.4). This concludes the proof.

Lemma 7** (Division Estimate)**

Let $\phi$ solve (13). For $K_{0}$ sufficiently large,

[TABLE]

Proof. We have the following identity

[TABLE]

We group the latter two terms together via

[TABLE]

where

[TABLE]

By Lemma 18, the $\Omega$ contribution is positively signed. We thus need to estimate the nonlinear part in $\Omega^{R}$:

[TABLE]

We now proceed to prove the final inequality above after equation (20) by estimating all five of the $L^{\infty}_{\psi}$ terms above. First, upon invoking (44.1),

[TABLE]

Above, we have also estimated $A$ via

[TABLE]

Second, upon invoking (44.4) and that $u\sim\eta$, $\bar{u}_{yy}\sim\eta^{2}\langle x\rangle^{-1}$ near $\eta=0$, we estimate

[TABLE]

Third, again upon invoking (44.4),

[TABLE]

Fourth, again upon invoking (44.4),

[TABLE]

Fifth, again upon invoking (44.4),

[TABLE]

Inserting these estimates into (20) yields the estimate shown beneath (20). This concludes the proof.

3 Middle Tier: $H^{k}$ for $1\leq k\leq K_{1}$

At the $H^{1}$ level, the linearized equation changes and so requires a new treatment. Taking one $x$ derivative of (13), we obtain

[TABLE]

The point here is that $u^{(1)}$ can be separated into $u^{(1)}=:\bar{u}^{(1)}+\rho^{(1)}$. While the $\rho^{(1)}$ contribution is quadratic, the $\bar{u}^{(1)}$ contribution is linear and highest order in $\phi$. To see this, we use the equation to rewrite $\phi_{\psi\psi}$ via:

[TABLE]

Lemma 8** (Energy Estimate)**

Let $\phi$ solve the equation (13). Let $0<\delta<<1$ be arbitrary. Then the following inequality is valid:

[TABLE]

Proof. Step 1: Induction on Weights We use induction on the weights, which amounts to applying three successive weighted multipliers:

[TABLE]

where $\omega_{1}^{1}<\omega_{1}^{2}=\sigma_{1}$. Call the right-hand side of (23) $R(x)$. One obtains the following two inequalities:

[TABLE]

Integrating from $x=\infty$ one obtains

[TABLE]

The establishment of (24) and (25) is in an identical fashion to the top order weight, so we omit it and just focus on the top order weight. We assume, thus, inductively that (26) is true.

Step 2: Top order weight: We apply the multiplier $\phi^{(1)}\langle x\rangle^{2-2\sigma_{1}}$ to (22). The first three terms have been treated already, with the modification that the $\partial_{x}$ term contributes a factor of:

[TABLE]

The main new leading order contribution is the fourth, which is

[TABLE]

The key point is that the first term above is nonnegative because $\bar{u}_{x}<0$ for Blasius solutions.

[TABLE]

We estimate the $\rho$ contribution from (27), which enables us to use the smallness of $\varepsilon$:

[TABLE]

where we have invoked estimate (44.4).

We now need to estimate (22.5):

[TABLE]

For the first term in (28), we do not have any smallness, so we take advantage of the fact that one of the terms, $\phi$, is lower order:

[TABLE]

Above, we have used the estimate $\bar{u}\gtrsim\eta$ and $u\gtrsim\eta$ on $\eta\lesssim 1$ and (21). We have also used that $|\frac{\bar{u}_{x}}{\bar{u}}|\lesssim\langle x\rangle^{-1}$.

For the second term from (28), we use the smallness of $\varepsilon$, as this term is cubic. We treat two different cases based on the location of $\xi$.

[TABLE]

First,

[TABLE]

Above, we have performed the Hardy-type inequality,

[TABLE]

The sixth term, (22.6), is almost exactly analogous to the fifth term which we just treated. The only exception is the control of the nonlinear part of $A_{x}$, which we expand here:

[TABLE]

We thus estimate the nonlinear term:

[TABLE]

This concludes the proof.

Lemma 9** (Division Estimate)**

Let $\phi$ be a solution to (13). Then the following estimate is valid:

[TABLE]

Proof. We take inner product of (22) with $\phi^{(1)}\frac{\langle\psi\rangle}{u}\langle x\rangle^{2-2\sigma_{1}}$. The first three terms from (22) work in the same manner as Lemma 7, while the fourth is a further positive contribution.

We thus estimate (22.5).

[TABLE]

We have used estimate (21) for $\|A\|_{L^{\infty}_{\psi}}$ and (44.5) for $\|\frac{u^{(j)}}{u}\|_{L^{\infty}_{\psi}}$.

The sixth term, (22.6), works almost exactly analogously to the previous term. This concludes the proof.

A nearly identical sequence of estimates is performed for the $2$ through $K_{1}$ order of $\partial_{x}$:

Lemma 10** (Energy Estimate)**

Let $2\leq k\leq K_{1}<<K_{0}$, and let $\phi$ solve the equation (13). Let $0<\delta<<1$ be arbitrary. Then the following inequality is valid:

[TABLE]

Lemma 11** (Division Estimate)**

Let $2\leq k\leq K_{1}<<K_{0}$, and let $\phi$ solve the equation (13). Let $0<\delta<<1$ be arbitrary. Then the following inequality is valid:

[TABLE]

The proofs of these lemmas are essentially identical to the proofs of Lemmas 8 and 9, so we omit them.

4 Highest Tier: $H^{k}$ for $K_{1}<k\leq K_{0}$

We take $\partial_{x}^{k}$ to the equation (13) to obtain

[TABLE]

We will simplify notations by setting $\phi^{(k)}:=\partial_{x}^{k}\phi$, in which case the above equation reads

[TABLE]

Lemma 12** (Energy Estimate)**

Let $\phi$ be a solution to (13). Then the following estimate is valid:

[TABLE]

Proof. We apply the weighted multiplier $\phi^{(k)}\langle x\rangle^{2(l-\omega_{l})}$ for $l=0,...,k$ to equation (32). Again, we write only the $l=k$ case, with the $l<k$ cases being carried out by the induction on weights argument as in Lemma 8.

The first three terms from (32) are estimated nearly identically to the lower order energy estimates. The only difference is the following perturbative term for which we integrate by parts:

[TABLE]

The first term above is majorized by

[TABLE]

We have appealed to estimate (45.1), and the definition of the $X$ norm in (11).

The second term above is easily majorized by

[TABLE]

Above, we have appealed to estimate (44.4).

For the next terms from (32), we begin by considering the case when $j=\text{min}\{j,k-j\}$. Consider the following term, which we integrate by parts in $\psi$:

[TABLE]

Above, $\chi(\xi)$ is a normalized cut-off function, equal to $1$ on $\xi\leq 1$ and equal to [math] on $\xi\geq 2$. We use the notation $\chi(\xi)^{C}:=1-\chi(\xi)$.

We now estimate the terms appearing above. First,

[TABLE]

Above, we have used $uu^{(j)}_{\psi}\sim u^{(j)}_{y}$ according to the chain rule, and subsequently (45.2). Note that $j\leq K_{0}-2$ as required by (45.2) for $k$ sufficiently large because $j=\min\{j,k-j\}$. We have also used the following Hardy type inequality:

[TABLE]

Note that we have also used that $E_{j}(x)I_{k-j}(x)=I_{k-j}(x)$ by definition, and also that $k-j\leq k-1$ because $j\geq 1$ and $j=\min\{j,k-j\}\leq k-1$.

Second, in the region where $\chi(\xi)^{C}$ is supported, $\xi\gtrsim 1$ and so $u\gtrsim 1$. We are thus free to add in factor of $u$ which we do via:

[TABLE]

We have invoked the same estimates as in (35.2), again admissible as $j\leq K_{0}-2$.

Third,

[TABLE]

Above, we have invoked estimate (44.5).

Fourth,

[TABLE]

This concludes the treatment of this term for $1\leq j=\min\{j,k-j\}$. We now treat this term for $j=\max\{j,k-j\}$. Instead of integrating by parts, we may treat the following:

[TABLE]

The far-field term is estimated:

[TABLE]

Above, we use (41.6), which is admissible since $l$ can be the top order, $K_{0}$ in (41) (to deal with $j=k$ here). We also use (46.1), admissible because $k-j<K_{0}-2$.

We estimate the localized contribution via

[TABLE]

Above, we have used that $u\sim\eta\sim\sqrt{\xi}$ in the region where $\xi\lesssim 1$. We have used (41.6), again admissible since $l$ can be equal to the top order in (41.6) (to match $j=k$ case here). We have also used the enhanced $L^{\infty}$ decay, (46.2), in turn admissible because $k-j<K_{0}-3$.

We now move to the final term:

[TABLE]

We must again split into the cases when $j=\min\{j,k-j\}$ and when $j=\max\{j,k-j\}$. This is largely analogous to the previous term, and so we treat the most difficult case which is when $j=\max\{j,k-j\}$ and $\xi\lesssim 1$. First, we identify the most singular term in $\partial_{x}^{j}A$ as

[TABLE]

This concludes the proof.

Lemma 13** (Division Estimate)**

Let $\phi$ be a solution to (13). Then the following estimate is valid:

[TABLE]

Proof. We apply the inductively weighted “division-multiplier” $\phi^{(k)}\frac{1}{u}\langle\psi\rangle\langle x\rangle^{2(l-\omega^{k}_{l})}$. The proof follows in essentially the same way as the baseline Division Estimate, and we thus treat the new commutator terms. First, consider

[TABLE]

We estimate the $\chi(\xi)$ term, the other being straightforward. First, we will treat the $j=1$ case. In this case, one takes $\partial_{x}^{k-1}$ of equation (10) to generate the equality

[TABLE]

Of these terms, the highest order is $\phi^{(k)}$, which, when inserted into (38) produces the positive term (we do not need to consider $\chi(\xi)$ here)

[TABLE]

The lower order terms, $L^{(k-1)}$ are inserted into (38) to produce

[TABLE]

Next, assume $2\leq j=\min\{j,k-j\}$. Then,

[TABLE]

Above, we have used that $j\geq 2$ so that $k-j\leq K_{0}-2$. We have subsequently applied (41.11).

Second, assume $j=\max\{j,k-j\}$. In this case, we estimate the localized contribution via

[TABLE]

Above, we have used (41.7) which crucially allows us to not lose any derivatives ($l$ can be taken equal to $K_{0}$ in (41.7)), and the decay $L^{\infty}$ estimates, (46.1) and (46.2).

We next treat

[TABLE]

For $1\leq j=\min\{j,k-j\}$, we estimate

[TABLE]

Next, we consider the case when $j=\max\{j,k-j\}$. For this case, we must expand $\partial_{x}^{j}A$ as in (29). For simplicity, we treat the term containing the highest order derivative on the unknown, $\rho$, which reads

[TABLE]

Inserting this into (39), we obtain

[TABLE]

The lower order terms from (40) are treated in the same manner. This concludes the proof.

Our scheme of a-priori estimates, (16), (18), (23), (30), (33), (37) , immediately yield the following:

Proposition 14** (Global Existence in $X$)**

Given initial data $u(1,\cdot)$ such that standard compatibility conditions are satisfied, and such that $\phi:=u^{2}-\bar{u}^{2}$ is rapidly decaying at $\infty$. Then the unique global solution $u$ guaranteed by Theorem 1 satisfies $\phi\in X$, where $\phi=u^{2}-\bar{u}^{2}$.

5 Embeddings

The reader should recall the specification of the $X$ norm given in (11).

Lemma 15** ($L^{2}$ Estimates)**

For $0\leq j\leq K_{0}-1$, $0\leq l\leq K_{0}$, $1\leq m\leq K_{0}$, $1\leq n\leq K_{0}-1$,

[TABLE]

Proof. The first two terms, (41.1) and (41.2) are part of the $X$ norm. We next move to (41.8). By using the Agmon inequality in the $x$ direction and subsequently Hardy inequality, we obtain

[TABLE]

since $j\leq K_{0}-1$. The same exact proof works for (41.9).

We may now estimate the third term, (41.3). Since $u\gtrsim 1$ on $\xi\gtrsim 1$, (41.3) on $\xi\gtrsim 1$ follows from (41.1). We can thus restrict to $\xi\lesssim 1$, in which case we use that $u^{2}\gtrsim\eta^{2}\gtrsim\xi=\frac{\psi}{\sqrt{x}}$ on the region where $\xi\lesssim 1$:

[TABLE]

Above, we have used the Hardy inequality in the $\psi$ direction, admissible because $\phi^{(j)}|_{\psi=0}=0$. We have also used (41.8), which is the reason we must restrict $j\leq K_{0}-1$.

For (41.4), we split $u^{(n)}=\bar{u}^{(n)}+\rho^{(n)}$. As $\bar{u}^{(n)}$ trivially satisfies this inequality (since $n\geq 1$), we must treat $\rho^{(n)}$. By using the identity $\rho=\frac{\phi}{u+\bar{u}}$, we obtain

[TABLE]

from where we obtain

[TABLE]

from here the result follows from the corresponding $\phi^{(n)}$ estimate, which we can use because $n\leq N_{0}-1$. The same proof works for (41.5) upon noticing that we can put an extra factor of $\frac{1}{u}$ in (42).

For the term (41.10), we have, using the equation (10),

[TABLE]

Clearly, we may upgrade to the general $j\leq K_{0}-1$ case. The proof of (41.11) works in an identical manner. This concludes the proof.

A key feature we take advantage of is that decay is enhanced in the region $\xi\lesssim 1$. Note that this type of enhanced decay is not available at the top two orders of $\partial_{x}$ (as seen by the restriction on $j$ below).

Lemma 16** ($L^{2}(\xi\lesssim 1)$ Estimates)**

For $\alpha=0,1,2$, and for $0\leq j\leq K_{0}-2$,

[TABLE]

Proof. We address the $j=0$ case, the general $j$ case being analogous. We begin with rearranging (10) to obtain

[TABLE]

Above, we have used that, for $j\leq k-1$, $\|\langle\sqrt{\psi}\rangle\phi^{(j)}_{\psi}\|_{L^{2}_{\psi}}\lesssim\langle x\rangle^{-j-(\frac{1}{2}-)}$ according to (41.7).

We localize the $\phi^{(j)}$ estimate via

[TABLE]

which implies

[TABLE]

For the enhanced localized $\phi_{\psi}$ estimate, we have

[TABLE]

This concludes the proof.

Lemma 17** ($L^{\infty}$ Estimates)**

Let $0\leq j\leq K_{0}-2$,

[TABLE]

Proof. We start with (44.3), for which a standard interpolation gives

[TABLE]

upon using (41.1) and (41.8).

We now estimate (45.1). We simply interpolate (integrating from $\psi=\infty$):

[TABLE]

upon using (41.8), (41.10), and (43).

We now move to (44.1). Clearly we may restrict to the region $\xi\lesssim 1$, in which case $u^{2}\sim\eta^{2}\sim\xi$. Since $\phi|_{\psi=0}=0$, we may write

[TABLE]

The proofs of (44.2) and (44.4) are identical.

We move to (45.2). Recall that

[TABLE]

and thus the estimate follows upon noticing that it holds for both $\phi^{(j)}_{\psi}$ and $\bar{u}^{(j)}_{y}$.

Finally, to estimate the $\phi_{\psi\psi}$ term, we use the equation. The general $j$ case follows in a similar manner, so we deal with $j=0$:

[TABLE]

In a similar fashion,

[TABLE]

We treat the first term, as the second term above is analogous. We estimate

[TABLE]

This concludes the proof.

6 Global Positivity of $\Omega$

We now analyze the quantity

[TABLE]

Lemma 18

For $[\bar{u},\bar{v}]$ Blasius solutions, $\Omega\geq 0$.

Proof. Since $\bar{u}$ is a solution to the Prandtl equation, which yields the identity

[TABLE]

As $[\bar{u},\bar{v}]$ are Blasius solutions, we may further invoke the self-similar structure and subsequently the self-similar ODE satisfied by $f$ to rewrite

[TABLE]

For Blasius solutions, $f^{\prime\prime}>0$, and the question, therefore, reduces to establishing nonnegativity of the quantity $-\eta f^{\prime}+2f$, which we thus name $\omega$.

First, as $f^{\prime}\rightarrow 1$ and $f\rightarrow\eta$ at $\eta=\infty$, clearly

[TABLE]

We now analyze a sufficiently small neighborhood of $\eta=0$. The following Taylor expansions are valid:

[TABLE]

Multiplying the first quantity, $f$, by $2$ we see that the $f^{\prime\prime}$ terms match. The $f^{\prime\prime\prime}(0)$ terms have a factor of $\frac{1}{3}$ versus $\frac{1}{2}$ for $\eta f^{\prime}$. In general, the $f^{(n)}(0)$ terms have a factor of $2\frac{1}{n!}$ which is less than the factor of $\frac{1}{(n-1)!}$. Upon realizing that $f^{\prime\prime\prime}(0)=f^{(4)}=0$ and then the first nonzero term is $f^{(5)}(0)<0$, the positivity of $\omega>0$ for $\eta<<1$ follows. In turn, this follows from differentiating the Blasius ODE to obtain

[TABLE]

We must now analyze $w$ for $\eta$ in the “in-between” regions. We need to check that $\omega$ cannot change sign. This would be implied if $\omega$ was non-decreasing. Taking one derivative:

[TABLE]

The aim is to check the right-hand side is nonnegative. Again, we can check that this quantity is zero at $y=0$ and $1$ at $y=\infty$, and thus would be nonnegative if it were to be monotonically increasing. Taking a further derivative, we obtain $-\eta f^{\prime\prime\prime}>0$.

7 Weighted Nash Inequality and Optimal Decay

Lemma 19

Solutions $\phi\in X$ to the system (13) satisfy the following Nash-type inequality

[TABLE]

Proof. We first localize based on $\xi=\frac{\psi}{\sqrt{x}}$. Fix a $\tau$ to be selected later. Then by triangle inequality we split

[TABLE]

For the localized portion, we need to condition on whether or not $\tau<1$ or $\tau>1$. We integrate by parts via

[TABLE]

We estimate the former term above term via

[TABLE]

More specifically, in the case when $\rho<1$

[TABLE]

The $o(1)$ term is absorbed to the left-hand side of (49).

In the case when $\rho>1$, we must estimate $\xi^{\frac{1}{4}}\leq\tau^{\frac{1}{4}}\sqrt{u}$. To see that this is true, first assume $\xi\leq 1$. Then $\xi^{\frac{1}{4}}\lesssim\sqrt{u}\lesssim\sqrt{u}\tau^{\frac{1}{4}}$ because $\tau>1$ by assumption. Next, suppose $\xi\geq 1$. Then $\xi^{\frac{1}{4}}\leq\tau^{\frac{1}{4}}\lesssim\tau^{\frac{1}{4}}\sqrt{u}$ because $u\gtrsim 1$ on the region when $\xi\geq 1$.

For the second term in (49), we estimate identically to the far-field term from (48), which we now treat.

For the far-field term, we estimate via

[TABLE]

In summary, we have thus established the inequality

[TABLE]

where $\varphi(\tau)$ is the piecewise function equal to $\tau^{\frac{3}{2}}$ on $\tau<1$ and $\tau^{2}$ on $\rho\geq 1$.

We now select

[TABLE]

The key point is that $\tau$ is homogeneous in $r$, and therefore we may consistently enforce when $\tau<1$ and $\tau>1$ because these are equivalent to $r<1$ and $r>1$.

To conclude, we note that by definition of the $X$ norm, the weighted quantities $\|\phi\sqrt{\psi}\|_{L^{2}_{\psi}}$ are conserved in $x$ for solutions to (13). This immediately gives (47).

We are now ready to establish the optimal decay rates.

Proof of Theorem 3. Using (47) in (16), letting $\alpha(x):=\|\phi\|_{L^{2}_{\psi}}^{2}$, we obtain either one of the two ODEs ($\cdot=\partial_{x}$) :

[TABLE]

at each $x\in\mathbb{R}_{+}$. This immediately implies that $|\alpha|\lesssim\langle x\rangle^{-\frac{1}{2}}$, which means that $\|\phi\|_{L^{2}_{\psi}}\lesssim\langle x\rangle^{-\frac{1}{4}}$.

We may $x$-differentiate (47) and use them in the higher order energy estimates (23), (33) in exactly the same fashion which yields $\|\phi^{(j)}\|_{L^{2}_{\psi}}\lesssim\langle x\rangle^{-j-\frac{1}{4}}$. From here, one repeats all of the above $X$-norm estimates with self-similar weights $\langle\eta\rangle$, which follows in an identical manner. The estimates (LABEL:est.main) follow from standard interpolation, the relations

[TABLE]

and Lemmas 4, 4’ in [Ser66] to replace the stream function variable $\psi$ with the physical variable $y$ (one takes the parameter $a$ in Lemmas $4,4^{\prime}$ of [Ser66] to be $\langle x\rangle^{-\frac{1}{2}}$).