Local well-posedness to the thermal boundary layer equations in Sobolev space

1.   Introduction

The concept of the boundary layer was first proposed by Ludwig Prandtl in 1904 (^[19]). For the incompressible viscous fluid satisfying the non-slip boundary condition, Prandtl obtained a degenerate parabolic equation coupled with the elliptic equation, namely the famous Prandtl equation, to describe the fluid motion in the boundary layer.

Since Prandtl boundary layer theory was put forward, many mathematicians have devoted themselves to establishing its mathematical theory (cf. ^{[2,3,5,7,8,9,17,18,23,25,27,29,30,31,34,35,36]}). Oleinik ^[17] performed the first rigorous mathematical systematic work by showing that under the monotonic condition of the boundary normal tangential velocity field, local well-posedness of the Prandtl system can be proved in two-dimensional by using the Crocco transformation. This well-posedness result was also obtained in the Sobolev spaces by using energy method (cf. ^[1,16]). The key ingredient in the proof is a nonlinear cancellation mechanism that can be used to eliminate the problematic terms in the equations.

Without the monotonicity condition, Caflisch and Sammartino ^[21,22] established the local well-posedness in the framework of analytic functions. If the initial data is neither monotonic in the normal variable nor analytic, E and Engquist ^[4] constructed a finite time blowup solution to the Prandtl equations. See also the instability results of Gérard-Varet and Dormy ^[6]. Recently, there are also many important studies on the boundary layer problem for some more complex fluids, such as the MHD system and viscoelastic equations. Interested readers can refer to ^{[11,12,13,14,15,20,24,32,33]} for more details.

In reality, most fluids have thermal conductivity, so the study of heat-conducting viscous fluid has important theoretical significance and application background. The main object of this paper is to establish the local well-posedness of the thermal boundary layer equations for two-dimensional incompressible heat conducting flow with non-slip boundary condition. Namely, we will consider the following system in the two-dimensional half space $\Omega: = \mathbb{T} \times \mathbb{R}^{+} = \{(x, y) \mid x \in \mathbb{R} / \mathbb{Z}, 0 < y < \infty\}$

where $(u, v)$ is the velocity field, and $\theta$ is the absolute temperature. The $U(t, x), \Theta(t, x)$ and $P(t, x)$ are the traces at the boundary $\{y = 0\}$ of the tangential velocity, temperature, and pressure of the outer inviscid flow with heat conduction, respectively. The reference temperature $\theta_{\infty}$ is assumed to be a positive constant in this paper. The states $U, \Theta$, and $P$ are interrelated through

The mathematical theory of the thermal boundary layer equations was first studied by Wang and Zhu in ^[26], where they proved the local existence and uniqueness of solutions under the assumption of analyticity. In ^[28], they also proved finite time blowup of the solutions if the monotonic condition is violated. On back flow of boundary layers in two-dimensional unsteady incompressible heat conducting flow be studied in ^[29]. Recently, Liu, Wang and Yang ^[10] developed energy method to prove the well-posedness of a viscous layer problem when the tangential velocity is monotonically increasing in the normal variable. In this paper, we are going to show the local well-posedness of the system (1.1) in Sobolev space under the monotonic condition. This extends the Oleinik local well-posedness theory to the thermal boundary layer equations.

To state the main result, we first introduce some notations and the function spaces in which the initial-boundary value problem (1.1) will be solved under the strictly monotonic assumption on the tangential velocity in the normal variable

First, $C$ is a genetic constant which may change from line to line throughout this paper. We denote the tangential derivative operator by

and the full derivative operator is given by

We also use the following notations

and

Second, $H^s(\Omega)$ and $H^s(\mathbb{T})$ is the usual Sobolev space on spatial domain $\Omega$ and $\mathbb{T}$ respectively. We also define the weighted Sobolev space $H^{s, \gamma}(\Omega)$ by

with $u(t, x, y):[0, T]\times\Omega \to \mathbb{R}$ and

Finally, we set $\tilde{\theta} = \theta-\Theta$ and define the space $H_{\mu, \delta}^{s, \gamma}$ for $(\omega, \tilde{\theta}): [0, T]\times\Omega \rightarrow \mathbb{R}$ by

with $s\ge 6, \gamma\ge1, \mu > \gamma+\frac{1}{2}$ and $\delta \in \left (0, 1 \right)$.

Remark 1.1. The condition $\mu > \gamma+\frac{1}{2}$ is indispensable for the definition of the space $H_{\mu, \delta}^{s, \gamma}$. Actually, if $\mu \leq \gamma + \frac{1}{2}$, one can check that $H_{\mu, \delta}^{s, \gamma}$ is an empty set. For example, taking $\mu = \gamma = 1, \alpha = 0,$ we find that $\left\|\langle y\rangle \omega\right\|_{L^{2}} < \infty, \langle y\rangle\omega\geq\delta, |\langle y\rangle\omega|\leq \frac{1}{\delta}$ can not hold at the same time. The same hypothesis is also explained by Masmoudi and Wong (see Remark 2.1 in ^[16]). The reason for introducing the weighted space $H_{\mu, \delta}^{s, \gamma}$ is to give the control of terms like $\frac{\partial_{x}\omega}{\omega}, \frac{\partial_{x}\theta}{\omega}, \frac{\partial_{x}^{2}\omega}{\omega}, \frac{\partial_{x}^{2}\theta}{\omega}$.

Before state the main result, we assume $\theta_{b} = 0$ throughout this paper for the sake of simplicity. We claim that the result still holds if we have a non-trivial $\theta_{b}$. Moreover, it is easy to find that the vorticity $\omega$ satisfies

Now, we are ready to state the main results of this paper in the following theorem.

Theorem 1.2. Given any even integer $s\geq 6$, real numbers $\gamma\ge1, \mu > \gamma+\frac{1}{2}, \delta \in \left (0, 1 \right)$, assume the following conditions on the initial data and the outer flow $U$ and $\Theta$:

$i)$ The initial data $u_{0}-U \in H^{s, \gamma-1}(\Omega)$ and $(\omega_0, \tilde{\theta}_{0}) \in H_{\mu, 2 \delta}^{s, \gamma}$. Here, the time derivatives of the initial data is expressed by solving equation (1.1) and (1.4) repeatly for $\partial_t^k(\omega, \tilde{\theta})$ and substituting the initial data into the result, for example:

with $v_0 = -\int_0^y\partial_xu_0dy.$

$ii)$The outer flow $U$ and $\Theta$ is supposed to satisfy

Then there exists a time $T: = T\left(s, \gamma, \delta, \|(w_{0}, \tilde{\theta}_0)\|_{H^{s, \gamma}}, U, \Theta\right) > 0$ such that the initial-boundary value problem (1.1) has a unique classical solution $(u, v, \theta)$ satisfying

and

where $H^{s}-w$ is the space $H^{s}$ endowed with its weak topology.

Remark 1.3. If the Dirichlet boundary condition for temperature is given, some boundary terms cannot be handled in the proof. In this paper, we give the Neumann boundary condition to the temperature $\theta$ and it is interesting to investigate the Dirichlet boundary condition case. Another interesting question is how to extend the results to fractional problems.

Remark 1.4. We assume $s\geq6$ in Theorem 1.2 mainly because we need to derive the uniform upper bound and lower bound of the solutions, and $s$ needs to be an even number. Moreover, we didn't get the result similar to (5.3) in ^[16], so we need to assume $s\geq6$ to get our results.

Remark 1.5. From the definition of $H_{\mu, \delta}^{s, \gamma}$, we can see that both the vorticity and the temperature enjoy some decay properties with respect to $\langle y\rangle$ at the far field $y = +\infty$. We refer to Appendix C of ^[16] for more details about the far-fields behavior of the vorticity, and the decay rates of the temperature can be obtained similarly.

Let us briefly describe the strategy of the proof of our main theorem. As mentioned earlier, we will use the energy method developed by Masmoudi and Wong ^[16] to prove the local well-posedness of the thermal boundary layer equations. To do this, we first need to construct a regularized system by adding the viscous terms $\varepsilon\partial^2_{x}u$ and $\varepsilon\partial_{x}^2\theta$ to the original equations. This will make the system no longer degenerate and the local existence of the regularized system can be established by using the classical local well-posedness theory of the hyperbolic-parabolic system. Next, to construct local solutions of the original system, we will derive the uniform-in-$\varepsilon$ estimates of the solutions to the regularized system, which is the main part of this paper. The uniform estimates are divided into two parts. The first part is the weighted $L^{2}$ estimates on $D^{\alpha}(\omega, \tilde{\theta})$ with $|\alpha| \leq s, |\beta| \leq s-1$ and the second part is to get the estimate of $\partial_\chi^{\beta}(\omega, \tilde{\theta})$ with $|\beta| = s$.

Different from classical Prandtl equations ^[16] where only spatial derivatives of the solutions need to be estimated, here we give the control of both spatial and time derivatives of the solutions, because the time derivatives of $\theta$ will be involved in estimating the boundary integral $\int_{\mathbb{T}} \partial_{y} D^{\alpha} \omega D^{\alpha} \omega dx$ (see Lemma 3.3 for example). Similarly, the control of $\partial_t^k\omega$ is also needed when we encountered with the boundary term $\int_{\mathbb{T}} \partial_{y} D^{\alpha} \tilde{\theta} D^{\alpha} \tilde{\theta}dx$. However, estimating $D^{\alpha}(\omega, \tilde{\theta})$ will bring us new difficulties when we have the term $\int_{\mathbb{T}} \partial_{y} D^{\alpha} \omega D^{\alpha} \omega dx$ with $|\alpha| = |(\beta_1, 0, m)| = s$ and $m = 2k+1$ is odd. Since there are no $x$-derivatives here, we can not use integrating by parts to reduce the $s+1$ order derivatives which will prevent us from using trace estimate to estimate the boundary term. The remedy is to replace $\partial_y^{m+1}\omega$ and $\partial_y^{m}\omega$ by using the equations repeatedly to get

where $\mathcal{P}_{k}$ and $\mathcal{Q}_{k}$ are low order terms. Now, as we already have $x$-derivatives in the above boundary integral, we can use integrating by parts and the trace lemma to control it.

The estimate of $\partial_\chi^{\beta}(\omega, \tilde{\theta})$ mainly based on the nonlinear cancellation method invented in ^[16]. Here we note that we only need the monotonic assumption on the tangential velocity $u$ in the normal variable but have no restrictions on the absolute temperature $\theta$. With the uniform estimates of the solutions, we show that the solutions of the regularized system actually exists in a time interval $[0, T]$ independent of $\varepsilon$. Moreover, we can use the Aubin-Lions lemma to extract a solution sequence and prove that the limit of this sequence is the solution of the original thermal boundary layer equations. Thus the existence of the local solution is constructed. Finally, the uniqueness of the solution is also proved by the energy estimate.

The rest of the paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we first introduce the regularized system (3.1) and the regularized vorticity system (3.3) in order to construct the approximate solutions. Then we give the uniform estimates of the solutions. The local-in-time existence and uniqueness of the solution to the initial-boundary value problem (1.1) will be proved in Sections 4 and Section 5 respectively based on the uniform weighted estimates derived in Section 3. The proof of some useful inequalities and the derivation of some equations will be given in the Appendix.

2.   Preliminaries

In this section, we introduce some notations and collect some preliminary results which will be used in the rest part of this paper.

As the Prandtl system, the key point for obtaining the energy estimates of solutions is to eliminate the terms $v\partial_{y} u$ and $v\partial_{y}\theta$ appeared in the first and the second equations of (1.1) respectively. Recalling that in ^[10] (see also ^[1,16] for a similar transformation), the authors introduce $\omega = \partial_{y}(\frac{u}{\partial_{y}u})$ and $\bar{\theta} = \theta-\frac{\partial_{y}\theta}{\partial_{y}u}u$. Here a little different from ^[10], we define

then we can introduce a weighted norm for the vorticity

and a weighted norm for the absolute temperature

Obviously, the main difference between norms $\|\cdot\|_{H^{s, \gamma}(\Omega)}$ and $\|\cdot\|_{H_{g}^{s, \gamma}(\Omega)}$, $\|\cdot\|_{H_{h}^{s, \gamma}(\Omega)}$ is that the weighted $L^{2}$ norm of $\partial_{\chi}^{\beta} \omega$ and $\partial_{\chi}^{\beta} \tilde{\theta}$ with $|\beta| = s$ is replaced by that of $g_{\beta}$, $h_{\beta}$. As we will see later, by estimating the weighted norms (2.1) and (2.2) of the solutions, we can avoid the loss of $x$-derivative through a delicate nonlinear cancellation.

Moreover, similar to those in ^[16], one can show that $\|\omega\|_{H^{s, \gamma}(\Omega)}$ and $\|\omega\|_{H_{g}^{s, \gamma}(\Omega)}$ are almost equivalent. That is, for any $(\omega, \tilde{\theta})\in H_{\mu, \delta}^{s, \gamma}(\Omega)$, there exists a positive constant $C$ such that

and

Remark 2.1. Although $h_{\beta}$ is similar to $g_{\beta}$ in form, the weighted norm $\|\tilde{\theta}\|_{H_{h}^{s, \gamma}}$ does not share the almost equivalent relationship with the norm $\|\tilde{\theta}\|_{H^{s, \gamma}}$. However, the above inequality (2.4) is enough to solve the problem.

Next, let us introduce several useful inequalities which will be frequently used in this paper. We omit the proofs of these inequalities for the sake of simplicity and interested readers may refer to ^[11] and ^[16] and the references therein for more details.

Lemma 2.2 (Hardy type inequality). Let $u: \Omega \rightarrow \mathbb{R}.$ Then

$i)$ if $\gamma > -\frac{1}{2}$ and $\lim \limits_{y \rightarrow +\infty} u(x, y) = 0$, we have

$ii)$ if $\gamma < -\frac{1}{2}$, we have

Lemma 2.3 (Sobolev type inequality). Let $u: \Omega \rightarrow \mathbb{R}.$ Then there exists a positive constant $C$ such that

Lemma 2.4 (Trace estimate). Let $u, v: \Omega \rightarrow \mathbb{R}.$ If $\lim\limits_{y \to +\infty}(uv)(x, y) = 0$, then

Lemma 2.5 (Aubin-Lions). Let $\Omega_{0} \subset \Omega \subset \Omega_{1}$ be three Banach spaces, with compact embedding $\Omega_{0} \subset \Omega$ and continuous embedding $\Omega \subset \Omega_{1}$. Let $p, q \geq 1$, then

is compactly embedded into $L^{p}([0, T]; \Omega)$.

Finally, we also need the following lemma which will be used to control certain $L^{2}$ and $L^{\infty}$ norms of $u, v, \omega, \tilde{\theta}, g_{\beta}, h_{\beta}$ and their derivatives in terms of the weighted norms $\|\omega(t, \cdot)\|_{H_{g}^{s, \gamma}}$ and $\|\tilde{\theta}(t, \cdot)\|_{H_{h}^{s, \gamma}}$. The proof of this lemma is given in the Appendix A.

Lemma 2.6. Let the vector field $(u, v)$ defined on $\Omega$ satisfy the condition $\partial_{x} u+\partial_{y} v = 0,$ the Dirichlet boundary condition $u|_{y = 0} = v|_{y = 0} = 0$ and $\lim\limits_{y \to +\infty}u = U.$ If $(\omega, \tilde{\theta}) \in H^{s, \gamma}_{\mu, \delta}$ for some constants $s\geq 6, \gamma \geq 1, \mu > \gamma + \frac{1}{2}$ and $\delta \in(0, 1)$, then we have the following estimates:

A) Weighted $L^2$ estimates.

$(i)$ For all $\left | \beta \right | = 0, 1, \cdots, s$,

$(ii)$ For all $\left | \beta \right | = 0, 1, \cdots, s-1$,

$(iii)$ For all $|\beta| \leq s$,

and

$(iv)$ For all $|\beta| = 1, 2, \cdots, s$,

and

B) Weighted $L^{\infty}$ estimates.

$(v)$ For all $|\beta| = 0, 1, \cdots, s-1$,

$(vi)$ For all For all $|\beta| = 0, 1, \cdots, s-2$,

$(vii)$ For all $|\alpha| \leq s-2$,

3.   Uniform estimates to the regularized system

In this section, in order to prove the local-in-time existence of the initial-boundary value problem (1.1), we consider the following regularized equations for any $\varepsilon > 0$:

The states $U^{\varepsilon}$ and $P^{\varepsilon}$ are interrelated through

By a direct calculation, we find that the regularized vorticity $\omega^{\varepsilon}: = \partial_{y}u^{\varepsilon}$ and $\tilde{\theta}^{\varepsilon} = \theta^{\varepsilon}-\Theta^{\varepsilon}$ satisfies the following regularized system

Here, the velocity field $(u^{\varepsilon}, v^{\varepsilon})$ is given by

Now, the regularized system (3.3) constitutes a hyperbolic-parabolic equations. For any fixed $\varepsilon > 0$, the well-posedness can be established in a standard way. Actually, we have

Lemma 3.1 (Local Existence of the Regularized Equations). Let $s\ge 6$ be an even integer, $\gamma\ge1, \mu > \gamma+\frac{1}{2}$, $\delta \in \left (0, \frac{1}{2} \right)$, and $\varepsilon \in (0, 1)$. If $(\omega_{0}, \theta_{0}) \in H_{\mu, 2\delta}^{s+12, \gamma}$, then there exist a time

and a solution

to the regularized system (3.3). Moreover, the velocity $(u^\varepsilon, v^\varepsilon)$ and the absolute temperature $\theta^\varepsilon$ satisfy the regularized system (3.1) as well.

By Lemma 3.1, we have obtained the local existence of solution in $[0, T]$ which depends on $(s, \gamma, \delta, \omega_{0}, \theta_{0}, U, \Theta)$ as well as the parameter $\varepsilon > 0$. To get a solution in a time interval independent of $\varepsilon$ for the original system, we need to derive the uniform-in-$\varepsilon$ estimates of the solutions. From now on, we omit the superscript $\varepsilon$ of the solution for the sake of simplicity.

The proof of the uniform estimates will be divided into four parts. First, we will give the weighted estimates of $D^\alpha(\omega, \tilde{\theta})$ with $|\alpha|\leq s$ and $|\beta|\leq s-1$ in Subsection 3.1. Then we study the estimates of $D^\beta_{\chi}(\omega, \tilde{\theta})$ with $|\beta| = s$ in Subsection 3.2. In Subsection 3.3, the weighted $H^s$ estimates of the solution are obtained by combining the estimates in the last two parts. Finally, to ensure that our solution belongs to the function space $H^{s, \gamma}_{\mu, \delta}$, we also need to deal with the $L^\infty$ estimates of the solution and this will be given in Subsection 3.4.

3.1.   Weighted $L^{2}$ estimates on $D^{\alpha}(\omega, \tilde{\theta})$ with $|\alpha| \leq s, |\beta| \leq s-1$

The main goal of this part is to prove:

Theorem 3.2. Let $s \geq 6$ be an even integer, $\gamma \geq 1, \mu > \gamma + \frac{1}{2}, \delta \in(0, 1)$, and $\varepsilon \in(0, 1].$ If

and $(u, v, \omega, \tilde{\theta})$ solves (3.1) and (3.3), then we have

where $C$ is a constant independent of $\varepsilon$ and $t$.

Proof. Applying the operator $D^{\alpha} = \partial_{\chi}^{\beta} \partial_{y}^{m}$ for $\alpha = \left(\beta, m \right) = (\beta_{1}, \beta_{2}, m)$ with $|\alpha| \leq s, |\beta| \leq s-1$ to the equation $(3.3)_{1}, (3.3)_{2}$, multiplying by $\langle y\rangle^{2 \gamma+2 m} D^{\alpha} \omega$, $\langle y\rangle^{2 \gamma+2 m} D^{\alpha} \tilde{\theta}$ respectively, then integrating over $\Omega$, we have

and

Now, we will give the estimates of $J_i$ and $K_i$ as follows. First of all, for $J_1$, it holds that

where an integration by parts in the $x$-variable is used. For $J_2$, utilizing integration by parts in the $y$-variable, we have

Clearly, $J_{2}^{1}$ can be controlled by using the Cauchy inequality

However, the estimate of the boundary integral $J_{2}^{2}$ is very complicated. To handle it, we need to introduce the following lemma and the proof of this lemma is given in the Appendix B.

Lemma 3.3. (Reduction of Boundary Data). Under the hypotheses of Theorem 3.2, we have

For any $2 \leq k \leq \frac{s}{2}$, we have

where $\mathcal{P}_{k}$ denotes a polynomial

Now, we claim that

We prove the above claim in two cases $|\alpha|\leq s-1$ and $|\alpha| = s$. When $|\alpha|\leq s-1$, we find

While when $|\alpha| = s$, we must have $m\geq 1$ since $|\beta|\leq s-1$. Two cases need to be considered here.

Case 1: $m = 2k$ is an even number.

In this case, we can use Lemma 3.3 to obtain

Therefore, applying the trace estimate in Lemma 2.3 and the Cauchy-Schwarz inequality to the above equation, we get

Case 2: $m = 2k+1$ is an odd number.

In this case, since $s$ is even, we have $|\beta| = |\beta_{1}|+|\beta_{2}|\geq 1$.

(1) When $\beta_{2} \geq 1$. Using integration by parts in $x$, we have

Now, the term $\partial_{x}D^{\alpha}\omega|_{y = 0} = \partial_{\chi}^{\beta+e_{2}}\partial_{y}^{2m+1}\omega|_{y = 0}$ has an odd number of $y$ derivatives. Hence, we can apply Lemma 3.2 to reduce the order of the right hand side of (3.11). Similar to the Case 1, we can further apply Lemma 2.4 to eventually obtain the following estimates:

(2) When $\beta_{2} = 0$. There is no derivative of $x$ here, so we cannot integrate by parts with respect to $x$ directly. However, we can substitute the $y-$derivative of $\omega$ by using Lemma 3.3 and

which is obtained by applying $\partial_{y}^{2k}$ to $(3.3)_1$ and $\mathcal{Q}_{k}$ is a polynomial of $D^\alpha(\omega, \tilde{\theta})$ with $|\alpha|\leq 2k+1$. This substitution gives us the derivative with respect to $x$, so we have

The difficulty lies in the term $\varepsilon^{2k+2}\int_{\mathbb{T}} \partial_{t}^{\beta_{1}} \partial_{x}^{2k+2}\omega \partial_{t}^{\beta_{1}} (\partial_{t}-\varepsilon^{2} \partial_{x}^{2})^{k} (\partial_{x} P-\theta)$, since the others can be estimated directly by the trace estimate. Integrating by parts with respect to $x$, we have

Now, the order of the derivatives of $(\omega, \theta)$ in (3.13) is no larger than $\beta_1+2k+1 = s$, so we can further apply Lemma 2.4 to eventually obtain the following estimates:

Thus, combining estimates (3.10), (3.12), (3.14), we prove (3.9). As a result, $J_{2}$ can be estimated as

For the term $J_{3}$, we can use integration by parts and the equation $\partial_{x}u+\partial_{y}v = 0$ to get

Next, we can estimate the term $J_{4}$ by using Cauchy-Schwarz inequality directly:

Finally, we will show the estimate of

where $\sigma = (\bar{\sigma}, \bar{m}) = (\sigma_{1}, \sigma_{2}, \bar{m})$. For $J_{5}^{1}$, we have two cases:

Case 1: $\bar{m} = 0$, and $|\bar{\sigma}| \leq s-1$. Here we have

Case 2: $\bar{m}\geq 1$. In this case, by Lemma 2.4, we find

While for $J_{5}^{2}$, we need to consider the following four cases:

Case 1: $\bar{m} = 0, \sigma_{2} = s-1$,

For the other three cases, by using $\partial_{x}u+\partial_{y}v = 0$ and Lemma 2.4, one has

Case 2: $m = 0, \sigma_{2}\leq s-2$,

Case 3: $m = 1$,

Case 4: $m\leq 2$,

Combining estimates (3.15)-(3.20), we get

In the following part, we will give the estimates of the right-hand side terms of (3.6). First, for $K_1$, it directly follows from an integration by parts in the $x$-variable that

Integrating by parts in the $y$-variable in $K_2$, we have

Here $K_{2}^{1}$ can be controlled by the Cauchy-Schwarz inequality

Similar to the estimate of $J_2^2$, the boundary integral $K_{2}^{2}$ is controlled in the following two cases: $|\alpha|\leq s-1$ and $|\alpha| = s$. When $|\alpha|\leq s-1$, we can use the basic trace estimate directly to control $K_{2}^{2}$ since the order of the derivatives in the boundary integral is no larger than $s$. While for the case $|\alpha| = s$, we have to appeal to the boundary reduction argument as before. Actually, we have the following lemma.

Lemma 3.4. (Reduction of Boundary Data). Under the hypotheses of Theorem 3.2, for any $2 \leq k \leq\frac{s}{2}$, we have

$\mathcal{H}_{k}$ denotes a polynomial, and

The proof of this Lemma is based on an elementary use of the original equation (1.1), so we just omit it. With this Lemma, we can give the estimate of $K_2^2$ in a similar fashion with $J_2^2$. By direct calculations, one has

Collecting the estimates (3.23), (3.24) and (3.25), we obtain

For $K_{3}$ we have

The estimate of $K_{4}$ is similar to $J_{5}$. Namely, we can use a similar strategy to get

The term $K_5$ can be estimated by using Lemma 2.6 directly:

Similarly, $K_6$ satisfies

Finally, putting all the above estimates of $J_i$ and $K_i$ into (3.5) and (3.6), we can obtain the desired estimate (3.4). This completes the proof of Theorem 3.2.

3.2.   Weighted $L^{2}$ estimates on $g_{\beta}, h_{\beta}$ with $|\beta|=s$

In this subsection, we will derive the weighted $L^{2}$ estimates on $g_{\beta}$ and $h_{\beta}$ by using standard energy method. We need to derive the evolution equations of $g_{\beta}$ and $h_{\beta}$ first.

Let $a = \frac{\partial_{y} \omega}{\omega}$, $b = \frac{\partial_{y} \tilde{\theta}}{\omega}$, then after a tedious but straightforward calculation (see Appendix C), we have

and

Theorem 3.5. Let $s \geq 6$ be an even integer, $\gamma \geq 1, \mu > \gamma + \frac{1}{2}, \delta \in(0, 1)$, and $\varepsilon \in(0, 1].$ If

and $(u, v, \omega, \tilde{\theta})$ solves (3.1) and (3.3), then we have

where $C$ is a constant independent of $\varepsilon$ and $t$.

Proof. Multiplying the equation (3.26) by $\left \langle y \right \rangle ^{2\gamma}g_{\beta}$ and integrating the resulting equation over $\Omega$, we have

where $g_{i} = \partial_{\chi}^{i} \omega-\frac{\partial_{y} \omega}{\omega} \partial_{\chi}^{i}(u-U).$

Then we shall estimate $L_i$ term by term as follows. It directly follows from an integration by parts in the $x$-variable that

Integrating by parts in the $y$-variable, we have

$L_{2}^{1}$ is controlled by the Cauchy inequality

For the boundary integral $L_{2}^{2}$, a direct calculation yields

This combined with $\partial_{y}\omega|_{y = 0} = \partial_{x}P-\theta|_{y = 0}+\theta_{\infty}$ and $u|_{y = 0} = 0$ gives

Substituting (3.32) to $L_{2}^{2}$ and using Lemma 2.4, we get

where $|\frac{\partial_{y}^{2}\omega}{\omega}| \leq \frac{1}{\delta^{2}}$ is used. Combining (3.29)-(3.33), we have

For $L_{3}$, by integrating by parts in the $x$-variable and $y$-variable, and using the equation $\partial_{x}u+\partial_{y}v = 0$, we have

The term $L_{4}$ is estimated as follows:

Since $\delta \leq \langle y\rangle^{\mu} \omega \leq \delta^{-1}, \quad \sum_{|\alpha| \leq 2}\left|\langle y\rangle^{\mu+m} D^{\alpha} \omega\right|^{2} \leq \delta^{-2}$, we have $\left\|\langle y\rangle \partial_{x} a\right\|_{L^{\infty}} \leq \delta^{-2}$. Note that $\omega \partial_{y}\left(\frac{\partial_{\chi}^{\beta+e_{2}} \tilde{u}}{\omega}\right) = \partial_{x} g_{\beta}+\partial_{x} a \partial_{\chi}^{\beta} \tilde{u}$, and since $\mu > \gamma+\frac{1}{2}$, we can use Lemma $2.2$ to get

Thus, we get

Since $\|\partial_{y}a\|_{L^{\infty}}\leq C$, we have

The estimates of $L_6-L_{13}$ are straightforward, so we omit the details for simplicity. Actually, we have

Now let us estimate $h_{\beta}$. Multiplying the equation (3.27) by $\left \langle y \right \rangle ^{2\gamma}h_{\beta}$ and integrating the resulting equation over $\Omega$, we have

where $h_{i} = \partial_{\chi}^{i} \tilde{\theta}-\frac{\partial_{y} \tilde{\theta}}{\omega} \partial_{\chi}^{i}(u-U)$.

Each $M_i$ need to be estimated now. However, since the estimate of $M_i$ is similar to $L_i$, we only give a skeleton of the proof. From integration by parts in the $x$-variable, we have

Similar to the estimate of $L_2$, we have

where $M_{2}^{1}$ is controlled by the Cauchy inequality

Using the boundary conditions $\partial_{y}\tilde{\theta}|_{y = 0} = 0$ and $u|_{y = 0} = 0$, we have

Substituting (3.36) to $M_{2}^{2}$ and using Lemma 2.4, we get

Combining (3.34)–(3.37), we have

We can give the control of $M_3$ as

The term $M_{4}$ can be controlled in a similar fashion as $L_{4}$, we just skip the details to get

By tedious but straightforward calculations, we find

Here we have used $\|\langle y \rangle a \|_{L^{\infty}}\leq C, \|\langle y \rangle b \|_{L^{\infty}}\leq C, \|\langle y \rangle \frac{\partial_{y}^{2}\tilde{\theta}}{\omega} \|_{L^{\infty}}\leq C, \|\langle y \rangle \frac{\partial_{y}^{2}\omega}{\omega} \|_{L^{\infty}}\leq C, \|\langle y \rangle ^{2} ab\|_{L^{\infty}} \leq C, \|\langle y \rangle ^{2}b^{2}\|_{L^{\infty}} \leq C, \|\langle y \rangle ^{2}\partial_{y}(ab)\|_{L^{\infty}} \leq C$, since $(\omega, \tilde{\theta}) \in H_{\mu, \delta}^{s, \gamma}$. For $M_{12}$, we have

By direct calculations, one can show that

The term $M_{17}$ can be estimated by using Lemma 2.6 directly:

The last term $M_{18}$ can also be estimated by using Lemma 2.6:

Finally combining all the above estimates, we obtain the estimate (3.28), and this completes the proof of Theorem 3.5.

3.3.   Weighted $H^{s}$ estimates on $\omega$ and $\tilde{\theta}$

In this subsection, we can derive the weighted $H^{s}$ estimates on $\omega, \tilde{\theta}$ by employing Theorem 3.2 and Theorem 3.5. The aim of this subsection is to derive the growth rate control (3.26) on the weighted $H^{s}$ energy of $\omega, \tilde{\theta}$.

Theorem 3.6. Let $s \geq 6$ be an even integer, $\gamma \geq 1, \mu > \gamma + \frac{1}{2}, \delta \in(0, 1)$, and $\varepsilon \in(0, 1].$ If

and $(u, v, \omega, \tilde{\theta})$ solves (3.1) and (3.3), then we have

where $C$ is a constant independent of $\varepsilon$ and $t$. The function $Y(t)$ is expressed by

Proof. Combining estimates (3.4) and (3.28), we find

Then it follows from the comparison principle of ordinary differential equations that

provided

which truly holds if $t > 0$ is chosen small enough. This completes the proof of Theorem 3.6.

3.4.   Uniform existence of the regularized system

In this subsection, we are going to prove the uniform existence of the solutions to the regularized system. To this end, we need to derive the uniform upper bound and lower bound of the solutions.

For $|\alpha|\leq 2$, applying the operator $D^{\alpha} = \partial_{\chi}^{\beta} \partial_{y}^{m}$ to the equation $(3.3)_{1}$ and multiplying $\langle y \rangle ^{\mu+m}$, we have

From Lemma 2.6, when $|\sigma| \leq 2$ with $\sigma = (\sigma_{1}, \sigma_{2}, \bar{m})$, we have

Then, by direct calculation and using the above inequalities, we get

Integrating (3.41) with respect to $t$, we have

where (3.42) is used. Specifically, when $\alpha = (0, 0, 0)$ and $m = 0$, we have

Applying the operator $D^{\alpha} = \partial_{\chi}^{\beta} \partial_{y}^{m}$ to the equation $(3.3)_{2}$ and multiplying $\langle y \rangle ^{\mu+m}$, we have

Similar to (3.42), we get from the above equation that

which further gives

Now the uniform existence of the regularized system (3.3) can be stated as follows.

Theorem 3.7. Let $s \geq 6$ be an even integer, $\gamma \geq 1, \mu > \gamma + \frac{1}{2}, \delta \in(0, 1)$ and $\varepsilon \in(0, 1)$. If $(\omega_{0}, \tilde{\theta}_{0}) \in H_{\mu, 2 \delta}^{s, \gamma}$ and $|Y(t)| \leq M$ for any $t\geq0$, then there exists a positive $T$ independent of $\varepsilon$, such that the regularized system (3.3) has solutions

Moreover, for any $t \in[0, T]$, the solutions satisfy the following uniform estimates.

i) Uniform weighted $H^{s}$ estimates

ii) Uniform weighted $L^{\infty}$ upper bound

iii) Uniform weighted $L^{\infty}$ lower bound

Proof. Since $Y(t)\leq M$, if we take

then by inequality (3.38), we have

for $t \in\left[0, T_{1}\right]$. This gives (3.46). Note that $(\omega_{0}, \tilde{\theta}_{0}) \in H_{\mu, 2\delta}^{s, \gamma}$, we have

Choosing $T_{2}$ as

then from (3.43), we have for all $t \in\left[0, T_{2}\right]$,

Next, due to $\langle y\rangle^{\mu} \omega_{0} \geq 2 \delta$, if $T_{3}$ is chosen as

then from (3.44), we have for all $t \in\left[0, T_{3}\right]$, $\langle y\rangle^{\mu} \omega \geq \delta$. Similarly, if we choose

then we can get from (3.45) that for all $t \in\left[0, T_{4}\right]$,

Finally, letting $T: = \min\big\{T_{1}, T_{2}, T_{3}, T_{4}\big\}$, we complete the proof of the theorem.

4.   Local-in-time existence

In this section, we will establish the local existence of solution to the original system (1.1) by compactness argument. Using the almost equivalence relation (2.3)-(2.4) and the uniform weighted $H^{s}$ estimate (3.46), we have

Furthermore, we also know that $\partial_{t} (\omega^{\varepsilon}, \tilde{\theta}^{\varepsilon})$ and $\partial_{t} \tilde{u}^{\varepsilon}$ are uniformly bounded in $L^{\infty}([0, T]; H^{s-2, \gamma})$ and $L^{\infty}([0, T]; H^{s-2, \gamma-1})$, respectively. By Aubin-Lions Lemma 2.5 and compact embedding of $H^{s, \gamma}$ in $H_{l o c}^{s^{\prime}} (s^{\prime} < s)$, we can find limit function

such that, after taking a subsequence, as $\varepsilon_{k} \rightarrow 0^{+}$:

Using the local uniform convergence of $\partial_{x} u^{\varepsilon_{k}}$, we also have the pointwise convergence of $v^{\varepsilon_{k}}$, as $\varepsilon_{k} \rightarrow 0^{+}$

Combining (4.4)-(4.5), one may justify the pointwise convergence of all terms in the regularized equation (3.1). Passing to the limit $\varepsilon_{k} \rightarrow 0^{+}$ in (3.1), we know that the limit function $(u, v, \tilde{\theta})$ solves the problem in the classical sense. Thus the local existence of solutions is obtained and we complete the proof of our main Theorem 1.2.

5.   Uniqueness

The purpose of this section is to prove the uniqueness of $H^{s, \gamma}_{\mu, \delta}$ solutions constructed in Section 4. Assume $\left(u_{1}, v_{1}, \theta_{1}\right)$ and $\left(u_{2}, v_{2}, \theta_{2}\right)$ are two solutions to the initial-boundary value problem (1.1) and $\omega_{i} = \partial_{y} u_{i}(i = 1, 2)$. Setting $\bar{u} =$ $u_{1}-u_{2}, \bar{v} = v_{1}-v_{2}, \bar{\theta} = \theta_{1}-\theta_{2}, \bar{\omega} = \omega_{1}-\omega_{2}$, we obtain the following equations

Furthermore, set $\varpi = \bar{\omega}-a_{2} \bar{u}$ and $\vartheta = \bar{\theta}-b_{2} \bar{u}$ with $a_{2} = \frac{\partial_{y} \omega_{2}}{\omega_{2}}$ and $b_{2} = \frac{\partial_{y} \tilde{\theta}_{2}}{\omega_{2}}.$ By direct calculations, we get

Since $(\omega_{2}, \tilde{\theta}_{2}) \in H_{\mu, \delta}^{s, \gamma}$, it follows from the weighted $L^{\infty}$ bounds on $\omega_{2}, \theta_{2}$ of Theorem 3.7 that

and

Multiplying the equations $(5.2)_{1}$ and $(5.2)_{2}$ by $2\varpi$ and $2\vartheta$, respectively, then integrating the resulting equations over $\Omega$, and using Lemma 2.6 we obtain

Since $\delta \leq \langle y \rangle ^{\mu} \omega_{2} \leq \delta^{-1}$, $\left.\bar{u}\right|_{y = 0} = 0$ and $\varpi = \omega_{2} \partial_{y}\left(\frac{\bar{u}}{\omega_{2}}\right)$, we can use the Hardy inequality of Lemma $2.2$ to obtain

Substituting (5.4) into (5.3), we get

Applying Gronwall's inequality to (5.5), we obtain

which further gives

provided that $u_{1}|_{t = 0} = u_{2}|_{t = 0}$ and $\theta_{1}|_{t = 0} = \theta_{2}|_{t = 0}$. As a result, we have $\varpi \equiv 0$ and $\vartheta \equiv 0$. Note that $\bar{u}$ can be expressed by $\bar{u} = \omega_{2} \int_{0}^{y} \frac{\varpi }{\omega_{2}} d y$, we get $\bar{u} \equiv 0$.

It is easy to see $\theta_{1} = \theta_{2}$ due to $\bar{\theta} = \vartheta +b_{2} \bar{u} = 0.$ Finally we get $v_{1} = v_{2}$ from $(1.1)_{3}$ and $\bar{u} = 0$. This completes the proof of the uniqueness.

6.   Conclusions

In this paper, we study the local well-posedness of the thermal boundary layer equations for the two-dimensional incompressible heat conducting flow by using a new weighted energy method. Our results show that we only need the monotonic assumption on the tangential velocity $u$ in the normal variable but have no restrictions on the absolute temperature $\theta$. Furthermore, this analytical approach can be applied to the boundary layer problems involving more complex fluids.

Appendix A.   Some inequalities

In this appendix, we will prove the inequalities given in Lemma 2.6. Here we give a proof for the reader's convenience.

Proof. Only need to prove $\beta_{2} = 0$. In other words, we only prove when it's all derivatives with respect to $t$, other cases can be found in ^[16].

$(i)$It follows from the definition of $\|u\|_{H^{s, \gamma-1}}$ that $\| \langle y \rangle ^{\gamma-1} \partial_{t}(u-U)\| \leq \|(u-U)\|_{H^{s, \gamma-1}}$, so it is a direct consequence of the almost equivalence inequality (2.3).

$(ii)$ Using Lemma 2.2 and (2.5), we have

which is inequality (2.6).

$(iii)$Inequality (2.7) and (2.8) follows directly from the defnition of $\|\omega\|_{H^{s, \gamma}_{g}}, \|\tilde{\theta}\|_{H^{s, \gamma}_{h}}$ and inequality (2.3)-(2.4).

$(iv)$ Since $(\omega, \tilde{\theta}) \in H_{\mu, \delta}^{s, \gamma}$, so we know that $\|\langle y \rangle \frac{\partial_{y}\tilde{\theta}}{\omega}\|_{L^{\infty}} \leq \delta^{-2}, \|\langle y \rangle \frac{\partial_{y}\omega}{\omega}\|_{L^{\infty}} \leq \delta^{-2}$. Thus

which is inequality for $|\beta| \leq s-1$. When $|\beta| = s$, the better upper bound in (2.9) and (2.10) follows directly from the definition of $\|\omega\|_{H_{g}^{s, \gamma}}, \|\tilde{\theta}\|_{H_{h}^{s, \gamma}}$.

$(v)$Using Lemma 2.3, (2.5) and (2.7), we have

which implies inequality (2.11).

$(vi)$ Applying triangle inequality, Lemma 2.3, $\partial_{x}u+\partial_{y}v = 0$ and $\omega = \partial_{y}u$, we have

which implies inequality (2.12), because of (2.5)-(2.8).

$(vii)$ The inequality follows directly from Lemma 2.3 and inequality (2.7)-(2.8).

Appendix B.   Proof of Lemma 3.3

Proof. In order to illustrate the idea, let us derive the formula (3.8) for the case $k = 2$ as follows. Applying $\partial_y^3$ to the vorticity equation $(3.3)_{1}$ and evaluating at $y = 0$, we obtain, by using $(3.7)_{2}$ and $\left.u\right|_{y = 0} = \left.v\right|_{y = 0} = \partial_{y}\theta|_{y = 0} = 0$, that

Since the last four terms on the right-hand side are our desired forms, we only need to deal with the terms $(\partial_{t}-\varepsilon^{2} \partial_{x}^{2})(\omega \partial_{x} \omega)|_{y = 0}$ and $(\partial_{t}-\varepsilon^{2} \partial_{x}^{2})(\partial_{y}^{2}\theta)|_{y = 0}$. Using the evolution equations for $\omega, \partial_{x} \omega$ and $\partial_{y}^{2}\theta$ as well as $u|_{y = 0} = v|_{y = 0} = \partial_{y}\theta|_{y = 0} = 0$, one may check that

Assuming that the lemma holds for $k = n$, we will show that it also holds for $k = n + 1$. Applying $\partial_y^{2n+1}$ to the vorticity equation and evaluating the resulting equation at $y = 0$ yields

Thanks to the induction hypothesis, we have

This completes the proof of Lemma 3.3.

Appendix C.   Equations for $a, b, g_{\beta},$ and $h_{\beta}$

In this appendix, we will derive the evolution equations for $a, b, g_{\beta},$ and $h_{\beta}$. The equations satisfied by $(\tilde{u}, \omega) = (u-U, \partial_{y}u)$ is

Equation for $a$:

Differentiating the equation $(6.1)_{2}$ with respect to $y$, we have

which implies

Note that

Substituting (6.3) into (6.2), we have (where $g_{e_{2}} = \partial_{x}\omega-a\partial_{x}\tilde{u}$):

Equation for $g_{\beta}$:

Differentiating the equations (6.1) with $\partial_{\chi}^{\beta}$ respectively, one has

and

Subtracting (6.4) $\times a$ from (6.5), we have

and then we get the equation satisfied by $g_{\beta}$

Similarly, we can write down the evolution equation of $h_{\beta}: = \partial_{\chi}^{\beta} \tilde{\theta}-\frac{\partial_{y} \tilde{\theta}}{\omega} \partial_{\chi}^{\beta}(u-U)$ as follows.

Equation for $b$:

The equations satisfied by $(\tilde{u}, \tilde{\theta}) = (u-U, \theta-\Theta)$ is

Differentiating the equation $(6.6)_{2}$ with respect to $y$, we have

and

Since

we get

Equation for $h_{\beta}$:

Differentiating the equations (6.6) with $\partial_{\chi}^{\beta}$ respectively, one has

and

Subtracting (6.9) $\times b$ from (6.10), we have

and then we get the equation satisfied by $h_{\beta}$

Acknowledgments

The research of Xu was supported by National Natural Science Foundation of China (Grant No. 12001506), the Natural Science Foundation of Shandong Province (Grant No. ZR2020QA014), and Project funded by China Postdoctoral Science Foundation (Grant No. 2021T140633, 2021M693028).

Conflict of interest

The authors declare no conflicts of interest.