Global well-posedness of the 2-D magnetic Prandtl model in the Prandtl

Abstract

In this paper, we prove the global well-posedness of the 2-D magnetic Prandtl model in the mixed Prandtl/Hartmann regime when the initial data is a small perturbation of the Hartmann layer in Sobolev space.

Keywords

Magnetic Prandtl equation Hartmann layer MHD equations global stability

1. Introduction

In this paper, we consider a 2-D magnetic Prandtl model in $R_{+}^{2}$ : $\begin{matrix} (1.1) & \{\begin{matrix} \partial_{t} u + u \partial_{x} u + v \partial_{y} u - \partial_{y}^{2} u = \partial_{y} b, \\ \partial_{y} u + \partial_{y}^{2} b = 0, \\ \partial_{x} u + \partial_{y} v = 0, \\ u |_{y = 0} = v |_{y = 0} = b |_{y = 0} = 0, \\ {lim}_{y \to + \infty} u (t, x, y) = u^{\infty}, {lim}_{y \to + \infty} b (t, x, y) = b^{\infty}, \end{matrix} \end{matrix}$ where $(u, v)$ denotes the velocity field and b denotes the tangential magnetic field. This model is derived from the incompressible MHD system under a transverse magnetic field on the boundary for the parameters of the system in the mixed Prandtl/Hartmann regime. See [8] for a formal derivation.

Thanks to the boundary conditions and $\partial_{y} u + \partial_{y}^{2} b = 0$ , we get $\begin{array}{l} \partial_{y} b = - (u - u^{\infty}) . \end{array}$ Substituting it into (1.1), we obtain a Prandtl type equation with an extra damping term $(u - u^{\infty})$ : $\begin{matrix} (1.2) & \{\begin{matrix} \partial_{t} u + u \partial_{x} u + v \partial_{y} u - \partial_{y}^{2} u + (u - u^{\infty}) = 0, \\ \partial_{x} u + \partial_{y} v = 0, \\ u |_{y = 0} = v |_{y = 0} = 0 and {lim}_{y \to + \infty} u (t, x, y) = u^{\infty} . \end{matrix} \end{matrix}$ The damping term in the system is induced by the combined effect of the magnetic diffusivity and transversal magnetic field. This term should have no effect on the local behavior of the solution. So, it seems not difficult to generalize the following well-posedness results for the classical Prandtl system to the system (1.2).

Local well-posedness in Sobolev space for monotonic data [1,3,13,14];

Local well-posedness for analytic data [12,16,20];

Ill-posedness in Sobolev space and local well-posedness in Gevrey class for a class of non-monotonic data [4,6,7,11];

Global existence of weak solution for monotonic data [18].

Moreover, E and Engquist proved that the analytic solution can blow up in a finite time [5]. See [10] for the extension to van Dommelen–Shen type singularity. However, the damping term should play an important role for the long time behaviour of the solution.

In a recent work [20], Zhang and the fouth author proved the long time well-posedness of the classical Prandtl equation with small analytic initial data. More precisely, if the initial data $‖ e^{\frac{1 + y^{2}}{8}} e^{| D_{x} |} u_{0} ‖_{B^{\frac{1}{2}, 0}} ⩽ ε$ and $u^{\infty} = ε^{5 / 3}$ , then the lifespan of the solution is greater than $ε^{- \frac{4}{3}}$ . In [9], Ignatova and Vicol obtained a larger lifespan $exp \frac{ε^{- 1}}{ln ε^{- 1}}$ with small analytical data of size $O (ε)$ , whose analytical width $τ_{ε} \to \infty$ as $ε \to 0$ .

With the damping term and $u^{\infty} = 0$ , Ren and Zhang proved the global well-posedness of the system (1.2) for small analytic data and finite time blowup for a class of large analytic data [15]. When $u^{\infty} > 0$ , Xie and Yang proved the global stability of the Hartmann layer $u^{s} = u^{\infty} (1 - e^{- y})$ for analytic perturbations [17]. The Hartmann layer derived from the incompressible MHD equations in the Hartmann regime(see [8]) is just a steady solution of the system (1.2).

When the initial data is a small perturbation of the Hartmann layer, it is still monotonic in y variable so that the system is locally well-posed in Sobolev space. Then an important question is:

Whether the system is globally well-posed in Sobolev space in the case when the initial data is near the Hartmann layer.

For the classical Prandtl system, when the initial data is close to a monotonic shear flow, Xu and Zhang [19] proved a long time existence result in Sobolev space. However, it remains open whether one can obtain an explicit life span of the solution when the data is a small perturbation of a monotonic shear flow in Sobolev space in the sprit of the results in [9,20] for small analytic data.

For analytic perturbation, the key point for global result is to show that the analytic band cannot deteriorate to zero. Due to the damping term $u - u^{\infty}$ , the authors in [17] control the analytic band globally. For Sobolev perturbation, the problem becomes highly nontrivial. More precisely, we study the global stability of the Hartmann layer $u^{s} = u^{\infty} (1 - e^{- y})$ . Without lose of generality, we take $u^{\infty} = 1$ and introduce the perturbation $\tilde{u} = u - u^{s}$ , which satisfies $\begin{matrix} (1.3) & \{\begin{matrix} \partial_{t} \tilde{u} + u \partial_{x} \tilde{u} + v \partial_{y} u - \partial_{y}^{2} \tilde{u} + \tilde{u} = 0, \\ \partial_{x} \tilde{u} + \partial_{y} v = 0, \\ \tilde{u} |_{y = 0} = v |_{y = 0} = 0 and {lim}_{y \to + \infty} \tilde{u} (t, x, y) = 0, \\ \tilde{u} |_{t = 0} = {\tilde{u}}_{0} (x, y) . \end{matrix} \end{matrix}$ To obtain the well-posedness in Sobolev space, we need to introduce a good unknown as in [1,13] defined by $\begin{array}{l} w = \partial_{y} (T_{\frac{1}{\partial_{y} u}} \tilde{u}), T_{a} b is a paraproduct . \end{array}$ Then w satisfies $\begin{matrix} (1.4) & \{\begin{matrix} \partial_{t} w + T_{u} \partial_{x} w - \partial_{y}^{2} w = \partial_{y} ({\tilde{F}}_{1} - 2 w) + F_{2}, \\ (\partial_{y} w + {\tilde{F}}_{1} - 2 w) |_{y = 0} = 0 and {lim}_{y \to + \infty} w = 0, \end{matrix} \end{matrix}$ where ${\tilde{F}}_{1}, F_{2}$ are some nonlinear terms (see Section 3). To prove the global well-posedness, we encounter two main difficulties:

Unlike the system (1.2), the new system (1.4) for good unknown w has no obvious damping mechanism;

Although $\partial_{y} u > 0$ for any $y \in [0, + \infty)$ , it decays exponentially to zero as $y \to \infty$ so that the good unknown w becomes degenerate at $y = + \infty$ . For this, we have to give a global in time control for the behavior of $\partial_{y} u$ .

In fact, these also explain why the question of giving an explicit life span for the classical Prandtl system is still open. To overcome these difficulties, our proof introduces the following three key ingredients:

The exponential point-wise decay estimate in y variable for the lower order derivatives of the velocity such as $\partial_{y} \tilde{u}, \partial_{y}^{2} \tilde{u}, \partial_{x} \tilde{u}, \partial_{x} \partial_{y} \tilde{u}$ . A key observation is that the decay rate in y does not slow down as the time evolves due to the damping term. This fact is crucial for resovling the second difficulty. The proof of the point-wise bounds relies on the high order energy bounds.

High order energy estimate without the derivative loss in x variable. For this, we introduce a good unknown w and use the paralinearized method in order to avoid using the Nash–Moser iteration as in [1]. More importantly, we reveal the damping mechanism hidden in the new system (1.4).

High order energy estimates in y variable. We still use the velocity equation. Here we don’t need to take care of the derivative loss in x variable, since we only take one order derivative in x variable.

To state our result, we introduce the following weighted Sobolev space. For $k, ℓ \in N$ , the space $H_{ω}^{k, ℓ}$ consists of all functions $f \in L_{ω}^{2}$ satisfying $\begin{array}{l} ‖ f ‖_{H_{ω}^{k, ℓ}}^{2} = \sum_{α = 0}^{k} \sum_{β = 0}^{ℓ} {‖ \partial_{x}^{α} \partial_{y}^{β} f ‖}_{L_{ω}^{2}}^{2} < + \infty, \end{array}$ where $‖ f ‖_{L_{ω}^{p}} = ‖ ω (y) f (x, y) ‖_{L^{p}}$ with $ω (y)$ a positive weight function.

Now our main result is stated as follows.

Theorem 1.1.
Assume that ${\tilde{u}}_{0} \in H_{μ}^{3, 1} \cap H_{μ}^{1, 2} (μ (y) = e^{\frac{y}{2}})$ with ${\tilde{u}}_{0} |_{y = 0} = 0$ satisfies $\begin{array}{l} | \partial_{x} {\tilde{u}}_{0} (x, y) | + | \partial_{x} \partial_{y} {\tilde{u}}_{0} (x, y) | + | \partial_{y}^{2} {\tilde{u}}_{0} (x, y) | ⩽ ε e^{- y} for (x, y) \in R_{+}^{2}, \\ ‖ {\tilde{u}}_{0} ‖_{H_{μ}^{3, 1}} + ‖ {\tilde{u}}_{0} ‖_{H_{μ}^{1, 2}} ⩽ ε . \end{array}$ Then there exists $ε_{0} > 0$ such that for any $ε \in (0, ε_{0})$ , the magnetic Prandtl model ( 1.2 ) has a unique global in time solution. Moreover, there holds that for any $(t, x, y) \in [0, \infty) \times R_{+}^{2}$ , $\begin{array}{l} c e^{- y} ⩽ \partial_{y} u (t, x, y) ⩽ C e^{- y}, \\ | \partial_{x} \tilde{u} (t, x, y) | + | \partial_{x} \partial_{y} \tilde{u} (t, x, y) | + | \partial_{y}^{2} \tilde{u} (t, x, y) | ⩽ C ε e^{- y}, \\ e^{\frac{t}{4}} ({‖ \tilde{u} (t) ‖}_{H_{μ}^{3, 1}}^{2} + {‖ \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2}) + \int_{0}^{t} e^{\frac{s}{4}} {‖ \tilde{u} (s) ‖}_{H_{μ}^{1, 3}}^{2} d s ⩽ C ε^{2} . \end{array}$

2. Exponential pointwise decay estimates

In this section, we derive the exponential pointwise decay estimates in the y variable. To this end, we introduce $\begin{array}{l} \bar{u} (t, x, y) = \tilde{u} (t, x + t, y), \bar{v} (t, x, y) = v (t, x + t, y) . \end{array}$ Then $\bar{u}$ satisfies $\begin{matrix} (2.1) & \{\begin{matrix} \partial_{t} (e^{t} \bar{u}) - \partial_{y}^{2} (e^{t} \bar{u}) = - e^{t} F (t, x, y), \\ \bar{u} |_{y = 0} = 0 and {lim}_{y \to + \infty} \bar{u} (t, x, y) = 0, \\ \bar{u} |_{t = 0} = {\tilde{u}}_{0} (x, y), \end{matrix} \end{matrix}$ where $F (t, x, y) = (\bar{u} + u^{s} - 1) \partial_{x} \bar{u} + \bar{v} \partial_{y} (\bar{u} + u^{s})$ . We define $\begin{array}{l} (2.2) & A (t) = sup_{(s, x, y) \in [0, t] \times R_{+}^{2}} e^{y} (\sum_{k = 0}^{2} | \partial_{y}^{k} \bar{u} (s, x, y) | + \sum_{k = 0}^{1} | \partial_{y}^{k} \partial_{x} \bar{u} (s, x, y) |) . \end{array}$

Proposition 2.1.
Let $\tilde{u}$ be a smooth solution of ( 1.3 ) on $[0, T]$ with the initial data $u_{0}$ satisfying $\begin{array}{l} | \partial_{x} {\tilde{u}}_{0} (x, y) | + | \partial_{y} \partial_{x} {\tilde{u}}_{0} (x, y) | + \sum_{k = 0}^{2} | \partial_{y}^{k} {\tilde{u}}_{0} (x, y) | ⩽ ε e^{- y} . \end{array}$ Then there exists a constant $C > 0$ independent of T so that for any $t \in [0, T]$ , $\begin{array}{l} A (t) ⩽ & C ε + C (1 + A (t)) (\int_{0}^{t} E {(s)}^{\frac{1}{2}} d s + {(\int_{0}^{t} e^{\frac{s}{4}} E {(s)}^{2} d s)}^{\frac{1}{4}}) \\ + C (1 + A (t)) sup_{s \in [0, t]} E {(s)}^{\frac{3}{8}} (\int_{0}^{t} e^{\frac{s}{4}} D {(s)) d s)}^{\frac{1}{8}}, \end{array}$ where $\begin{array}{l} (2.3) & E (t) = {‖ \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2} + {‖ \tilde{u} (t) ‖}_{H_{μ}^{3, 1}}^{2}, D (t) = {‖ \partial_{y} \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2} . \end{array}$

Let us first give the following pointwise estimates of F.
Lemma 2.2.
It holds that $\begin{array}{l} {‖ e^{y} F ‖}_{L^{\infty}} ⩽ C (1 + A (t)) E {(t)}^{\frac{1}{2}}, \\ {‖ e^{y} \partial_{x} F ‖}_{L^{\infty}} ⩽ C (1 + A (t)) E {(t)}^{\frac{1}{2}}, \\ {‖ e^{y} \partial_{y} F ‖}_{L^{\infty}} ⩽ C (1 + A (t)) (E {(t)}^{\frac{3}{8}} D {(t)}^{\frac{1}{8}} + E {(t)}^{\frac{1}{2}}) . \end{array}$
Proof.
A direct calculation shows that $\begin{array}{l} {‖ e^{y} F ‖}_{L^{\infty}} & ⩽ {‖ e^{y} ((u^{s} - 1) \partial_{x} \bar{u} + \bar{u} \partial_{x} \bar{u} + \bar{v} \partial_{y} u^{s} + \bar{v} \partial_{y} \bar{u}) ‖}_{L^{\infty}} \\ ⩽ ({‖ e^{y} (u^{s} - 1) ‖}_{L^{\infty}} + {‖ e^{y} \bar{u} ‖}_{L^{\infty}}) ‖ \partial_{x} \bar{u} ‖_{L^{\infty}} + ‖ \bar{v} ‖_{L^{\infty}} ({‖ e^{y} \partial_{y} u^{s} ‖}_{L^{\infty}} + {‖ e^{y} \partial_{y} \bar{u} ‖}_{L^{\infty}}) \\ ⩽ C (1 + A (t)) {‖ \bar{u} (t) ‖}_{H^{2, 1}} + C {‖ \bar{u} (t) ‖}_{H^{2, 0}} (1 + A (t)) \\ ⩽ C (1 + A (t)) E {(t)}^{\frac{1}{2}}, \end{array}$ and $\begin{array}{l} {‖ e^{y} \partial_{y} F ‖}_{L^{\infty}} & ⩽ {‖ e^{y} ((u^{s} - 1) \partial_{x} \partial_{y} \bar{u} + \bar{u} \partial_{x} \partial_{y} \bar{u} + \bar{v} \partial_{y}^{2} u^{s} + \bar{v} \partial_{y}^{2} \bar{u}) ‖}_{L^{\infty}} \\ ⩽ ({‖ e^{y} (u^{s} - 1) ‖}_{L^{\infty}} + {‖ e^{y} \bar{u} ‖}_{L^{\infty}}) ‖ \partial_{x} \partial_{y} \bar{u} ‖_{L^{\infty}} + ‖ \bar{v} ‖_{L^{\infty}} ({‖ e^{y} \partial_{y}^{2} u^{s} ‖}_{L^{\infty}} + {‖ e^{y} \partial_{y}^{2} \bar{u} ‖}_{L^{\infty}}) . \end{array}$ By Sobolev embedding and the interpolation, we have $\begin{array}{l} ‖ \partial_{x} \partial_{y} \bar{u} ‖_{L^{\infty}} & ⩽ C ‖ \partial_{x} \partial_{y} \tilde{u} ‖_{L_{x}^{2} H_{y}^{1}}^{\frac{1}{2}} ‖ \partial_{x} \partial_{y} \tilde{u} ‖_{H_{x}^{1} H_{y}^{1}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{x} \partial_{y} \tilde{u} ‖_{L_{x}^{2} H_{y}^{1}}^{\frac{1}{2}} ‖ \partial_{x} \partial_{y} \tilde{u} ‖_{H_{x}^{2} L_{y}^{2}}^{\frac{1}{4}} ‖ \partial_{x} \partial_{y} \tilde{u} ‖_{L_{x}^{2} H_{y}^{2}}^{\frac{1}{4}} \\ ⩽ ‖ \tilde{u} ‖_{H_{μ}^{1, 2}}^{\frac{1}{2}} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{3, 0}}^{\frac{1}{4}} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 2}}^{\frac{1}{4}} \\ ⩽ E {(t)}^{\frac{3}{8}} D {(t)}^{\frac{1}{8}} . \end{array}$ Then we infer that $\begin{array}{l} {‖ e^{y} \partial_{y} F ‖}_{L^{\infty}} ⩽ C (1 + A (t)) (E {(t)}^{\frac{3}{8}} D {(t)}^{\frac{1}{8}} + E {(t)}^{\frac{1}{2}}) . \end{array}$

A simple calculation yields $\begin{array}{l} \partial_{x} F = \partial_{x} \bar{u} \partial_{x} \bar{u} + (\bar{u} + u^{s} - 1) \partial_{x}^{2} \bar{u} + \partial_{x} \bar{v} \partial_{y} (u^{s} + \bar{u}) + \bar{v} \partial_{y} \partial_{x} \bar{u} . \end{array}$ Then we have $\begin{array}{l} {‖ e^{y} \partial_{x} F ‖}_{L^{\infty}} & ⩽ C (1 + A (t)) (‖ \partial_{x} \bar{u} ‖_{L^{\infty}} + {‖ \partial_{x}^{2} \bar{u} ‖}_{L^{\infty}} + ‖ \partial_{x} \bar{v} ‖_{L^{\infty}}) + C ‖ v ‖_{L^{\infty}} {‖ e^{y} \partial_{x} \partial_{y} \bar{u} ‖}_{L^{\infty}} \\ ⩽ C (1 + A (t)) ‖ \bar{u} ‖_{H_{μ}^{3, 1}} ⩽ C (1 + A (t)) E {(t)}^{\frac{1}{2}} . \end{array}$ This proves the second inequality. □

Now we prove Proposition 2.1. Proof.
The solution of (2.1) could be written as $\begin{array}{l} \bar{u} (t, x, y) = & \frac{1}{2 \sqrt{π t}} e^{- t} \int_{0}^{+ \infty} (e^{- \frac{{(y - y^{'})}^{2}}{4 t}} - e^{- \frac{{(y + y^{'})}^{2}}{4 t}}) {\tilde{u}}_{0} (x, y^{'}) d y^{'} \\ (2.4) & - e^{- t} \int_{0}^{t} \frac{1}{2 \sqrt{π (t - s)}} \int_{0}^{+ \infty} (e^{- \frac{{(y - y^{'})}^{2}}{4 (t - s)}} - e^{- \frac{{(y + y^{'})}^{2}}{4 (t - s)}}) e^{s} F (s, x, y^{'}) d y^{'} ≜ I_{1} + I_{2} . \end{array}$

Using the assumption and ${\tilde{u}}_{0} |_{y = 0} = 0$ , we get by integration by parts that $\begin{array}{l} | e^{y} (\partial_{x} + \partial_{x} \partial_{y} + \sum_{k = 0}^{2} \partial_{y}^{k}) I_{1} (t, x, y) | \\ ⩽ \frac{e^{y}}{2 \sqrt{π t}} e^{- t} \int_{0}^{+ \infty} (e^{- \frac{{(y - y^{'})}^{2}}{4 t}} - e^{- \frac{{(y + y^{'})}^{2}}{4 t}}) \\ \times (| \partial_{x} {\tilde{u}}_{0} (x, y^{'}) | + | \partial_{y} \partial_{x} {\tilde{u}}_{0} (x, y^{'}) | + \sum_{k = 0}^{2} | \partial_{y}^{k} {\tilde{u}}_{0} (x, y^{'}) |) d y^{'} \\ ⩽ \frac{C ε}{2 \sqrt{π t}} e^{y} e^{- t} \int_{0}^{+ \infty} (e^{- \frac{{(y - y^{'})}^{2}}{4 t}} - e^{- \frac{{(y + y^{'})}^{2}}{4 t}}) e^{- y^{'}} d y^{'} \\ ⩽ \frac{C ε}{\sqrt{π t}} e^{y} e^{- t} \int_{0}^{+ \infty} e^{- \frac{{(y - y^{'})}^{2}}{4 t}} e^{- y^{'}} d y^{'} \\ ⩽ C ε e^{y} e^{- t} \int_{- \frac{y}{2 \sqrt{t}}}^{+ \infty} e^{- ξ^{2}} e^{- (2 \sqrt{t} ξ + y)} d ξ ⩽ C ε \int_{- \infty}^{+ \infty} e^{- η^{' 2}} d η^{'} ⩽ C ε . \end{array}$

It follows from Lemma 2.2 that $\begin{array}{l} | e^{y} I_{2} (t, x, y) | & ⩽ e^{y} e^{- t} \int_{0}^{t} \frac{1}{\sqrt{π (t - s)}} \int_{0}^{+ \infty} e^{- \frac{{(y - y^{'})}^{2}}{4 (t - s)}} e^{s} e^{- y^{'}} | e^{y^{'}} F (s, x, y^{'}) | d y^{'} d s \\ ⩽ C e^{y} e^{- t} \int_{0}^{t} {‖ e^{y} F (s) ‖}_{L_{x, y}^{\infty}} e^{s} (\int_{- \frac{y}{2 \sqrt{t - s}}}^{+ \infty} e^{- ξ^{2} - (2 \sqrt{t - s} ξ + y)} d ξ) d s \\ ⩽ C \int_{0}^{t} {‖ e^{y} F (s) ‖}_{L_{x, y}^{\infty}} (\int_{- \infty}^{+ \infty} e^{- ξ^{' 2}} d ξ^{'}) d s \\ ⩽ C \int_{0}^{t} {‖ e^{y} F (s) ‖}_{L_{x, y}^{\infty}} d s ⩽ C (1 + A (t)) \int_{0}^{t} E {(s)}^{\frac{1}{2}} d s . \end{array}$ In a similar way, we have $\begin{array}{l} | e^{y} \partial_{x} I_{2} (t, x, y) | & ⩽ C (1 + A (t)) \int_{0}^{t} E {(s)}^{\frac{1}{2}} d s, \end{array}$ and $\begin{array}{l} | e^{y} \partial_{y} I_{2} (t, x, y) | & ⩽ C e^{y} e^{- t} \int_{0}^{t} \frac{1}{2 \sqrt{π (t - s)}} \int_{0}^{+ \infty} e^{- \frac{{(y - y^{'})}^{2}}{4 (t - s)}} e^{s} e^{- y^{'}} | e^{y^{'}} \partial_{y} F (s, x, y^{'}) | d y^{'} d s \\ ⩽ C \int_{0}^{t} {‖ e^{y} \partial_{y} F (s) ‖}_{L_{x, y}^{\infty}} d s . \end{array}$ Thus, we have $\begin{array}{l} | e^{y} \partial_{y} I_{2} (t, x, y) | & ⩽ C (1 + A (t)) (\int_{0}^{t} E {(s)}^{\frac{3}{8}} e^{- \frac{s}{32}} (e^{\frac{s}{32}} D {(s)}^{\frac{1}{8}}) d s + \int_{0}^{t} E {(s)}^{\frac{1}{2}} d s) \\ ⩽ C (1 + A (t)) (sup_{s \in [0, t]} E {(s)}^{\frac{3}{8}} {(\int_{0}^{t} e^{\frac{s}{4}} D (s) d s)}^{\frac{1}{8}} + \int_{0}^{t} E {(s)}^{\frac{1}{2}} d s) . \end{array}$

A direct calculation gives $\begin{array}{l} | e^{y} \partial_{x} \partial_{y} I_{2} (t, x, y) | \\ ⩽ C e^{y} e^{- t} \int_{0}^{t} \frac{1}{2 \sqrt{π (t - s)}} \int_{0}^{+ \infty} | \partial_{y} e^{- \frac{{(y - y^{'})}^{2}}{4 (t - s)}} - \partial_{y} e^{- \frac{{(y + y^{'})}^{2}}{4 (t - s)}} | e^{s} e^{- y^{'}} | e^{y^{'}} \partial_{x} F (s, x, y^{'}) | d y^{'} \\ ⩽ C e^{y} e^{- t} \int_{0}^{t} {(t - s)}^{- \frac{1}{2}} {‖ e^{y} \partial_{x} F (s) ‖}_{L_{x, y}^{\infty}} e^{s} \\ \times (\int_{- \frac{y}{2 \sqrt{t - s}}}^{+ \infty} | ξ | e^{- ξ^{2} - (2 \sqrt{t - s} ξ + y)} d ξ + \int_{\frac{y}{2 \sqrt{t - s}}}^{\infty} η e^{- η^{2} - (2 \sqrt{t - s} η - y)} d η) d s \\ ⩽ C \int_{0}^{t} {(t - s)}^{- \frac{1}{2}} {‖ e^{y} \partial_{x} F (s) ‖}_{L_{x, y}^{\infty}} \\ \times (\int_{- \frac{y}{2 \sqrt{t - s}} + \sqrt{t - s}}^{+ \infty} (| ξ^{'} | + \sqrt{t - s}) e^{- ξ^{' 2}} d ξ^{'} + e^{2 y} \int_{\frac{y}{2 \sqrt{t - s}} + \sqrt{t - s}}^{+ \infty} (η^{'} - \sqrt{t - s}) e^{- η^{' 2}} d η^{'}) d s \\ ⩽ C \int_{0}^{t} (1 + {(t - s)}^{- \frac{1}{2}}) (1 + e^{2 y} e^{{(\frac{y}{2 \sqrt{t - s}} + \sqrt{t - s})}^{2}}) {‖ e^{y} \partial_{x} F (s) ‖}_{L_{x, y}^{\infty}} d s \\ ⩽ C \int_{0}^{t} (1 + {(t - s)}^{- \frac{1}{2}}) {‖ e^{y} \partial_{x} F (s) ‖}_{L_{x, y}^{\infty}} d s . \end{array}$ Here we use the fact $\int_{a}^{\infty} e^{- z^{2}} d z ⩽ C e^{- a^{2}} (a > 0)$ with C independent of a. Thus, we infer from Lemma 2.2 that $\begin{array}{l} | e^{y} \partial_{x} \partial_{y} I_{2} (t, x, y) | ⩽ & C (1 + A (t)) \int_{0}^{t} (1 + {(t - s)}^{- \frac{1}{2}}) E {(s)}^{\frac{1}{2}} d s \\ ⩽ & C (1 + A (t)) {(\int_{0}^{t} e^{\frac{s}{4}} E {(s)}^{2} d s)}^{\frac{1}{4}} . \end{array}$

Similarly, we have $\begin{array}{l} | e^{y} \partial_{y}^{2} I_{2} (t, x, y) | \\ ⩽ C (1 + A (t)) (\int_{0}^{t} (1 + {(t - s)}^{- \frac{1}{2}}) (E {(s)}^{\frac{3}{8}} D {(s)}^{\frac{1}{8}} + E {(s)}^{\frac{1}{2}}) d s) \\ ⩽ C (1 + A (t)) (sup_{s \in [0, t]} E {(s)}^{\frac{3}{8}} {(\int_{0}^{t} (1 + {(t - s)}^{- \frac{4}{7}}) e^{- \frac{s}{28}} d s)}^{\frac{7}{8}} \\ \times {(\int_{0}^{t} e^{\frac{s}{4}} D (s) d s)}^{\frac{1}{8}} + {(\int_{0}^{t} e^{\frac{s}{4}} E {(s)}^{2} d s)}^{\frac{1}{4}}) \\ ⩽ C (1 + A (t)) (sup_{s \in [0, t]} E {(s)}^{\frac{3}{8}} {(\int_{0}^{t} e^{\frac{s}{4}} D (s) d s)}^{\frac{1}{8}} + {(\int_{0}^{t} e^{\frac{s}{4}} E {(s)}^{2} d s)}^{\frac{1}{4}}) . \end{array}$

In summary, we conclude the proof of the proposition. □

3. High order energy estimate in x variable

In this section, we assume that $\begin{array}{l} (3.1) & A (t) ⩽ C_{1} ε ⩽ \frac{1}{100}, \end{array}$ where $C_{1}$ is a constant determined later. We introduce a weight $ν (y) = e^{- \frac{y}{2}}$ and denote by C a constant independent of T and $C_{1}$ .

3.1. Paralinearization and good unknown

Due to the derivative loss induced by the term $v \partial_{y} \tilde{u}$ , we cannot make the energy estimate to the equation (1.3) directly. Motivated by [1,13], we need to introduce a good unknown. To avoid the Nash–Moser iteration, we will use the paralinearized method.

Using Bony’s decomposition (A.1), we get $\begin{array}{l} \partial_{t} \tilde{u} + T_{u} \partial_{x} \tilde{u} + T_{\partial_{y} u} v - \partial_{y}^{2} \tilde{u} + \tilde{u} = f, \end{array}$ where $\begin{array}{l} (3.2) & f = - R_{\partial_{x} \tilde{u}} \tilde{u} - R_{v} \partial_{y} \tilde{u} . \end{array}$ Notice that there is no derivative loss for the terms in f. We denote $\begin{array}{l} a = {(\partial_{y} u)}^{- 1}, b = \partial_{y} u, c = \frac{\partial_{y}^{2} u}{\partial_{y} u}, \\ \tilde{a} = {(\partial_{y} u)}^{- 1} - {(\partial_{y} u^{s})}^{- 1}, \tilde{b} = \partial_{y} \tilde{u}, \tilde{c} = c + 1 . \end{array}$

To eliminate the trouble term $T_{\partial_{y} u} v$ , we introduce a good unknown w defined by $\begin{array}{l} (3.3) & w = \partial_{y} (T_{{(\partial_{y} u)}^{- 1}} \tilde{u}) = \partial_{y} T_{a} \tilde{u} . \end{array}$ A direct calculation shows that $\begin{matrix} (3.4) & \{\begin{matrix} \partial_{t} w + T_{u} \partial_{x} w - \partial_{y}^{2} w + w = \partial_{y} F_{1} + F_{2}, \\ (\partial_{y} w + F_{1}) |_{y = 0} = 0 and {lim}_{y \to + \infty} w = 0, \\ w |_{t = 0} = w_{0} (x, y), \end{matrix} \end{matrix}$ where $F_{1}$ and $F_{2}$ are given by $\begin{array}{l} F_{1} = T_{(\partial_{t} - \partial_{y}^{2}) a} \tilde{u} - 2 T_{\partial_{y} a} \partial_{y} \tilde{u} - [T_{a}, T_{u}] \partial_{x} \tilde{u} + (T_{a b} - T_{a} T_{b}) v + T_{a} f + T_{u} T_{\partial_{x} a} \tilde{u}, \\ F_{2} = (T_{b a} - T_{b} T_{a}) \partial_{x} \tilde{u} + T_{\partial_{y} u} T_{\partial_{x} a} \tilde{u} . \end{array}$

Using the equation (1.2), we find $\begin{matrix} \partial_{t} a - \partial_{y}^{2} a - 2 \frac{\partial_{y}^{2} u}{\partial_{y} u} \partial_{y} a = \frac{\partial_{y} (F + u)}{{(\partial_{y} u)}^{2}}, \end{matrix}$ where $F = u \partial_{x} \tilde{u} + v \partial_{y} u$ . Then we have $\begin{array}{l} T_{(\partial_{t} - \partial_{y}^{2}) a} \tilde{u} - 2 T_{\partial_{y} a} \partial_{y} \tilde{u} \\ = T_{\frac{\partial_{y} u}{{(\partial_{y} u)}^{2}}} \tilde{u} + T_{\frac{\partial_{y} F}{{(\partial_{y} u)}^{2}}} \tilde{u} + 2 T_{\frac{\partial_{y}^{2} u}{\partial_{y} u} \partial_{y} a} \tilde{u} + 2 T_{\frac{\partial_{y}^{2} u}{\partial_{y} u} a} \partial_{y} \tilde{u} \\ = T_{a} \tilde{u} + T_{\frac{\partial_{y} F}{{(\partial_{y} u)}^{2}}} \tilde{u} + 2 T_{c} w + 2 (T_{c \partial_{y} a} - T_{c} T_{\partial_{y} a}) \tilde{u} + 2 (T_{c a} - T_{c} T_{a}) \partial_{y} \tilde{u} \\ = T_{a} \tilde{u} - 2 w + 2 T_{\tilde{c}} w + T_{\frac{\partial_{y} F}{{(\partial_{y} u)}^{2}}} \tilde{u} + 2 (T_{c \partial_{y} a} - T_{c} T_{\partial_{y} a}) \tilde{u} + 2 (T_{c a} - T_{c} T_{a}) \partial_{y} \tilde{u} . \end{array}$ Then we can write $F_{1}$ as $\begin{array}{l} (3.5) & F_{1} = {\tilde{F}}_{1} - 2 w + T_{a} \tilde{u}, \end{array}$ where $\begin{array}{l} {\tilde{F}}_{1} = & T_{\frac{\tilde{u} \partial_{x} \partial_{y} \tilde{u} + v \partial_{y}^{2} \tilde{u}}{{(\partial_{y} u)}^{2}}} \tilde{u} + 2 T_{\tilde{c}} w + 2 (T_{c \partial_{y} a} - T_{c} T_{\partial_{y} a}) \tilde{u} + 2 (T_{c a} - T_{c} T_{a}) \partial_{y} \tilde{u} \\ - [T_{a}, T_{u}] \partial_{x} \tilde{u} + (T_{a b} - T_{a} T_{b}) v + T_{u} T_{\partial_{x} a} \tilde{u} + T_{a} f \\ ≜ & I_{1} + \dots + I_{8} . \end{array}$ Therefore, $\partial_{y} F_{1} = w - 2 \partial_{y} w + \partial_{y} {\tilde{F}}_{1}$ . Thanks to $T_{a} \tilde{u} |_{y = 0} = 0$ , we can rewrite (3.4) as $\begin{matrix} (3.6) & \{\begin{matrix} \partial_{t} w + T_{u} \partial_{x} w - \partial_{y}^{2} w = \partial_{y} ({\tilde{F}}_{1} - 2 w) + F_{2}, \\ (\partial_{y} w + {\tilde{F}}_{1} - 2 w) |_{y = 0} = 0, and {lim}_{y \to + \infty} w = 0, \\ w |_{t = 0} = w_{0} (x, y) . \end{matrix} \end{matrix}$

3.2. Relationship between w and u

The following facts could be deduced from the definitions of $a, b, c$ .

Lemma 3.1.
It holds that for any $(t, x, y) \in [0, T] \times R_{+}^{2}$ , $\begin{array}{l} \frac{1}{2} e^{- y} ⩽ \partial_{y} u ⩽ 2 e^{- y}, | e^{- y} \partial_{y} a | ⩽ 2, \\ | e^{y} \tilde{b} | ⩽ A (t), | e^{- y} \tilde{a} | + | e^{- y} \partial_{y} \tilde{a} | ⩽ C A (t), | \tilde{c} | ⩽ C A (t) . \end{array}$

The following lemma states the relation between $‖ w ‖_{H_{ν}^{3, 0}}$ and $‖ \tilde{u} ‖_{H_{μ}^{3, 1}}$ . For this, we will use the following formulas: $\begin{array}{l} (3.7) & \tilde{u} = T_{b} \int_{0}^{y} w d y^{'} + (T_{b a} - T_{b} T_{a}) \tilde{u}, \\ (3.8) & \partial_{y} \tilde{u} = T_{b} w + (T_{b a} - T_{b} T_{a}) \partial_{y} \tilde{u} - T_{b} T_{\partial_{y} a} \tilde{u} . \end{array}$ Lemma 3.2.
There exists $ε_{1} > 0$ so that if $ε ⩽ ε_{1}$ , then we have $\begin{array}{l} c_{2} ‖ w ‖_{H_{ν}^{3, 0}} ⩽ ‖ \tilde{u} ‖_{H_{μ}^{3, 1}} & ⩽ C_{2} ‖ w ‖_{H_{ν}^{3, 0}} . \end{array}$ Here $c_{2}, C_{2}$ are constants independent of $T, C_{1}$ .
Proof.
It follows from Lemma A.1 and Lemma 3.1 that $\begin{array}{l} {‖ e^{- \frac{y}{2}} w ‖}_{H^{3, 0}} & ⩽ C {‖ e^{- y} a ‖}_{L^{\infty}} {‖ e^{\frac{y}{2}} \partial_{y} \tilde{u} ‖}_{H^{3, 0}} + C {‖ e^{- y} \partial_{y} a ‖}_{L^{\infty}} {‖ e^{\frac{y}{2}} \tilde{u} ‖}_{H^{3, 0}} \\ ⩽ C ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{3, 0}} + C ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} ⩽ C ‖ \tilde{u} ‖_{H_{μ}^{3, 1}} . \end{array}$

By (3.7), we write $\begin{array}{l} e^{\frac{y}{2}} \tilde{u} & = T_{e^{y} b} \int_{0}^{y} e^{- \frac{y - y^{'}}{2}} (e^{- \frac{y^{'}}{2}} w) d y^{'} + (T_{\tilde{b} \tilde{a}} - T_{\tilde{b}} T_{\tilde{a}}) (e^{\frac{y}{2}} \tilde{u}) . \end{array}$ Here we used the fact that $(T_{b a} - T_{b} T_{a}) = (T_{\tilde{b} \tilde{a}} - T_{\tilde{b}} T_{\tilde{a}})$ . Then by Lemma A.1, Lemma 3.1 and Young’s inequality, we get $\begin{array}{l} {‖ e^{\frac{y}{2}} \tilde{u} ‖}_{H^{3, 0}} & ⩽ C {‖ e^{y} b ‖}_{L^{\infty}} {‖ e^{- \frac{| y |}{2}} ‖}_{L^{1}} {‖ e^{- \frac{y}{2}} w ‖}_{H^{3, 0}} + C {‖ e^{y} \tilde{b} ‖}_{L^{\infty}} {‖ e^{- y} \tilde{a} ‖}_{L^{\infty}} {‖ e^{\frac{y}{2}} \tilde{u} ‖}_{H^{3, 0}} \\ ⩽ C ‖ w ‖_{H_{ν}^{3, 0}} + C A {(t)}^{2} ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} . \end{array}$ Thanks to (3.1), taking ε small enough, we can deduce that $\begin{array}{l} (3.9) & ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} & ⩽ C ‖ w ‖_{H_{ν}^{3, 0}} . \end{array}$

Using (3.8), we get by Lemma A.1 and Lemma 3.1 that $\begin{array}{l} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{3, 0}} ⩽ & C {‖ e^{y} b ‖}_{L^{\infty}} {‖ e^{- \frac{y}{2}} w ‖}_{H^{3, 0}} + C {‖ e^{y} \tilde{b} ‖}_{L^{\infty}} {‖ e^{- y} \tilde{a} ‖}_{L^{\infty}} {‖ e^{\frac{y}{2}} \partial_{y} \tilde{u} ‖}_{H^{3, 0}} \\ + C {‖ e^{y} \tilde{b} ‖}_{L^{\infty}} {‖ e^{- y} \partial_{y} \tilde{a} ‖}_{L^{\infty}} {‖ e^{\frac{y}{2}} \tilde{u} ‖}_{H^{3, 0}} \\ ⩽ & C ‖ w ‖_{H_{ν}^{3, 0}} + C A {(t)}^{2} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{3, 0}} + C A {(t)}^{2} ‖ \tilde{u} ‖_{H_{μ}^{3, 0}}, \end{array}$ which along with (3.1) and (3.9) implies $\begin{array}{l} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{3, 0}} ⩽ C ‖ w ‖_{H_{ν}^{3, 0}} . \end{array}$ This completes the proof of the lemma. □

3.3. Energy estimates

Proposition 3.3.
Let w be a smooth solution of ( 3.4 ). There exists $ε_{2} > 0$ so that if $ε ⩽ ε_{2}$ , then for any $t \in [0, T]$ , $\begin{array}{l} \frac{d}{d t} {‖ w (t) ‖}_{H_{ν}^{3, 0}}^{2} + \frac{1}{2} {‖ w (t) ‖}_{H_{ν}^{3, 0}}^{2} + {‖ \partial_{y} w (t) ‖}_{H_{ν}^{3, 0}}^{2} ⩽ 0 . \end{array}$

We first present the estimates of the source terms $F_{1}, F_{2}$ .
Lemma 3.4.
It holds that for any $t \in [0, T]$ , $\begin{array}{l} {‖ {\tilde{F}}_{1} (t) ‖}_{H_{ν}^{3, 0}} + {‖ F_{2} (t) ‖}_{H_{ν}^{3, 0}} ⩽ C A (t) {‖ w (t) ‖}_{H_{ν}^{3, 0}} . \end{array}$
Proof.
By Lemma A.1 and Lemma 3.1, we get $\begin{array}{l} ‖ I_{1} ‖_{H_{ν}^{3, 0}} + ‖ I_{2} ‖_{H_{ν}^{3, 0}} & ⩽ {‖ e^{- y} \frac{u \partial_{x} \partial_{y} \tilde{u} + v \partial_{y}^{2} u}{{(\partial_{y} u)}^{2}} ‖}_{L^{\infty}} ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} + 2 ‖ \tilde{c} ‖_{L^{\infty}} ‖ w ‖_{H_{ν}^{3, 0}} \\ ⩽ C (1 + A (t)) A (t) ‖ \tilde{u} ‖_{H_{ν}^{3, 0}} + C A (t) ‖ w ‖_{H_{ν}^{3, 0}} . \end{array}$ By Lemma A.1 and Lemma 3.1 again, we have $\begin{array}{l} ‖ I_{3} ‖_{H_{ν}^{3, 0}} + ‖ I_{4} ‖_{H_{ν}^{3, 0}} & ⩽ C ‖ \tilde{c} ‖_{L^{\infty}} ({‖ e^{- y} \partial_{y} \tilde{a} ‖}_{L^{\infty}} + {‖ e^{- y} \tilde{a} ‖}_{L^{\infty}}) (‖ \tilde{u} ‖_{H_{μ}^{3, 0}} + ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{3, 0}}) \\ ⩽ C A {(t)}^{2} ‖ \tilde{u} ‖_{H_{μ}^{3, 1}} . \end{array}$ Similarly, we have $\begin{array}{l} ‖ I_{7} ‖_{H_{ν}^{3, 0}} & ⩽ ‖ u ‖_{L^{\infty}} {‖ e^{- y} \partial_{x} a ‖}_{L^{\infty}} ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} ⩽ C (1 + A (t)) A (t) ‖ \tilde{u} ‖_{H_{μ}^{3, 0}}, \\ ‖ I_{8} ‖_{H_{ν}^{3, 0}} & ⩽ {‖ e^{- y} a ‖}_{L^{\infty}} (‖ \partial_{x} \tilde{u} ‖_{L^{\infty}} ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} + ‖ v ‖_{L^{\infty}} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{3, 0}}) \\ ⩽ C A (t) ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} + C A (t) ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{3, 0}} . \end{array}$ By Lemma A.2 and Lemma 3.1, we have $\begin{array}{l} ‖ I_{5} ‖_{H_{ν}^{3, 0}} + ‖ I_{6} ‖_{H_{ν}^{3, 0}} ⩽ & ‖ \tilde{u} ‖_{W_{x}^{1, \infty} L_{y}^{\infty}} {‖ e^{- y} \tilde{a} ‖}_{W_{x}^{1, \infty} L_{y}^{\infty}} ‖ \partial_{x} \tilde{u} ‖_{H_{μ}^{2, 0}} \\ + {‖ e^{\frac{y}{2}} \partial_{y} \tilde{u} ‖}_{W_{x}^{1, \infty} L_{y}^{2}} {‖ e^{- y} \tilde{a} ‖}_{W_{x}^{1, \infty} L_{y}^{\infty}} ‖ v ‖_{H_{x}^{2} L_{y}^{\infty}} \\ ⩽ & C A {(t)}^{2} ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} . \end{array}$

Summing up the estimates of $I_{1} - I_{8}$ , we get by Lemma 3.2 that $\begin{array}{l} ‖ {\tilde{F}}_{1} ‖_{H_{ν}^{3, 0}} ⩽ C A (t) ‖ w ‖_{H_{ν}^{3, 0}} . \end{array}$ Similarly, we have $\begin{array}{l} ‖ F_{2} ‖_{H_{ν}^{3, 0}} & ⩽ {‖ e^{- y} \tilde{a} ‖}_{W_{x}^{1, \infty} L_{y}^{\infty}} ‖ \tilde{b} ‖_{W_{x}^{1, \infty} L_{y}^{\infty}} ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} + ‖ \partial_{y} u ‖_{L^{\infty}} {‖ e^{- y} \partial_{x} a ‖}_{L^{\infty}} ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} \\ ⩽ C A {(t)}^{2} ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} + C A (t) ‖ \tilde{u} ‖_{H_{μ}^{3, 0}} ⩽ C A (t) ‖ w ‖_{H_{ν}^{3, 0}} . \end{array}$ This finished the proof of the lemma. □

Now we prove Proposition 3.3. Proof.
Taking the $H_{ν}^{3, 0}$ inner product between (3.6) and w, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ w ‖_{H_{ν}^{3, 0}}^{2} + {(- \partial_{y}^{2} w - \partial_{y} ({\tilde{F}}_{1} - 2 w), w)}_{H_{ν}^{3, 0}} = {(- T_{u} \partial_{x} w, w)}_{H_{ν}^{3, 0}} + {(F_{2}, w)}_{H_{ν}^{3, 0}} . \end{array}$ Thanks to $(\partial_{y} w + {\tilde{F}}_{1} - 2 w) |_{y = 0} = 0$ , we get by integration by parts that $\begin{array}{l} {(- \partial_{y}^{2} w - \partial_{y} ({\tilde{F}}_{1} - 2 w), w)}_{H_{ν}^{3, 0}} & = {(- \partial_{y} (\partial_{y} w + {\tilde{F}}_{1} - 2 w), w e^{- y})}_{H^{3, 0}} \\ = {((\partial_{y} w + {\tilde{F}}_{1} - 2 w), \partial_{y} (w e^{- y}))}_{H^{3, 0}} \\ = ((\partial_{y} w - 2 w + {\tilde{F}}_{1}), {(\partial_{y} w - w) e^{- y}))}_{H^{3, 0}} \\ = ‖ \partial_{y} w ‖_{H_{ν}^{3, 0}}^{2} + 2 ‖ w ‖_{H_{ν}^{3, 0}}^{2} - 3 {(\partial_{y} w, w)}_{H_{ν}^{3, 0}} + {({\tilde{F}}_{1}, \partial_{y} w - w)}_{H_{ν}^{3, 0}}, \end{array}$ and $\begin{array}{l} {(\partial_{y} w, w)}_{H_{ν}^{3, 0}} = - \frac{1}{2} ‖ w ‖_{H_{x}^{3}}^{2} |_{y = 0} + \frac{1}{2} ‖ w ‖_{H_{ν}^{3, 0}}^{2} . \end{array}$ This shows that $\begin{array}{l} {(- \partial_{y}^{2} w - \partial_{y} ({\tilde{F}}_{1} - 2 w), w)}_{H_{ν}^{3, 0}} ⩾ ‖ \partial_{y} w ‖_{H_{ν}^{3, 0}}^{2} + \frac{1}{2} ‖ w ‖_{H_{ν}^{3, 0}}^{2} + {({\tilde{F}}_{1}, \partial_{y} w - w)}_{H_{ν}^{3, 0}} . \end{array}$

A simple calculation gives $\begin{array}{l} {(T_{u} \partial_{x} w, w)}_{H_{ν}^{3, 0}} = & \sum_{k = 0}^{3} {(\partial_{x}^{k} T_{u} \partial_{x} w, \partial_{x}^{k} w)}_{L_{ν}^{2}} \\ = & \sum_{k = 0}^{3} {(T_{u} \partial_{x}^{k} \partial_{x} w, \partial_{x}^{k} w)}_{L_{ν}^{2}} + \sum_{k = 1}^{3} {([\partial_{x}^{k}, T_{\tilde{u}}] \partial_{x} w, \partial_{x}^{k} w)}_{L_{ν}^{2}} \\ = & - \frac{1}{2} \sum_{k = 0}^{3} {(T_{\partial_{x} \tilde{u}} \partial_{x}^{k} w, \partial_{x}^{k} w)}_{L_{ν}^{2}} + \frac{1}{2} \sum_{k = 0}^{3} {((T_{u} - T_{u}^{*}) \partial_{x}^{k + 1} w, \partial_{x}^{k} w)}_{L_{ν}^{2}} \\ + \sum_{k = 1}^{3} {([\partial_{x}^{k}, T_{\tilde{u}}] \partial_{x} w, \partial_{x}^{k} w)}_{L_{ν}^{2}} \\ ≜ & D_{1} + D_{2} + D_{3} . \end{array}$ From Lemma A.1 and Lemma A.2, we deduce that for $i = 1, 2, 3$ , $\begin{array}{l} D_{i} ⩽ C \sum_{k = 0}^{3} ‖ \partial_{x} \tilde{u} ‖_{L^{\infty}} {‖ \partial_{x}^{k} w ‖}_{L_{ν}^{2}}^{2} ⩽ C A (t) ‖ w ‖_{H_{ν}^{3, 0}}^{2} . \end{array}$

Summing up, we conclude by Lemma 3.4 that $\begin{array}{l} \frac{d}{d t} ‖ w ‖_{H_{ν}^{3, 0}}^{2} + ‖ w ‖_{H_{ν}^{3, 0}}^{2} + 2 ‖ \partial_{y} w ‖_{H_{ν}^{3, 0}}^{2} ⩽ C A (t) (‖ w ‖_{H_{ν}^{3, 0}}^{2} + ‖ \partial_{y} w ‖_{H_{ν}^{3, 0}}^{2}), \end{array}$ which gives our result by taking ε small enough so that $C A (t) ⩽ C C_{1} ε ⩽ \frac{1}{2}$ . □

4. High order energy estimate in y variable

In Proposition 3.3, we just present the high order derivative estimates of the solution in the x variable. To close the energy estimate, we need to derive the high order derivative estimates in the y variable. For this part, we don’t need to use the monotonicity. Again we make the assumption (3.1).

Proposition 4.1.
Let $\tilde{u}$ be a smooth solution of ( 1.3 ) in $[0, T]$ . It holds that for any $t \in [0, T]$ , $\begin{array}{l} \frac{d}{d t} {‖ \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2} + {‖ \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2} + {‖ \partial_{y} \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2} \\ ⩽ (\frac{1}{8} + C A (t) + C {‖ \tilde{u} (t) ‖}_{H_{μ}^{2, 0}}^{2}) {‖ \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2} + C {‖ \tilde{u} (t) ‖}_{H_{μ}^{2, 1}}^{2} + \frac{1}{4} {‖ \partial_{y} \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2} . \end{array}$ Here C is a constant independent of $C_{1}, T$ .
Proof.
The proof is split into three steps.

Step 1. $H_{μ}^{1, 0}$ estimate.

Taking $H_{μ}^{1, 0}$ inner product between (1.3) and $\tilde{u}$ , we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} - {(\partial_{y}^{2} \tilde{u}, \tilde{u})}_{H_{μ}^{1, 0}} + ‖ \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} = - {(u \partial_{x} \tilde{u}, \tilde{u})}_{H_{μ}^{1, 0}} - {(v \partial_{y} u, \tilde{u})}_{H_{μ}^{1, 0}} . \end{array}$

First of all, we get by integration by parts that $\begin{array}{l} - {(\partial_{y}^{2} \tilde{u}, \tilde{u})}_{H_{μ}^{1, 0}} & = ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} + {(\partial_{y} \tilde{u}, \tilde{u})}_{H_{μ}^{1, 0}} \\ ⩾ \frac{1}{2} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} - \frac{1}{2} ‖ \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} . \end{array}$ By integration by parts again and $\partial_{x} \tilde{u} + \partial_{y} v = 0$ , we get $\begin{array}{l} {(u \partial_{x} \tilde{u}, \tilde{u})}_{H_{μ}^{1, 0}} + {(v \partial_{y} u, \tilde{u})}_{H_{μ}^{1, 0}} = & - \frac{1}{2} {(v \tilde{u}, \tilde{u})}_{L_{μ}^{2}} - \frac{1}{2} {(v \partial_{x} \tilde{u}, \partial_{x} \tilde{u})}_{L_{μ}^{2}} + {(v \partial_{y} u^{s}, \tilde{u})}_{L_{μ}^{2}} \\ + {(\partial_{x} \tilde{u} \partial_{x} \tilde{u}, \partial_{x} \tilde{u})}_{L_{μ}^{2}} + {(\partial_{x} v \partial_{y} u, \partial_{x} \tilde{u})}_{L_{μ}^{2}} . \end{array}$ It is easy to see that $\begin{array}{l} \frac{1}{2} {(v \tilde{u}, \tilde{u})}_{L_{μ}^{2}} + \frac{1}{2} {(v \partial_{x} \tilde{u}, \partial_{x} \tilde{u})}_{L_{μ}^{2}} ⩽ C A (t) ‖ \tilde{u} ‖_{H_{μ}^{1, 0}}^{2}, \\ {(v \partial_{y} u^{s}, \tilde{u})}_{L_{μ}^{2}} ⩽ C ‖ \tilde{u} ‖_{H_{μ}^{1, 0}}^{2}, \\ {(\partial_{x} \tilde{u} \partial_{x} \tilde{u}, \partial_{x} \tilde{u})}_{L_{μ}^{2}} ⩽ C A (t) ‖ \tilde{u} ‖_{H_{μ}^{1, 0}}^{2}, \\ {(\partial_{x} v \partial_{y} u, \partial_{x} \tilde{u})}_{L_{μ}^{2}} ⩽ C (1 + A (t)) ‖ \tilde{u} ‖_{H_{μ}^{2, 0}} ‖ \tilde{u} ‖_{H_{μ}^{1, 0}} . \end{array}$ This shows that $\begin{array}{l} {(u \partial_{x} \tilde{u}, \tilde{u})}_{H_{μ}^{1, 0}} + {(v \partial_{y} u, \tilde{u})}_{H_{μ}^{1, 0}} ⩽ C ‖ \tilde{u} ‖_{H_{μ}^{2, 0}}^{2} . \end{array}$ Thus, we deduce that $\begin{array}{l} (4.1) & \frac{d}{d t} ‖ \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} + ‖ \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} + ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} ⩽ C ‖ \tilde{u} ‖_{H_{μ}^{2, 0}}^{2} . \end{array}$

Step 2. $H_{μ}^{1, 1}$ estimate

We get by taking $\partial_{y}$ to (1.3) that $\begin{array}{l} (4.2) & \partial_{t} \partial_{y} \tilde{u} + u \partial_{x} \partial_{y} \tilde{u} + v \partial_{y}^{2} u - \partial_{y}^{3} \tilde{u} + \partial_{y} \tilde{u} = 0 . \end{array}$ Taking $H_{μ}^{1, 0}$ inner product between (4.2) and $\partial_{y} \tilde{u}$ , we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} - {(\partial_{y}^{3} \tilde{u}, \partial_{y} \tilde{u})}_{H_{μ}^{1, 0}} + ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} = - {(u \partial_{x} \partial_{y} \tilde{u} + v \partial_{y}^{2} u, \partial_{y} \tilde{u})}_{H_{μ}^{1, 0}} . \end{array}$ Due to $\partial_{y}^{2} \tilde{u} |_{y = 0} = 0$ , we get by integration by parts that $\begin{array}{l} - {(\partial_{y}^{3} \tilde{u}, \partial_{y} \tilde{u})}_{H_{μ}^{1, 0}} = {‖ \partial_{y}^{2} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} + {(\partial_{y}^{2} \tilde{u}, \partial_{y} \tilde{u})}_{H_{μ}^{1, 0}} ⩾ \frac{1}{2} {‖ \partial_{y}^{2} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} - \frac{1}{2} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} . \end{array}$ By integration by parts again and $\partial_{x} \tilde{u} + \partial_{y} v = 0$ , we get $\begin{array}{l} {(u \partial_{x} \partial_{y} \tilde{u} + v \partial_{y}^{2} u, \partial_{y} \tilde{u})}_{L_{μ}^{2}} & = - \frac{1}{2} {(v \partial_{y} \tilde{u}, \partial_{y} \tilde{u})}_{L_{μ}^{2}} + {(v \partial_{y}^{2} u^{s}, \partial_{y} \tilde{u})}_{L_{μ}^{2}} \\ ⩽ C A (t) ‖ \partial_{y} \tilde{u} ‖_{L_{μ}^{2}}^{2} + C ‖ \tilde{u} ‖_{H_{μ}^{1, 0}} ‖ \partial_{y} \tilde{u} ‖_{L_{μ}^{2}} ⩽ C ‖ \tilde{u} ‖_{H_{μ}^{1, 1}}^{2}, \end{array}$ and $\begin{array}{l} {(u \partial_{x}^{2} \partial_{y} \tilde{u} + v \partial_{x} \partial_{y}^{2} \tilde{u} + \partial_{x} \tilde{u} \partial_{x} \partial_{y} \tilde{u} + \partial_{x} v \partial_{y}^{2} u, \partial_{x} \partial_{y} \tilde{u})}_{L_{μ}^{2}} \\ = - \frac{1}{2} {(v \partial_{y} \partial_{x} \tilde{u}, \partial_{y} \partial_{x} \tilde{u})}_{L_{μ}^{2}} + {(\partial_{x} \tilde{u} \partial_{x} \partial_{y} \tilde{u} + \partial_{x} v \partial_{y}^{2} u, \partial_{x} \partial_{y} \tilde{u})}_{L_{μ}^{2}} \\ ⩽ C A (t) ‖ \partial_{x} \partial_{y} \tilde{u} ‖_{L_{μ}^{2}}^{2} + C (1 + A (t)) ‖ \partial_{x} \partial_{y} \tilde{u} ‖_{L_{μ}^{2}} ‖ \tilde{u} ‖_{H_{μ}^{2, 0}} ⩽ C ‖ \tilde{u} ‖_{H_{μ}^{2, 1}}^{2} . \end{array}$ Thus, we deduce that $\begin{array}{l} (4.3) & \frac{d}{d t} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} + {‖ \partial_{y}^{2} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} + ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 0}}^{2} ⩽ C ‖ \tilde{u} ‖_{H_{μ}^{2, 1}}^{2} . \end{array}$

Step 3. $H_{μ}^{1, 2}$ estimate

We first take $\partial_{y}^{2}$ to (1.3) to obtain $\begin{array}{l} (4.4) & \partial_{t} \partial_{y}^{2} \tilde{u} + \partial_{y} (u \partial_{x} \partial_{y} \tilde{u} + v \partial_{y}^{2} u) - \partial_{y}^{4} \tilde{u} + \partial_{y}^{2} \tilde{u} = 0 . \end{array}$ Then we take $H_{μ}^{1, 0}$ inner product with $\partial_{y}^{2} \tilde{u}$ to obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ \partial_{y}^{2} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} + {‖ \partial_{y}^{2} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} - {(\partial_{y}^{4} \tilde{u}, \partial_{y}^{2} \tilde{u})}_{H_{μ}^{1, 0}} = - {(\partial_{y} (u \partial_{x} \partial_{y} \tilde{u} + v \partial_{y}^{2} u), \partial_{y}^{2} \tilde{u})}_{H_{μ}^{1, 0}} . \end{array}$ We get by integration by parts and using $\partial_{y}^{2} \tilde{u} |_{y = 0} = 0$ that $\begin{array}{l} - {(\partial_{y}^{4} \tilde{u}, \partial_{y}^{2} \tilde{u})}_{H_{μ}^{1, 0}} ⩾ \frac{1}{2} {‖ \partial_{y}^{3} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} - \frac{1}{2} {‖ \partial_{y}^{2} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} . \end{array}$ Using $\partial_{x} \tilde{u} + \partial_{y} v = 0$ , we get by integration by parts that $\begin{array}{l} {(u \partial_{x} \partial_{y}^{2} \tilde{u} + v \partial_{y}^{3} u + \partial_{y} u \partial_{x} \partial_{y} \tilde{u} + \partial_{y} v \partial_{y}^{2} u, \partial_{y}^{2} \tilde{u})}_{L_{μ}^{2}} \\ = - \frac{1}{2} {(v \partial_{y}^{2} \tilde{u}, \partial_{y}^{2} \tilde{u})}_{L_{μ}^{2}} + {(v \partial_{y}^{3} u^{s} + \partial_{y} u \partial_{x} \partial_{y} \tilde{u} + \partial_{y} v \partial_{y}^{2} u, \partial_{y}^{2} \tilde{u})}_{L_{μ}^{2}} \\ ⩽ C A (t) {‖ \partial_{y}^{2} \tilde{u} ‖}_{L_{μ}^{2}}^{2} + C (1 + A (t)) ‖ \tilde{u} ‖_{H_{μ}^{2, 1}} {‖ \partial_{y}^{2} \tilde{u} ‖}_{L_{μ}^{2}} \\ ⩽ (\frac{1}{32} + C A (t)) {‖ \partial_{y}^{2} \tilde{u} ‖}_{L_{μ}^{2}}^{2} + C ‖ \tilde{u} ‖_{H_{μ}^{2, 1}}^{2} . \end{array}$ Similarly, we have $\begin{array}{l} {(u \partial_{x}^{2} \partial_{y}^{2} \tilde{u} + v \partial_{x} \partial_{y}^{3} \tilde{u} + \partial_{x} v \partial_{y}^{3} u + \partial_{y} u \partial_{x}^{2} \partial_{y} \tilde{u} + \partial_{x} \partial_{y} \tilde{u} \partial_{x} \partial_{y} \tilde{u} + \partial_{x} \partial_{y} v \partial_{y}^{2} u, \partial_{x} \partial_{y}^{2} \tilde{u})}_{L_{μ}^{2}} \\ = - \frac{1}{2} {(v \partial_{x} \partial_{y}^{2} \tilde{u}, \partial_{x} \partial_{y}^{2} \tilde{u})}_{L_{μ}^{2}} + {(\partial_{x} v \partial_{y}^{3} u + \partial_{y} u \partial_{x}^{2} \partial_{y} \tilde{u} + \partial_{x} \partial_{y} \tilde{u} \partial_{x} \partial_{y} \tilde{u} + \partial_{x} \partial_{y} v \partial_{y}^{2} u, \partial_{x} \partial_{y}^{2} \tilde{u})}_{L_{μ}^{2}} \\ ⩽ C A (t) {‖ \partial_{x} \partial_{y}^{2} \tilde{u} ‖}_{L_{μ}^{2}}^{2} + ‖ \partial_{x} v ‖_{L_{x}^{2} L_{y}^{\infty}} ({‖ e^{\frac{y}{2}} \partial_{y}^{3} \tilde{u} ‖}_{L_{x}^{\infty} L_{y}^{2}} + {‖ e^{\frac{y}{2}} \partial_{y}^{3} u^{s} ‖}_{L_{y}^{2}}) {‖ \partial_{x} \partial_{y}^{2} \tilde{u} ‖}_{L_{μ}^{2}} \\ + (1 + A (t)) {‖ \partial_{x}^{2} \partial_{y} \tilde{u} ‖}_{L_{μ}^{2}} {‖ \partial_{x} \partial_{y}^{2} \tilde{u} ‖}_{L_{μ}^{2}} + A (t) ‖ \partial_{x} \partial_{y} \tilde{u} ‖_{L_{μ}^{2}} {‖ \partial_{x} \partial_{y}^{2} \tilde{u} ‖}_{L_{μ}^{2}} \\ + (1 + A (t)) {‖ \partial_{x}^{2} \tilde{u} ‖}_{L_{μ}^{2}} {‖ \partial_{x} \partial_{y}^{2} \tilde{u} ‖}_{L_{μ}^{2}} \\ ⩽ \frac{1}{4} {‖ \partial_{y}^{3} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} + (\frac{1}{32} + C A (t) + C ‖ \tilde{u} ‖_{H_{μ}^{2, 0}}^{2}) {‖ \partial_{x} \partial_{y}^{2} \tilde{u} ‖}_{L_{μ}^{2}}^{2} + C ‖ \tilde{u} ‖_{H_{μ}^{2, 1}}^{2} . \end{array}$ Thus, we deduce that $\begin{array}{l} \frac{d}{d t} {‖ \partial_{y}^{2} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} + {‖ \partial_{y}^{2} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} + {‖ \partial_{y}^{3} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} \\ (4.5) & ⩽ (\frac{1}{8} + C A (t) + C ‖ \tilde{u} ‖_{H_{μ}^{2, 0}}^{2}) {‖ \partial_{y}^{2} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} + C ‖ \tilde{u} ‖_{H_{μ}^{2, 1}}^{2} + \frac{1}{4} {‖ \partial_{y}^{3} \tilde{u} ‖}_{H_{μ}^{1, 0}}^{2} . \end{array}$

Summing up (4.1)–(4.5), we conclude that $\begin{array}{l} \frac{d}{d t} ‖ \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} + ‖ \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} + ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} \\ ⩽ (\frac{1}{8} + C A (t) + C ‖ \tilde{u} ‖_{H_{μ}^{2, 0}}^{2}) ‖ \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} + C ‖ \tilde{u} ‖_{H_{μ}^{2, 1}}^{2} + \frac{1}{4} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} . \end{array}$ This completes the proof. □

5. Proof of Theorem 1.1

The proof of the local well-posedness of the system (1.2) is almost the same as the classical Prandtl system. So, we just use the bootstrap argument to prove that the local solution can be continued to a global one. To this end, we introduce the energy functional $\begin{array}{l} E (t) = {‖ u (t) ‖}_{H_{μ}^{3, 1}}^{2} + {‖ \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2}, D (t) = {‖ \partial_{y} \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2} . \end{array}$

Let us first assume that for any $t \in [0, T]$ , $\begin{array}{l} (5.1) & A (t) ⩽ C_{1} ε ⩽ \frac{1}{100}, \\ (5.2) & e^{\frac{t}{4}} E (t) + \int_{0}^{t} e^{\frac{s}{4}} D (s) d s ⩽ {(C_{2} ε)}^{2}, \end{array}$ where the constants $C_{1}, C_{2}$ are determined later, and $ε ⩽ min (ε_{1}, ε_{2})$ with $ε_{1}, ε_{2}$ given by Lemma 3.2 and Proposition 3.3 respectively.

By Proposition 2.1, (5.1) and (5.2), we have $\begin{array}{l} A (t) ⩽ & C ε + C (1 + A (t)) (\int_{0}^{t} E {(s)}^{\frac{1}{2}} d s + {(\int_{0}^{t} e^{\frac{s}{4}} E {(s)}^{2} d s)}^{\frac{1}{4}}) \\ + C (1 + A (t)) sup_{s \in [0, t]} E {(s)}^{\frac{3}{8}} {(\int_{0}^{t} e^{\frac{s}{4}} D (s) d s)}^{\frac{1}{8}} \\ ⩽ & C (1 + 4 C_{2}) ε . \end{array}$ Taking $C_{1} = 2 C (1 + 4 C_{2}) ε$ , we deduce that for any $t \in [0, T]$ , $\begin{array}{l} (5.3) & A (t) ⩽ \frac{C_{1} ε}{2} . \end{array}$

By Proposition 3.3, we have $\begin{array}{l} \frac{d}{d t} ‖ w ‖_{H_{ν}^{3, 0}}^{2} + \frac{1}{2} ‖ w ‖_{H_{ν}^{3, 0}}^{2} + ‖ \partial_{y} w ‖_{H_{ν}^{3, 0}}^{2} ⩽ 0, \end{array}$ which along with Lemma 3.2 implies $\begin{array}{l} (5.4) & e^{\frac{t}{4}} {‖ u (t) ‖}_{H_{μ}^{3, 1}}^{2} + \int_{0}^{t} e^{\frac{t}{4}} {‖ u (s) ‖}_{H_{μ}^{3, 1}}^{2} d s ⩽ C ‖ {\tilde{u}}_{0} ‖_{H_{μ}^{3, 1}}^{2} t \in [0, T) . \end{array}$

By Proposition 4.1, we have $\begin{array}{l} \frac{d}{d t} ‖ \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} + ‖ \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} + ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} \\ ⩽ (\frac{1}{8} + C A (t) + C ‖ \tilde{u} ‖_{H_{μ}^{2, 0}}^{2}) ‖ \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} + C ‖ \tilde{u} ‖_{H_{μ}^{2, 1}}^{2} + \frac{1}{4} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} . \end{array}$ Using (5.1) and (5.2), we can take ε small enough(if necessary) so that $\begin{array}{l} \frac{d}{d t} ‖ \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} + \frac{1}{2} ‖ \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} + \frac{1}{2} ‖ \partial_{y} \tilde{u} ‖_{H_{μ}^{1, 2}}^{2} ⩽ C ‖ \tilde{u} ‖_{H_{μ}^{2, 1}}^{2}, \end{array}$ which gives $\begin{array}{l} (5.5) & e^{\frac{t}{4}} {‖ \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2} + \int_{0}^{t} e^{\frac{s}{4}} {‖ \tilde{u} (s) ‖}_{H_{μ}^{1, 3}}^{2} d s ⩽ ‖ {\tilde{u}}_{0} ‖_{H_{μ}^{1, 2}}^{2} + C \int_{0}^{t} e^{\frac{s}{4}} {‖ \tilde{u} (s) ‖}_{H_{μ}^{2, 1}}^{2} d s . \end{array}$

It follows from (5.4) and (5.5) that $\begin{array}{l} e^{\frac{t}{4}} ({‖ \tilde{u} (t) ‖}_{H_{μ}^{3, 1}}^{2} + {‖ \tilde{u} (t) ‖}_{H_{μ}^{1, 2}}^{2}) + \int_{0}^{t} e^{\frac{s}{4}} {‖ \tilde{u} (s) ‖}_{H_{μ}^{1, 3}}^{2} d s \\ ⩽ C ‖ {\tilde{u}}_{0} ‖_{H_{μ}^{3, 1}}^{2} + ‖ {\tilde{u}}_{0} ‖_{H_{μ}^{1, 2}}^{2} ⩽ {(\sqrt{1 + C} ε)}^{2} . \end{array}$ Note that here C is independent of $C_{1}$ . So, we can take $C_{2} = 4 \sqrt{1 + C}$ so that $\begin{array}{l} (5.6) & e^{\frac{t}{4}} E (t) + \int_{0}^{t} e^{\frac{s}{4}} D (s) d s ⩽ \frac{{(C_{2} ε)}^{2}}{2} . \end{array}$

Now, (5.3) and (5.6) ensure that maximal existence time $T^{*} = + \infty$ of the solution and there hold (5.1) and (5.2) for any $t \in [0, + \infty)$ .

Footnotes

Paraproduct

We introduce the paraproduct decomposition in R. Choose two smooth functions $χ (τ)$ and $φ (τ)$ , which satisfy $\begin{array}{l} supp φ \subset {τ \in R : \frac{3}{4} ⩽ | τ | ⩽ \frac{8}{3}}, supp χ \subset {τ \in R : | τ | ⩽ \frac{4}{3}}, \end{array}$ and for any $τ \in R$ , $\begin{array}{l} χ (τ) + \sum_{j ⩾ 0} φ (2^{- j} τ) = 1 . \end{array}$ Then we define $\begin{aligned} Δ_{j} f = F^{- 1} (φ (2^{- j} ξ) \hat{f}), S_{j} f = F^{- 1} (χ (2^{- j} ξ) \hat{f}) for j ⩾ 0, \\ Δ_{- 1} f = S_{0} f, S_{j} f = S_{0} f for j < 0 . \end{aligned}$ The Bony’s paraproduct $T_{f} g$ is defined by $\begin{array}{l} T_{f} g = \sum_{j ⩾ - 1} S_{j - 1} f Δ_{j} g . \end{array}$ Then we have the following Bony’s decomposition $\begin{array}{l} (A.1) & f g = T_{f} g + R_{g} f, \end{array}$ where the remainder term $R_{g} f$ is defined by $\begin{array}{l} R_{g} f = \sum_{j ⩾ 0} Δ_{j} f S_{1} g + \sum_{j ⩾ 1, j^{'} ⩾ j - 1} Δ_{j^{'}} f Δ_{j} g . \end{array}$

Let us recall the classical paraproduct estimate and paraproduct calculus in Sobolev space [2]. We denote by $W^{s, p}$ the usual Sobolev spaces.

Acknowledgements

Z. Zhang is partially supported by NSF of China under Grant 11425103.

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