Well-posedness of strong solutions for the Vlasov equation coupled to non-Newtonian fluids in dimension three

TL;DR

This paper proves local and global well-posedness of strong solutions for a coupled Vlasov and non-Newtonian fluid system in three dimensions, under certain initial data conditions.

Contribution

It establishes the existence, uniqueness, and regularity of solutions for the coupled system, including global solutions for small initial data.

Findings

Local well-posedness of strong solutions

Global existence for small initial data

Conditions for regularity and uniqueness

Abstract

We consider the Cauchy problem for coupled system of Vlasov and non-Newtonian fluid equations. We establish local well--posedness of the strong solutions, provided that the initial data are regular enough. Global existence of unique strong solutions for any given time interval is shown as well if the initial data are sufficiently small.

Equations

$$\displaystyle\begin{aligned} &\partial_{t}f+v\cdot\nabla_{x}f+\nabla_{v}\cdot[(u-v)f]=0,\\ &\partial_{t}u-\nabla\cdot\big{(}G[|Du|^{2}]Du\big{)}+(u\cdot\nabla)u+\nabla p=-\int_{{\mathbb{R}}^{3}}(u-v)fdv,\\ &\mbox{div}\ u=0,\end{aligned}$$

$$~{}~{}Du=D_{ij}u:=\frac{1}{2}\Big{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Big{)},\ i,j=1,2,3.$$

$$f(x,v,0)=f_0(x,v),u(x,0)=u_0(x),x,v\in {\mathbb{R}}^3,$$

$$G[s]\ge m_0,G[s]+2G^{{}^{\prime }}[s]s\ge m_0,$$

$$|G^{(k)}[s]s^{\alpha }|\le C_k|G^{(k-1)}[s]|,\alpha \in \{0,1\},$$

$$G[s]=(m_0^{\frac{2}{q-2}}+s{)}^{\frac{q-2}{2}},2<q<\infty ,$$

$$G[s]=m_0+(\sigma +s{)}^{\frac{q-2}{2}},1<q<\infty ,\sigma >0.$$

$${\partial }_{t}f+v\cdot {\nabla }_{x}f+{\nabla }_v\cdot [\left(F_{CS}(f)+(u-v)\right)f]=0,$$

$${\partial }_{t}u-\nabla \cdot \left(G[|Du{|}^2]Du\right)+(u\cdot \nabla )u+\nabla p=-{\int }_{{\mathbb{R}}^3}(u-v)fdv,$$

$$F_{CS}(f)(t,x,v)={\iint }_{{\mathbb{T}}^3\times {\mathbb{R}}^3}(w-v)\psi (|x-y|)f(t,y,w)dydw$$

$$G[s]s\ge C({\mu }_{0}s+s^{\frac{q}{2}}),\left|G[s]\right|\le C({\mu }_0+s{)}^{\frac{q-2}{2}},{\mu }_0>0,$$

$$\left(G[|Dv{|}^2]Dv-G[|Dw{|}^2]Dw:Dv-Dw\right)\ge C\left({\mu }_0|Dv-Dw{|}^2+|Dv-Dw{|}^q\right),$$

$$\left(G[s]+G^{\prime }[s]s\right)s\approx ({\mu }_0+s{)}^{\frac{q}{2}}.$$

$$f\in L^{\infty }\left((0,T);L^1\cap L^{\infty }({\mathbb{T}}^3\times {\mathbb{R}}^3)\right),|v{|}^{2}f\in L^{\infty }\left([0,T];L^1({\mathbb{T}}^3\times {\mathbb{R}}^3)\right),$$

$$u\in C\left([0,T];L^2({\mathbb{T}}^3\right))\cap L^{\infty }\left(0,T;W^{1,2}({\mathbb{T}}^3)\right)\cap L^2\left(0,T;W^{2,2}({\mathbb{T}}^3)\right),$$

$$\nabla u\in L^{\infty }\left(0,T;L^q({\mathbb{T}}^3)\right)\cap L^q\left(0,T;L^{3q}({\mathbb{T}}^3)\right),u_t\in L^2\left(0,T;L^2({\mathbb{T}}^3)\right).$$

$$f\in L^{\infty }\left((0,T);L^1\cap L^{\infty }({\mathbb{T}}^3\times {\mathbb{R}}^3)\right),|v{|}^{2}f\in L^{\infty }\left([0,T];L^1({\mathbb{T}}^3\times {\mathbb{R}}^3)\right),$$

$$u\in L^{\infty }\left([0,T];L^2({\mathbb{T}}^3)\right)\cap L^q\left(0,T;W^{1,q}({\mathbb{T}}^3)\right),$$

$$L^2\left(0,T;W^{1,2}({\mathbb{T}}^3)\right),\text{mbox}if{\mu }_0>0.$$

$$\displaystyle\begin{aligned} &(i)~{}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}f_{0}\,dv\,dx=1,\quad\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}f_{0}\,dv\,dx<\infty,\quad f_{0}\geq 0.\cr&(ii)~{}\mbox{supp}_{v}f_{0}~{}\mbox{is bounded in $\mathbb{R}^{3}$ for each $x\in\mathbb{R}^{3}$}.\end{aligned}$$

$$(1+|x{|}^2+|v{|}^2)f_0(x,v)\in L^1({\mathbb{R}}^3\times {\mathbb{R}}^3),u_0\in H^3({\mathbb{R}}^3).$$

$$(1+|v{|}^p){\nabla }_v^{\alpha }{\nabla }_x^{\beta }{f}_0\in L_v^2{L}_x^2({\mathbb{R}}^3\times {\mathbb{R}}^3),0\le |\alpha |+|\beta |\le 2.$$

$$\left\{\begin{array}[]{ll}\|\varphi\|_{L^{q}(0,T;X)}:=\Big{(}\int_{0}^{T}\left\|\varphi(t)\right\|^{q}dt\Big{)}^{\frac{1}{q}}<\infty,\hskip 42.67912pt\mbox{if}\quad 1\leq q<\infty,\\ \vspace{-3mm}\\ \|\varphi\|_{L^{\infty}(0,T;X)}:=\displaystyle\sup_{t\in(0,T)}\left\|\varphi(t)\right\|<\infty,\quad\quad\ \mbox{if}\quad q=\infty.\\ \end{array}\right.$$

$$A:B=\sum _{i,j=1}^3{a}_{ij}{b}_{ij},\nabla A:\nabla B=\sum _{i,j=1}^3\nabla a_{ij}\cdot \nabla b_{ij},{\nabla }^{2}A:{\nabla }^{2}B=\sum _{i,j=1}^3{\nabla }^2{a}_{ij}:{\nabla }^2{b}_{ij}.$$

$${\mathcal{B}}_0\hookrightarrow \hookrightarrow \mathcal{B}\hookrightarrow {\mathcal{B}}_1,$$

$$W:=\{v\in L^{p_0}(I;{\mathcal{B}}_0)\text{with}\frac{dv}{dt}\in L^{p_1}(I;{\mathcal{B}}_1)\}$$

$$W\hookrightarrow \hookrightarrow L^{p_0}(I;\mathcal{B}).$$

$$\partial_{x_{\sigma_{l}(l)}}\partial_{x_{\sigma_{l}(l-1)}}\cdots\partial_{x_{\sigma_{l}(1)}}G\left[|Du|^{2}\right]=2\big{(}G^{\prime}\left[|Du|^{2}\right]Du:\partial_{x_{\sigma_{l}(l)}}\partial_{x_{\sigma_{l}(l-1)}}\cdots\partial_{x_{\sigma_{l}(1)}}Du\big{)}+E_{l},$$

$$E_{l}=2\big{(}\partial_{x_{\sigma(l)}}(G^{{}^{\prime}}[|Du|^{2}]Du):\partial^{l-1}Du\big{)}+\partial_{x_{\sigma(l)}}E_{l-1},\qquad\quad E_{1}=0,$$

$$|E_2|$$

$$|E_3|$$

Full text

1. Introduction

We study the following coupled system of Vlasov and non-Newtonian fluid equations in phase space ${\mathbb{R}}^{3}\times{\mathbb{R}}^{3}$:

[TABLE]

where $Du$ is the symmetric part of the velocity gradient, namely,

[TABLE]

In (LABEL:main), we indicate by $u:{\mathbb{R}}^{3}\times(0,\,T)\rightarrow{\mathbb{R}}^{3}$, $p:{\mathbb{R}}^{3}\times(0,\,T)\rightarrow{\mathbb{R}}$ and $f:{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}\times(0,\,T)\rightarrow{\mathbb{R}}$ the flow velocity vector, the scalar pressure and the density function of particles, respectively. In this article, we study the Cauchy problem of (LABEL:main) with

[TABLE]

and in particular, $u_{0}$ holds the compatibility condition, that is, $\mbox{div}\ u_{0}=0$.

Here, we make some assumptions on the viscous part of the stress tensor, $G[|Du|^{2}]$.

Assumption 1.1**.**

We suppose that $G:[0,\infty)\rightarrow[0,\infty)$ is a smooth function and satisfies the following the structure conditions: There is a positive constant $m_{0}$ such that for any $s\in[0,\infty)$

[TABLE]

where $G^{(k)}[\cdot]$ is the $k-$th derivative of $G$, and $m_{0}$ and $C_{k}$ are positive constants.

We remark that some typical types of $G[s]$ satisfying Assumption 1.1 are of the following power-law models, e.g.

[TABLE]

or

[TABLE]

We recall some known results for the case $G[s]=1$, namely, the fluid equations is of Newtonian case. Hamdache [10] proved the global existence of weak solutions to the time dependent Stokes system coupled with the Vlasov equation in a bounded domain. Later, existence of weak solution was extended to the Vlasov-Navier-Stokes system by Boudin et al. [5] in a periodic domain (refer also to [7, 8] for hydrodynamic limit problems). When the fluid is inviscid, Baranger and Desvillettes established the local existence of solutions to the compressible Vlasov-Euler equations [4].

In case of non-Newtonian fluid, recently, Mucha et. al. [15] investigated the Cucker–Smale flocking model coupled with an incompressible viscous generalized Navier-Stokes equations with $G$ given in (1.4), for $q\geq\frac{11}{5}$ in a periodic spatial domain $\mathbb{T}^{3}$. To be more precise, the following equations are considered:

[TABLE]

[TABLE]

where

[TABLE]

and $\psi(\cdot)$ the communication weight is positive, decreasing and $\|\psi\|_{C^{1}}<\infty$.

The viscosity part of stress tensor $G$ in [15] was assumed to satisfy the structure conditions, which are given as

[TABLE]

[TABLE]

[TABLE]

Under the assumption that a nonnegative $f_{0}\in(L^{\infty}\cap L^{1})(\mathbb{T}^{3}\times{\mathbb{R}}^{3})$ with compact support and $u_{0}\in W^{1,2}(\mathbb{T}^{3})$, existence of weak solutions was established for the system (1.5)-(1.6) in the class (see [15, Theorem 2.1])

[TABLE]

[TABLE]

[TABLE]

It was also shown via a Lyapunov functional approach that convergence of flocking states is made exponentially fast in time as well.

The second and third authors with their collaborators also proved, independently, global–in–time existence of weak solutions for the system (1.5)-(1.6) with power-law types of $G$ including the degenerate case i.e. $G[s]=(\mu_{0}+s)^{\frac{q-2}{2}}$ with $\mu_{0}\geq 0$ and $q>2$. More precisely, if the initial data $f_{0}\in L^{\infty}\cap L^{1}(\mathbb{T}^{3}\times{\mathbb{R}}^{3})$ with the compact support and $u_{0}\in L^{2}(\mathbb{T}^{3})$, weak solutions exist in the class

[TABLE]

[TABLE]

[TABLE]

As following almost the same arguments in [9], we also can establish existence of global-in-time weak solutions to (LABEL:main)

• (1.2), provided that $G$ satisfies structure conditions (1.3), (1.7) and (1.8) with $q>2$. Since its verification is rather tedious repetitions of the arguments in [9], we just give the statement without proof.

Theorem 1.2**.**

Let $T>0$. Suppose that initial data $f_{0}\in(L^{1}\cap L^{\infty})(\mathbb{R}^{3}\times\mathbb{R}^{3})$ and $u_{0}\in L^{2}(\mathbb{R}^{3})$ satisfy

[TABLE]

Assume further that $G$ satisfies (1.3), (1.7) and (1.8) with $q>2$. Then, there exists a global weak solution $(f,u)$ in the class (1.10)-(1.12) to the equations (LABEL:main)-(1.2).

Our main objective is to prove existence of strong solutions for the system (LABEL:main)-(1.2) where strong solutions are defined below (see Definition 1.3). More precisely, we establish the well-posedness of strong solutions locally in time (Theorem 1.4).

One of main tools is to use Schauder fixed point theorem via the weighted estimate for $f$ (Lemma 3.2 below). In particular, we use the optimal transport technique to show the stability of solutions to the Vlasov equation with respect to the velocity vector $u$ as well as the existence of solutions to the Vlasov equation. To prove the stability of solutions, we borrow the idea of the paper of Han-Kwan et. al. [11]. However, unlike [11], we exploit estimates on the Wasserstein distance instead of Loeper’s functional.

We introduce the notion of strong solutions to (LABEL:main)-(1.2).

Definition 1.3**.**

Let $T>0$ and $k\geq 3$. We say that $(f,u)$ is a strong solution to (LABEL:main)-(1.2) if the following conditions are satisfied:

• (i)

$f(x,\xi,t)\geq 0$* for all $(x,\xi,t)\in\mathbb{R}^{3}\times\mathbb{R}^{3}\times(0,T]$. *

• (ii)

$(1+|v|^{k})\nabla^{\alpha}_{v}\nabla_{x}^{\beta}f\in L^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$, where $0\leq|\alpha|+|\beta|\leq 2$.

• (iii)

$u\in C(0,T;H^{3}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{4}(\mathbb{R}^{3}))$.

• (iv)

$(f,u)$* solves the equations (LABEL:main) - (1.2) in the pointwise sense.*

Now, we are ready to state the main result.

Theorem 1.4**.**

Let $k\geq 3$. Suppose that

[TABLE]

Assume further that $f_{0}(x,v)\geq 0$ satisfies for $p>\max\left\{5,\frac{2k+3}{2}\right\}$

[TABLE]

Then, there exists $T_{*}>0$ such that the system (LABEL:main)-(1.2) has a unique strong solution $(f,u)$ on the time-interval $(0,T_{*}]$ in the sense of Definition 1.3.

Remark 1.5**.**

We note that the local time $T^{*}$ in Theorem 1.4 is dependent on the size of data in (1.13) and $\|u_{0}\|_{H^{3}({\mathbb{R}}^{3})}$.

Remark 1.6**.**

One can observe that the assumption (1.13) implies the following:

• $\bullet$

$f_{0}(x,v)\leq\frac{C_{2}}{1+|v|^{p}}$* due to the embedding w.r.t x- variable (see Section 3.1.2).*

• $\bullet$

$(1+|v|^{k})\nabla_{v}^{\alpha}\nabla_{x}^{\beta}f_{0}\in L^{2}({\mathbb{R}}^{3}\times{\mathbb{R}}^{3})$* if $k<(2p-3)/2$ by the Hölder inequality (see Proposition 3.3 below).*

These will be used later in the proof of Theorem 1.4.

Remark 1.7**.**

The novelty of this paper is that, compared to known results, one of main difficulties is of course caused by nonlinear structure of viscous part of fluid, which we successfully controlled somehow. Another improvement is that initial data of Vlasov equation is not assumed to be compactly supported, which makes arguments a bit complicated for solvability of the Vlasov equation. To the knowledge of authors, previous results seem to suppose compactly supported initial data, and we do not know, however, if the decay condition (1.13) is optimal or not.

This paper is organized as follows: In Section 2, we review some preliminary results. Section 3 is devoted to the study of Vlasov equation. In section 4, we prove Theorem 1.4 is presented. In Section 5, the convergence of two strong solutions of the fluid equation involving the drag force is shown.

2. Preliminary

We first introduce some notations. Let $(X,\|\cdot\|)$ be a normed space. By $L^{q}(0,T;X)$, we denote the space of all Bochner measurable functions $\varphi:(0,T)\rightarrow X$ such that

[TABLE]

For $1\leq q\leq\infty$, we mean by $W^{k,q}(\mathbb{R}^{3})$ the usual Sobolev space. In particular, for $q=2$, we write $W^{k,q}(\mathbb{R}^{3})$ as $H^{k}(\mathbb{R}^{3})$. Let $A=(a_{ij})_{i,j=1}^{3}$ and $B=(b_{ij})_{i,j=1}^{3}$ be matrix valued maps, we then denote

[TABLE]

The letter $C$ is used to represent a generic constant, which may change from line to line.

Next we recall the Aubin-Lions Lemma, which will be used for compactness (see e.g. [16]).

Lemma 2.1**.**

Let $\mathcal{B}$ be a Banach space and $\mathcal{B}_{i}$, $i=0,1$, separable, reflexive Banach spaces, and suppose that

[TABLE]

where $\hookrightarrow$ denotes continuous imbedding and $\hookrightarrow\hookrightarrow$ compact imbedding. Let

[TABLE]

for some finite interval $I\subset\mathbb{R}$, where $p_{i}$ satisfying $1<p_{i}<\infty$. Then, we have

[TABLE]

We also remind a version of Schauder’s fixed point theorem.

Lemma 2.2**.**

Let $V$ be a Banach space and $K$ be a nonempty convex closed subset of $V$. If $T:K\rightarrow K$ is a continuous mapping and $T(K)$ is contained in a compact subset of $K$, then $T$ has a fixed point.

Next, we introduce a priori estimate, which is one of key estimates involving third derivatives of $u$ (refer [12, Lemma 2.1]).

Lemma 2.3**.**

Let $l$ be a positive integer, $\tilde{\sigma}_{l}:\{1,2,\cdots,l\}\rightarrow\{1,2,\cdots,l\}$ a permutation of $\{1,2,\cdots,l\}$, and $\pi_{l}$ a mapping from $\{1,2,\cdots,l\}$ to $\{1,2,3\}$. Suppose that $u\in C^{\infty}({\mathbb{R}}^{3})\cap H^{l}({\mathbb{R}}^{3})$. Assume further that $G:[0,\infty)\rightarrow[0,\infty)$ is infinitely differentiable and satisfies properties given in (1.3). Then, the multi-derivative of $G$ can be rewritten as the following decomposition:

[TABLE]

where $\sigma_{l}:=\pi_{l}\circ\tilde{\sigma}_{l}$ and

[TABLE]

where $\partial^{l-1}:=\partial_{x_{\sigma(l-1)}}\cdots\partial_{x_{\sigma(1)}}$. Furthermore, we obtain the following.

$(1)$* $E_{2}$ and $E_{3}$ satisfy*

[TABLE]

$(2)$* For $1\leq\alpha\leq l$*

[TABLE]

[TABLE]

Next, we also introduce a monotonicity property of the viscous part of the stress tensor, which is useful for the uniqueness of strong solutions for fluid equations (see [12, Lemma 2.2]).

Lemma 2.4**.**

Let $v,w\in W^{1,2}(\mathbb{R}^{3})$. Under the Assumptions 1.1 on $G$, we have

[TABLE]

where $m_{0}$ is a positive constant in (1.3).

2.1. Review on Maximal functions

Let us first remind the notion of Maximal functions. For every $g\in L^{1}_{loc}(\mathbb{R}^{3})$, the associated maximal function, denoted $Mg$, is defined by

[TABLE]

where $B_{r}(x)$ is the ball of radius $r$ centered at $x$.

It is well known that if $g\in L^{p}(\mathbb{R}^{3})~{}(1<p\leq\infty)$ then so is $Mg$, and there is a constant $C_{p}>0$ such that

[TABLE]

We will use the following property of the maximal function (see [1, Lemma 3]).

Lemma 2.5**.**

If $g\in W^{1,p}(\mathbb{R}^{3})$ for $p\geq 1$ then, for a.e. $x,y\in\mathbb{R}^{3}$ one has

[TABLE]

2.2. Preliminary results on optimal mass transportation and Wasserstein space

In this subsection, we introduce the Wasserstein space and remind some properties of it. For more detail, readers may refer [2].

Definition 2.6**.**

Let $\mu$ be a probability measure on $\mathbb{R}^{d}$ and $T:\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$ a measurable map. Then, $T$ induces a probability measure $\nu$ on $\mathbb{R}^{d}$ which is defined as

[TABLE]

We denote $\nu:=T_{\#}\mu$ and say that $\nu$ is the push-forward of $\mu$ by $T$.

Definition 2.7**.**

Let us denote by $\mathcal{P}_{2}(\mathbb{R}^{d})$ the set of all Borel probability measures on $\mathbb{R}^{d}$ with a finite second moment. For $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, we consider

[TABLE]

where $\Gamma(\mu,\nu)$ denotes the set of all Borel probability measures on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ which has $\mu$ and $\nu$ as marginals, i.e.

[TABLE]

for every Borel set $A\subset\mathbb{R}^{d}.$

Equation (2.18) defines a distance on $\mathcal{P}_{2}(\mathbb{R}^{d})$ which is called the Wasserstein distance. Equipped with the Wasserstein distance, $\mathcal{P}_{2}(\mathbb{R}^{d})$ is called the Wasserstein space. It is known that the infimum in the right hand side of Equation (2.18) always achieved. We will denote by $\Gamma_{o}(\mu,\nu)$ the set of all $\gamma$ which minimize the expression.

Definition 2.8**.**

Let $\sigma:[a,b]\mapsto\mathcal{P}_{2}(\mathbb{R}^{d})$ be a curve. We say that $\sigma$ is absolutely continuous and denote it by $\sigma\in AC_{2}(a,b;\mathcal{P}_{2}(\mathbb{R}^{d}))$, if there exists $m\in L^{2}([a,b])$ such that

[TABLE]

If $\sigma\in AC_{2}(a,b;\mathcal{P}_{2}(\mathbb{R}^{d}))$, then the limit

[TABLE]

exists for $L^{1}$-a.e $t\in[a,b]$. Moreover, the function $|\sigma^{\prime}|$ belongs to $L^{2}(a,b)$ and satisfies

[TABLE]

for any $m$ satisfying (2.19). We call $|\sigma^{\prime}|$ by the metric derivative of $\sigma$.

Lemma 2.9** ([2], Theorem 8.3.1).**

If $\sigma\in AC_{2}(a,b;\mathcal{P}_{2}(\mathbb{R}^{d}))$, then there exists a Borel vector field $v:\mathbb{R}^{d}\times(a,b)\mapsto\mathbb{R}^{d}$ such that

[TABLE]

and the continuity equation

[TABLE]

holds in the sense of distribution sense.

Conversely, if a weak continuous curve $\sigma:[a,b]\mapsto\mathcal{P}_{2}(\mathbb{R}^{d})$ satisfies the the continuity equation for some Borel vector field $v_{t}$ with $||v_{t}||_{L^{2}(\sigma_{t})}\in L^{2}(a,b)$ then $\sigma:[a,b]\mapsto\mathcal{P}_{2}(\mathbb{R}^{d})$ is absolutely continuous and $|\sigma_{t}^{\prime}|\leq||v_{t}||_{L^{2}(\sigma_{t})}$ for $L^{1}$-a.e $t\in[a,b]$.*

Notation : In Lemma 2.9, we use notation $v_{t}:=v(\cdot,t)$ and $\sigma_{t}:=\sigma(t)$. Throughout this paper, we keep this convention, unless any confusion is to be expected, and a usual notation $\partial_{t}$ is adopted for temporal derivative, i. e. $f_{t}:=f(\cdot,t)$ and $\partial_{t}f:=\frac{\partial f}{\partial t}$.

Lemma 2.10**.**

For $i=1,~{}2$, let $\sigma^{i}\in AC_{2}(a,b;\mathcal{P}_{2}(\mathbb{R}^{d}))$ and $\sigma^{i}$ satisfies

[TABLE]

We set

[TABLE]

Then, we have $Q\in W^{1,2}(a,b)$ and, for a.e $t\in(a,b)$

[TABLE]

Proof.

Refer to Lemma 2.4.1 of [14]. ∎

3. Vlasov equation

In this section, for given $\bar{u}\in L^{\infty}\left(0,T;H^{3}(\mathbb{R}^{3})\right)\cap L^{2}\left(0,T;H^{4}(\mathbb{R}^{3})\right)$, we consider the following linearized system of the Vlasov type equation of (LABEL:main):

[TABLE]

which requires initial condition $f(x,v,0)=f_{0}(x,v)$.

3.1. Solutions of Vlasov Equation as a curve in Wasserstein space

3.1.1. ODE

For given $\bar{u}\in L^{\infty}(0,T;H^{3}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{4}(\mathbb{R}^{3}))$, we define a vector field $\Xi:[0,T]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}\times\mathbb{R}^{3}$ by

[TABLE]

We also define a flow map $\Phi:[0,T]\times[0,T]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\mapsto\mathbb{R}^{3}\times\mathbb{R}^{3}$ corresponding to the vector field $\Xi$ by

[TABLE]

where

[TABLE]

We note

[TABLE]

for $|V|\geq|\bar{u}|$. Hence, if $|v|\geq\|\bar{u}\|_{L^{\infty}(0,T:L^{\infty}(\mathbb{R}^{3}))}$ then $|V(t;s,x,v)|\leq|v|$ for all $(t,x)\in[s,T]\times\mathbb{R}^{3}$. Let $M$ be a positive number such that

[TABLE]

Then we know $\left\|\bar{u}\right\|_{L^{\infty}(0,T:L^{\infty}(\mathbb{R}^{3}))}\leq C_{1}M$ for some $C_{1}>0$ . Hence, we have

[TABLE]

This implies

[TABLE]

3.1.2. Flow map generating a solution of Vlasov Equation

Let $f_{0}\in L^{1}(\mathbb{R}^{3}\times\mathbb{R}^{3})$ be a nonnegative function and we may assume $\|f_{0}\|_{L^{1}}=1$ without loss of generality. That is, $f_{0}\in\mathcal{P}(\mathbb{R}^{3}\times\mathbb{R}^{3})$. We define $f:[0,T]\mapsto\mathcal{P}(\mathbb{R}^{3}\times\mathbb{R}^{3})$ by

[TABLE]

Then, $f$ is the unique solution of (refer to [2])

[TABLE]

We note that the change of variable formula combined with (3.23) gives

[TABLE]

We also see that (3.24) can be written as

[TABLE]

Suppose $f_{0}$ satisfies (1.13), that is there exists $C_{2}>0$ such that

[TABLE]

Then we have

[TABLE]

First of all, we note that if $|v|\geq C_{1}M\geq\|\bar{u}\|_{L^{\infty}(0,T;L^{\infty})}$, then $|V(0;t,x,v)|\geq|v|$. Hence,

[TABLE]

On the other hand, we get

[TABLE]

Combining (3.27), (3.28) and (3.29), we have

[TABLE]

where

[TABLE]

That is $\rho_{f}\in L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{3}))$. Similarly, we have $m_{2}f\in L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{3}))$ and

[TABLE]

where

[TABLE]

Suppose $f_{0}\in\mathcal{P}_{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$, that is

[TABLE]

Then, we have

[TABLE]

where we exploit (3.21) and (3.22) in the first inequality. This says that $t\mapsto f_{t}$ is a curve in the Wasserstein space $\mathcal{P}_{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$. Exploiting Lemma 2.9, we can show that it is an absolutely continuous curve in $\mathcal{P}_{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$ as follows:

[TABLE]

which implies

[TABLE]

For convenience, we set $f_{t}:=f(t)$ and thus, we have $t\mapsto f_{t}\in AC_{2}(0,T;\mathcal{P}_{2}(\mathbb{R}^{3}\times\mathbb{R}^{3}))$.

3.1.3. Estimation of Wasserstein distance

Compared to [11] in which authors used Loeper’s functional to estimate the distance between two solutions to the Vlasov equation, we estimate the Wasserstein distance between of two solutions to the Vlasov equation.

Lemma 3.1**.**

Let $\bar{u}_{i}\in L^{\infty}(0,T;H^{3}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{4}(\mathbb{R}^{3}))$ for $i=1,2$. Suppose that $f_{i}$ is a solution of the following equation associated with $\bar{u}_{i}$:

[TABLE]

where $f_{0}\in\mathcal{P}_{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$ is given. We set $Q(t):=\frac{1}{2}W_{2}^{2}(f_{1}(t),f_{2}(t))$. Then, we have

[TABLE]

Here, $\rho_{i}(t,x):=\int_{\mathbb{R}^{3}}f_{i}(t,x,v)dv$ for $i=1,2$.

Proof.

From Lemma 2.10, we have

[TABLE]

[TABLE]

where $d\gamma_{t}\in\Gamma_{o}(f_{1}(t),f_{2}(t))$. Hence, we have

[TABLE]

Exploiting Gronwall’s inequality to (3.32) with $Q(0)=0$, we have

[TABLE]

where

[TABLE]

This completes the proof. ∎

3.2. Some estimates on the Vlasov equation

In this subsection, we present the proof of solvability of the linear equation (3.20) and provide some estimates in a weighted Lebesgue spaces. In the next lemma, we consider the case that $f_{0}$ is compactly supported and smooth.

Lemma 3.2**.**

Let $T>0$, $k\in\mathbb{N}$ and $f_{0}\in C^{\infty}_{c}({\mathbb{R}}^{3}\times{\mathbb{R}}^{3})$ and $\bar{u}\in L^{\infty}(0;T;(H^{3}\cap C^{\infty})(\mathbb{R}^{3}))\cap L^{2}(0;T;(H^{4}\cap C^{\infty})(\mathbb{R}^{3}))$. Suppose that $f$ is the solution given in (3.25) to the equations (3.20). Then, we have $(1+|v|^{k})\nabla^{\alpha}_{v}\nabla_{x}^{\beta}f\in L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3}))$ and furthermore,

[TABLE]

[TABLE]

where $C$ is a constant depending only on $k$.

Proof.

Due to the assumptions of $f_{0}$ and $\bar{u}$, it is well-known that $f$ has compact support and smooth with respect to $(x,v)$ variables (see Section 3.1.1–3.1.2). Therefore, it suffices to prove the estimate (3.33). Indeed, we observe first that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

On the other hand, we compute that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $C$ is a constant depending on $k$.

From the estimate (3.34) and (3.35), it follows that

[TABLE]

[TABLE]

Next, testing (3.20) by $(1+|v|^{2k})f$ and integrating over $\mathbb{R}^{3}\times\mathbb{R}^{3}$ with the estimate (3.36), we obtain

[TABLE]

[TABLE]

Taking the differential operator $\nabla_{x}$ to (3.20), the equation can be rewritten as

[TABLE]

Testing (3.38) by $(1+|v|^{2k})\partial_{x}f$, we obtain

[TABLE]

[TABLE]

Exploiting (3.36), we estimate $I_{1}$ as follows:

[TABLE]

Using Hölder and Young’s inequalities, we have

[TABLE]

Hence, combining $I_{1}$ with $I_{2}$, we obtain

[TABLE]

[TABLE]

[TABLE]

where we use Sobolev embedding in last inequality. Again, taking the differential operator $\nabla^{2}_{x}$ to (3.38), it is rewritten as

[TABLE]

Multiplying (3.40) with $(1+|v|^{2k})\nabla^{2}_{x}f$ and integrating over the phase variables, we obtain

[TABLE]

[TABLE]

[TABLE]

Using (3.36), Hölder and Young’s inequalities, the terms $II_{1}$, $II_{2}$ and $II_{3}$ are estimated, respectively, as follows.

[TABLE]

[TABLE]

and

[TABLE]

Combining estimates $II_{1}$–$II_{3}$, we get

[TABLE]

[TABLE]

Taking $\nabla_{v}$ to (3.20), we have

[TABLE]

Again, testing $(1+|v|^{2k})\nabla_{v}f$ and integrating over the phase variables, we obtain

[TABLE]

[TABLE]

[TABLE]

where Young’s inequality is used for last term. Therefore,

[TABLE]

[TABLE]

Taking $\nabla_{v}$ to (3.38), we get

[TABLE]

[TABLE]

Testing (3.44) by $(1+|v|^{2k})\nabla_{v}\nabla_{x}f$ and using integration by parts, we obtain

[TABLE]

[TABLE]

Lastly, taking $\nabla_{v}$ to (3.42), we have

[TABLE]

Testing (3.46) by $(1+|v|^{2k})\partial^{2}_{v}f$ and integrating over the phase variables, by the direct calculations, we obtain

[TABLE]

[TABLE]

Summing up the estimates (3.37), (3.39), (3.41), (3.43), (3.45) and (3.47), we obtain

[TABLE]

[TABLE]

Put X(t):=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(1+|v|^{2k})\Big{(}|f|^{2}+|\nabla_{x}f|^{2}+|\nabla_{v}f|^{2}+|\nabla_{v}\nabla_{x}f|^{2}+|\nabla^{2}_{x}f|^{2}+|\nabla^{2}_{v}f|^{2}\Big{)}\,dvdx. Thus, we have

[TABLE]

Applying Grownwall’s inequality to (3.48) with $\bar{u}\in L^{\infty}(0;T;H^{3}(\mathbb{R}^{3}))\cap L^{2}(0;T;H^{4}(\mathbb{R}^{3}))$, we finally get the desired result, that is,

[TABLE]

[TABLE]

We complete the proof. ∎

Using the Lemma 3.2, we obtain similar results for more general data $f_{0}$ and $\bar{u}$. In particular, compact support of $f_{0}$ is not necessary.

Proposition 3.3**.**

Let $T>0$ and $k\geq 3$. For a given $\bar{u}\in L^{\infty}(0;T;H^{3}(\mathbb{R}^{3}))\cap L^{2}(0;T;H^{4}(\mathbb{R}^{3}))$ and the initial data $f_{0}$ satisfying

[TABLE]

there exists a unique solution $f$ to (3.20) such that $(1+|v|^{k})\nabla^{\alpha}_{v}\nabla_{x}^{\beta}f\in L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3}))$. Furthermore, we obtain the following estimate:

[TABLE]

[TABLE]

where $C$ is a constant depending only on $k$.

Proof.

For a proof, we introduce a cut-off function $\phi_{\delta}$ and mollifier function $\eta_{\varepsilon}$ as follows. Let $\phi\in C_{c}^{\infty}({\mathbb{R}}^{3}\times{\mathbb{R}}^{3})$ be a function such that

[TABLE]

and for $\delta>0$, we set $\phi_{\delta}(x,v):=\phi(\delta x,\delta v)$ for all $(x,v)\in{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}$. For $\varepsilon>0$, we define a mollifier $\theta_{\varepsilon}$ to satisfy

[TABLE]

Thus we let $\eta^{\varepsilon}(x,v)=\theta^{\varepsilon}(x)\theta^{\varepsilon}(v)$, and let

[TABLE]

Then, we define $(f_{\delta}^{\varepsilon},\bar{u}^{\varepsilon})$ as the unique solution on $[0,T)$ to the approximated equation of the linear equation (3.20):

[TABLE]

obtained by the method of characteristics (see Section 3.1.1–3.1.2). We set

[TABLE]

We then claim that $f^{\varepsilon}_{\delta 0}$ strongly converges to $f_{0}$ in $\mathcal{X}$, that is,

[TABLE]

The proof of (3.52) is as follows: As mentioned, due to the decay assumption (1.13) in Theorem 1.4, we prove that $(1+|v|^{k})\nabla_{v}^{\alpha}\nabla_{x}^{\beta}f_{0}\in L^{2}({\mathbb{R}}^{3}\times{\mathbb{R}}^{3})$. Indeed,

[TABLE]

[TABLE]

[TABLE]

where we use \int_{{\mathbb{R}}^{3}}\Big{|}\frac{1+|v|^{k}}{1+|v|^{p}}\Big{|}^{2}\,dv<\infty (since $p>\frac{2k+3}{2}$). Considering the estimate (3.53), due to the definition of $f^{\varepsilon}_{\delta 0}$, for any $\varsigma$ we can choose $R_{0}>0$ such that for a sufficiently small $\varepsilon$ and $\delta$

[TABLE]

On the other hand, we note that

[TABLE]

Via the relation (3.54) and (3.55), choosing sufficiently small $\varepsilon$ and $\delta$, we have for any $\varsigma$

[TABLE]

which implies (3.52). Also, due to Lemma 3.2, the approximated solution $(f_{\delta}^{\varepsilon},\bar{u}^{\varepsilon})$ for (3.51) satisfies the following estimate

[TABLE]

[TABLE]

When $\delta$ and $\varepsilon$ go to [math], there exist a weak limit $\bar{f}$ such that $f^{\varepsilon}_{\delta}\rightharpoonup\bar{f}$(at least up to a sequence) in the weighted space $\mathcal{X}$ and $\bar{f}$ is a solution to the equation (3.20) in a the sense of distribution and moreover the solution is unique by the standard argument. Hence, we briefly give a proof for a unique solvability of the solution $f$ to the linear equation (3.20) with the initial data (3.49) for a given $\bar{u}$. ∎

4. Proof of Theorems

In this section, we prove the existence of a local solution of the second equation of (LABEL:main). For given vector field $U:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}^{3}$ with $\mbox{div}\ U=0$ and tensor field $F:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}^{3}$, we first consider the following non-Newtonian Stokes type equations with drift term

[TABLE]

with the initial condition

[TABLE]

We show the local well-posedness of (4.56)-(4.57) in the next lemma.

Lemma 4.1**.**

Suppose that $U\in L^{\infty}(0,T;H^{3}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{4}(\mathbb{R}^{3}))$ and $F\in L^{2}(0,T;H^{2}(\mathbb{R}^{3}))$. There exists $T^{*}:=T^{*}(\|u_{0}\|_{H^{3}},\|U\|_{(L^{\infty}_{t}H^{3}_{x}\cap L^{2}_{t}H^{4}_{x})(Q_{T})},\|F\|_{L^{2}_{t}H^{2}_{x}(Q_{T})})>0$ such that there exists unique solutions $u\in L^{\infty}(0,T^{*};H^{3}(\mathbb{R}^{3}))\cap L^{2}(0,T^{*};H^{4}(\mathbb{R}^{3}))$ to the equation (4.56)-(4.57). Moreover, $u$ satisfies

[TABLE]

Proof.

The proof is similar to that in [13, Proposition 3.1]. The only difference is the control of the class of $F$. For the sake of convenience, we give a proof in Appendix. ∎

For proof of Theorem 1.4 using Schauder fixed point theorem (Lemma 2.2), we introduce a function space $X_{M}$ defined as follows:

[TABLE]

with the norm $\|u\|^{2}_{X}:=\|u\|^{2}_{L^{\infty}(0,T;H^{3}(\mathbb{R}^{3}))}+\frac{m_{0}}{2}\|u\|^{2}_{L^{2}(0,T;H^{4}(\mathbb{R}^{3}))}$.

**Proof of Theorem 1.4. ** First, we define a map $\Theta:X_{M}\rightarrow X_{M}$ by $\Theta(\bar{u})=(u)$:

[TABLE]

$\bullet$ (Step A: $\Theta$ is well-defined): We set

[TABLE]

[TABLE]

From the estimate (4.58) in Lemma 4.1, we know

[TABLE]

[TABLE]

Once we have $F\in L^{2}(0,T;H^{2}(\mathbb{R}^{3}))$, we can choose $0<T_{1}\leq T^{*}$ ($T^{*}$ given in Lemma 4.1) such that

[TABLE]

Now, we check $F\in L^{2}(0,T;H^{2}(\mathbb{R}^{3}))$. Indeed, we note that

[TABLE]

[TABLE]

[TABLE]

Using Jensen and Hölder’s inequalities, we have

[TABLE]

[TABLE]

Thus, we have

[TABLE]

[TABLE]

Similarly,

[TABLE]

Using the same method, we can check that

[TABLE]

Using the estimate (4.60), we get

[TABLE]

[TABLE]

Finally, using the estimate (3.33), we obtain

[TABLE]

[TABLE]

[TABLE]

which implies $F\in L^{2}(0,T_{1};H^{2}({\mathbb{R}}^{3}))$. Note that

[TABLE]

[TABLE]

where Korn’s inequality was used. Applying Grownwall’s inequality to (4.59), we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where we use the estimate (4.61) in second inequality with (4.62) and the result in Proposition 3.3 and Hölder’s inequality for $\bar{u}\in X_{M}$ in third inequality. If we choose a sufficiently small $0<\tilde{T}\leq T_{1}$ such that

[TABLE]

then $\|u\|_{X_{M}}<M$. Lastly, we also prove that $u_{t}\in L^{2}(0,\tilde{T};L^{2}(\mathbb{R}^{3}))$. It is enough to show $F:=\int_{{\mathbb{R}}^{3}}(\bar{u}-v)f\,dv\in L^{2}(0,\tilde{T};L^{2}({\mathbb{R}}^{3}))$ due to (A.90) and (A.91). Indeed, by the estimate (4.61) for the control of the drag force and $u\in L^{\infty}(0,\tilde{T};H^{3}(\mathbb{R}^{3}))\cap L^{2}(0,T_{1};H^{4}(\mathbb{R}^{3}))$, it follows that

[TABLE]

[TABLE]

[TABLE]

which implies $u_{t}\in L^{2}(0,\tilde{T};L^{2}(\mathbb{R}^{3}))$.

$\bullet$ (Step B: $\Theta$ is continuous w.r.t $L^{2}(0,\tilde{T}:L^{2}(\mathbb{R}^{3}))$–topology): From Corollary 5.2 in Section 6, we know that $\Theta:X_{M}\mapsto X_{M}$ is continuous w.r.t the topology inherited from $L^{2}(0,\tilde{T}:L^{2}(\mathbb{R}^{3}))$. Due to the Aubin-Lions Lemma, $X_{M}$ is a compact subset of $L^{2}(0,\tilde{T}:L^{2}(\mathbb{R}^{3}))$ and $X_{M}$ is clearly convex. Hence, the Schauder’s fixed point theorem gives us the existence of solutions for the system (LABEL:main) and the contraction property (5.70) says the uniqueness of solutions.

5. Contraction of the iteration with respect to $L^{2}(0,T;L^{2}(\mathbb{R}^{3}))$

Lemma 5.1**.**

Assume $u_{0}\in H^{3}(\mathbb{R}^{3})$ and, for $i=1,~{}2$, $\bar{u}_{i}\in L^{\infty}(0,T;H^{3}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{4}(\mathbb{R}^{3}))$. Let $f_{i}\in AC_{2}(0,T;\mathcal{P}_{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$ be the solution of (3.30) with a given $f_{0}\in\mathcal{P}_{2}$ satisfying (3.26) and set $\rho_{i}(t,x):=\int_{\mathbb{R}^{3}}f_{i}(t,x,v)dv$ for $i=1,2$. Suppose that $u_{i}\in L^{\infty}(0,T;H^{3}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{4}(\mathbb{R}^{3}))$ is the solution of

[TABLE]

with initial condition $u_{i}(0)=u_{0}$. Then, we have

[TABLE]

where

[TABLE]

and

[TABLE]

The estimate (5.64) implies that there exists $T\ll 1$ such that

[TABLE]

Proof.

We consider the equation for $\tilde{u}:=u_{1}-u_{2}$ and $\tilde{p}:=p_{1}-p_{2}$,

[TABLE]

and we obtain

[TABLE]

where

[TABLE]

By Hölder and Young’s inequalities, we note,

[TABLE]

and

[TABLE]

On the other hand, we have

[TABLE]

We note that

[TABLE]

Now we estimate

[TABLE]

and we exploit (2.16) and (2.17) to get

[TABLE]

Adding above estimates,

[TABLE]

[TABLE]

On the other hand,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Combining (5.65) and (5.66), we have

[TABLE]

where $C_{1}=C_{1}(\|\rho_{1}\|_{L^{\infty}(Q_{T})},\|\bar{u}_{2}\|_{L^{\infty}_{t}H^{3}_{x}(Q_{T})}),\|\tilde{u}\|_{L^{\infty}_{t}H^{3}_{x}(Q_{T})})$, and $C_{2}$ depends on $(\|m_{2}f_{1}\|_{L^{\infty}(Q_{T})}$, $\|m_{2}f_{2}\|_{L^{\infty}(Q_{T})}$, $\|\bar{u}_{2}\|_{L^{\infty}_{t}H^{3}_{x}(Q_{T})})$. Hence, we have

[TABLE]

where $C_{3}$ depends on $\|\rho_{1}\|_{L^{\infty}(Q_{T})}$ and $\|\bar{u}_{2}\|_{L^{\infty}_{t}H^{3}_{x}(Q_{T})}$. Plugging (3.31) into (5.68), we get

[TABLE]

Using Gronwall’s Lemma with $\tilde{u}(0)\equiv 0$, we obtain

[TABLE]

where

[TABLE]

From (5.69), we have

[TABLE]

which gives us

[TABLE]

for some small $T\ll 1$. ∎

Corollary 5.2**.**

Assume $u_{0}\in H^{3}(\mathbb{R}^{3})$ and $f_{0}\in\mathcal{P}_{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$ satisfying (3.26). Let

[TABLE]

and $\Theta:X_{M}\rightarrow X_{M}$ be defined by $u:=\Theta(\bar{u})$ where

[TABLE]

Then, for small $T\ll 1$, $\Theta$ is a contraction mapping with respect to the topology induced by the norm $L^{2}(0,T;L^{2}(\mathbb{R}^{3}))$. That is, we have

[TABLE]

for small $T\ll 1$.

Appendix A

In this section, as mentioned, we give a proof of Lemma 4.1.

**Proof of Lemma 4.1. ** We suppose that $u$ is regular. We then compute certain a priori estimates. First of all, we note that by the $L^{2}$-energy estimate with (1.3),

[TABLE]

$\bullet$ ($\|\nabla u\|_{L^{2}}$-estimate)  Taking derivative $\partial_{x_{i}}$ to (4.56) and multiplying $\partial_{x_{i}}u$,

[TABLE]

[TABLE]

Noting that

[TABLE]

we have

[TABLE]

[TABLE]

Taking $A=Du$ and $B=\partial_{x_{i}}Du$ in the last inequality, we use the following inequality:

[TABLE]

Indeed, if $G^{\prime}[|A|^{2}]\geq 0$, then

[TABLE]

In case that $G^{\prime}[|A|^{2}]<0$, we note that

[TABLE]

Applying the inequality (A.73) to (A.72), we obtain

[TABLE]

We will treat the term in righthand side caused by convection together later.

$\bullet$ ($\|\nabla^{2}u\|_{L^{2}}$-estimate)  Taking the derivative $\partial_{x_{j}}\partial_{x_{i}}$ on (4.56) and multiplying it by $\partial_{x_{j}}\partial_{x_{i}}u$,

[TABLE]

[TABLE]

We observe that

[TABLE]

where $\sigma:\{i,j\}\rightarrow\{i,j\}$ is a permutation of $\{i,j\}$. We separately estimate terms $I_{22}$ and $I_{23}$ in (A.76). Using Hölder, Young’s and Gagliardo-Nirenberg inequalities, we have for $I_{22}$

[TABLE]

[TABLE]

[TABLE]

where we used the condition (1.3).

For $I_{23}$, using Lemma 2.3, we compute

[TABLE]

The term $I_{231}$ is estimated as

[TABLE]

where we used the first inequality of (2.14). We combine estimates (A.75)-(A.77) to get

[TABLE]

[TABLE]

Similarly as in (A.74), we have

[TABLE]

[TABLE]

[TABLE]

$\bullet$ ($\|\nabla^{3}u\|_{L^{2}}$-estimate) For convenience, we denote $\partial^{3}:=\partial_{x_{k}}\partial_{x_{j}}\partial_{x_{i}}$. Similarly as before, taking the derivative $\partial^{3}$ on (4.56) and multiplying it by $\partial^{3}u$,

[TABLE]

[TABLE]

Direct computations show that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\sigma_{3}=\pi_{3}\circ\tilde{\sigma}_{3}$ such that $\tilde{\sigma}_{3}:\{i,j,k\}\rightarrow\{i,j,k\}$ is a permutation of $\{i,j,k\}$ and $\pi_{3}$ is a mapping from $\{i,j,k\}$ to $\{1,2,3\}$.

We separately estimate terms $I_{32}$, $I_{33}$ and $I_{34}$. We note first that

[TABLE]

For $I_{33}$, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where we use same argument as (A.80) in the fourth inequality. Finally, for $I_{34}$, using Lemma 2.3, we note that

[TABLE]

The second term in (A.82) is estimated as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where we use same argument as (A.81) in the third inequality. Adding up the estimates (A.79)-(A.83), we obtain

[TABLE]

[TABLE]

[TABLE]

Hence, we have

[TABLE]

[TABLE]

[TABLE]

Next, we estimate the terms caused by convection terms in (A.74), (A.78) and (A.84).

[TABLE]

where we use the following inequality:

[TABLE]

For the external force $F$,

[TABLE]

We combine (A.71), (A.74), (A.78) and (A.84) with (A.85) and (A.86) to conclude

[TABLE]

Furthermore, we have

[TABLE]

where $g:[0,\infty)\mapsto[0,\infty)$ is a nondecreasing function. We set $X(t):=\|u(t)\|_{H^{3}(\mathbb{R}^{3})}$ and it then follows from (A.87) and (A.88) that

[TABLE]

for some nondecreasing continuous function $f_{3}$, which immediately implies that there exists $T_{3}>0$ such that $\sup_{0\leq t\leq T_{3}}X(t)<\infty$.

We note that $\partial_{t}{u}\in L^{2}((0,T);L^{2}(\mathbb{R}^{3}))$. Indeed, we introduce the antiderivative of $G$, denoted by $\tilde{G}$, i.e. $\tilde{G}[s]=\int^{s}_{0}G[\tau]d\tau$. Multiplying $\partial_{t}u$ to (4.56), integrating it by parts, and using Hölder and Young’s inequalities, we have

[TABLE]

Again, integrating the estimate (A.89) over the time interval $[0,T]$, we obtain

[TABLE]

[TABLE]

Using Sobolev embedding, the second term in (A.90) is estimated as follows:

[TABLE]

[TABLE]

and due to the assumption for the external force $F$, we also get $\int_{0}^{T}\int_{\mathbb{R}^{3}}|F|^{2}\,dxdt<\infty$. Therefore, we obtain $\partial_{t}{u}\in L^{2}(0,T;L^{2}(\mathbb{R}^{3}))$.

For a uniqueness of a solution, we let $u_{1}$ and $u_{2}$ be strong solutions for the system (4.56). First of all, we rewrite the equation for $\tilde{u}:=u_{1}-u_{2}$ and $\tilde{p}:=p_{1}-p_{2}$.

[TABLE]

with $\nabla\cdot\tilde{u}=0$ and $\nabla\cdot U=0$. Multiplying $\tilde{u}$ on the both sides of the equation above and integrating on $\mathbb{R}^{3}$, we obtain

[TABLE]

where we use Lemma 2.4 and the divergence free condition. Applying Grownwall’s inequality to estimate (A.92), we get $\tilde{u}(x,0)=0$ in $L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{1}(\mathbb{R}^{3}))$. Hence this imply the uniqueness of a solution. Finally, we introduce Galerkin approximation procedure of the equation (4.56)–(4.57) to make up construction of solution. We omit the this part (see refer to [13, Proposition 3.1] for detailed proof).

Acknowledgments

Kyungkeun Kang’s work is supported by NRF-2019R1A2C1084685. Hwa Kil Kim’s work is supported by NRF-2021R1F1A1048231. Jae-Myoung Kim was supported by National Research Foundation of Korea Grant funded by the Korean Government (NRF-2020R1C1C1A01006521).