Searcharxiv⌕ Search

arXiv subjects

Wenke Tan

Publications and source records attributed to Wenke Tan.

12 recordsLinked to original sources
arXiv2026

Liouville theorems for the fractional Navier-Stokes equations with arbitrary asymptotic state at infinity

We mainly consider a Liouville-type problem for the three dimensional stationary fractional Navier-Stokes equations with arbitrary asymptotic state $u_\infty$ at infinity. When $u_\infty\neq 0$ and $\frac{1}{2}\leq s<1$, we prove a complete Liouville theorem by establishing some refined $L^p$ estimates for the velocity without relying on perturbation arguments. These new estimates are stronger than the $L^3$ estimates obtained by the classical perturbation framework, we thus can take $u$ as a test function and give a direct and simple proof of Liouville theorem while avoiding some technical fractional calculus. When $u_\infty\neq 0, s=\frac{1}{2}$ or $u_\infty=0,\frac{1}{2}\leq s\leq\frac{5}{6}$, we also prove a complete Liouville theorem by using frequency localization to overcome the obstacles coming from the non-local effects of $(-\Delta)^s$. We wish to emphasize that our method dealing with the case of $u_\infty=0$ is also applicable to dimension $n$ with $n\geq 2$ and $\frac{1}{2}\leq s\leq \frac{n+2}{6}$.

math.AP↗
arXiv2025

New Liouville type theorems for the stationary Navier-Stokes equations

We mainly research the Liouville type problem for the stationary Navier-Stokes equations (including the fractional case) in $\mathbb{R}^3$. We first establish a new formula for the Dirichlet integral of solutions and show that the globally defined quantity $\int_{\mathbb{R}^3}|\nabla u|^2dx$ is completely determined by the information of the solution $u$ at the origin in frequency space. From this character, we show some new Liouville type theorems for solutions of the stationary Navier-Stokes equations. Then we extend the obtained results for classical stationary Navier-Stokes equations to the stationary fractional Navier-Stokes equations for $\frac{1}{2}\leq s<1$, especially, we solve the Liouville type problem for $s=\frac{5}{6}$.

math.AP↗
arXiv2025

Liouville-type theorem for the stationary inhomogeneous Navier-Stokes equations

In this manuscript, a new Liouville-type theorem for the three-dimensional stationary inhomogeneous Navier-Stokes equations is established. We first localize the Dirichlet energy into the region near the origin in frequency spaces by two times localizations. The first localization is to eliminate the non-zero frequency part coming from the interaction between $ρu$ and $u$, the second one is to eliminate the non-zero frequency part coming from the interaction between $ρ$ and $u$. Based on the local formula of Dirichlet energy, we can establish suitable estimates on different frequency parts of $u$ and $ρ$, then show our new Liouville-type theorem.

math.AP↗
arXiv2025

New Liouville type theorems for the stationary MHD equations in $\mathbb{R}^3$

We research the Liouville type problem for the 3D stationary MHD equations in the frequency space. We establish two new Liouville type theorems for solutions with finite Dirichlet energy. Specifically, we show that the low-frequency part of the velocity field plays the leading role in a Liouville theory for MHD equations and then improve the results of Chae-Weng \cite{Chae-W}.

math.AP↗
arXiv2022

Energy conservation for the non-resistive MHD equations with physical boundaries

In this paper, we study the energy equality for weak solutions to the non-resistive MHD equations with physical boundaries. Although the equations of magnetic field $b$ are of hyperbolic type, and the boundary effects are considered, we still prove the global energy equality provided that $ u \in L^{q}_{loc}\left(0, T ; L^{p}(Ω)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } b \in L^{r}_{loc}\left(0, T ; L^{s}(Ω)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4 $. In particular, compared with the existed results, we do not require any boundary layer assumptions and additional conditions on the pressure $P$. Our result requires the regularity of boundary $\partialΩ$ is only Lipschitz which is the minimum requirement to make the boundary condition $b\cdot n$ sense. The proof is based on the important properties of weak solutions of the nonstationary Stokes system and the separate mollification of weak solutions from the boundary effect by considering a non-standard local energy equality and transform the boundary effects into the estimates of the gradient of cut-off functions.

math.AP↗
arXiv2022

On the energy equality for the 3D incompressible viscoelastic flows

In this paper, we study the problem of energy conservation for the solutions to the incompressible viscoelastic flows. First, we consider Leray-Hopf weak solutions in the bounded Lipschitz domain $Ω$ in $\mathbb{R}^d\,\, (d\geq 2)$. We prove that under the Shinbrot type conditions $ u \in L^{q}_{loc}\left(0, T ; L^{p}(Ω)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } {\bf F} \in L^{r}_{loc}\left(0, T ; L^{s}(Ω)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4 $, the boundary conditions $u|_{\partialΩ}=0,\,\,{\bf F}\cdot n|_{\partialΩ}=0$ can inhibit the boundary effect and guarantee the validity of energy equality. Next, we apply this idea to deal with the case $Ω= \mathbb{R}^d\,\,(d=2, 3, 4)$, and showed that the energy is conserved for $u\in L_{loc}^{q}\left(0,T;L_{loc}^{p}\left(\mathbb{R}^{d}\right)\right)$ with $ \frac{2}{q}+\frac{2}{p}\leq1, p\geq 4 $ and $ {\bf F}\in L_{loc}^{r}\left(0,T;L_{loc}^{s}\left(\mathbb{R}^{d}\right)\right)\cap L^{\frac{4d+8}{d+4}}\left(0,T;L^{\frac{4d+8}{d+4}}\left(\mathbb{R}^{d}\right)\right)$ with $\frac{2}{r}+\frac{2}{s}\leq1, s\geq 4 $. This result shows that the behavior of solutions in the finite regions and the behavior at infinite play different roles in the energy conservation. Finally, we consider the problem of energy conservation for distributional solutions and show energy equality for the distributional solutions belonging to the so-called Lions class $L^4L^4$.

math.AP↗
arXiv2022

The energy conservation for the Navier-Stokes equations on the Lipschitz domains

In this paper, we consider the energy conservation of the Leray-Hopf weak solution $u$ to the Navier-Stokes equations on bounded domains $Ω$ with Lipschitz boundary $\partialΩ$. We prove that although the boundary effect appears, the Shinbrot's condition $u\in L^q_{loc}((0,T];L^p(Ω))$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{2},p\geq 4$ still guarantees the validity of energy conservation of $u$, no boundary layer assumptions are required when dealing with domains with Lipschitz boundary. Compared to the existed methods, our critical strategies are that we first separate the mollification of weak solution from the boundary effect by considering non-standard local energy equality and transform the boundary effects into the estimates of the gradient of the cut-off functions, then by establishing a sharp $L^2L^2$ estimate for pressure $P$ and using the zero boundary condition, we obtain global energy equality by taking suitable cut-off functions. Our result provides a unified method to deal with domains with or without boundary and improves the corresponding results in \cite{C-L,Yu}.

math.AP↗
arXiv2021

The local characterizations of the singularity formation for the MHD equations

This paper characterizes the possible blow-up of solutions for the 3D magneto-hydrodynamics (MHD for short) equations. We first establish some $ε$-regularity criteria in $L^{q,\infty}$ spaces for suitable weak solutions, and then together with an embedding theorem from $L^{p,\infty}$ space into a Morrey type space to characterize the local behaviors of solutions near a potential singular point. More precisely, we show that if $z_{0}=\left(t_{0}, x_{0}\right)$ is a singular point, then for any $r>0$ it holds that $$ \limsup _{t \rightarrow t_{0}^{-}}\left(\left\|u(t, x)-u(t)_{x_{0}, r}\right\|_{L^{3, \infty}\left(B_{r}\left(x_{0}\right)\right)}+\left\|b(t, x)-b(t)_{x_{0}, r}\right\|_{L^{3, \infty}\left(B_{r}\left(x_{0}\right)\right)}\right)>δ^{*}; $$ $$ \limsup\limits _{t \rightarrow t_{0}^{-}}\left(t_{0}-t\right)^{\frac{1}μ} r^{\frac{2}ν-\frac{3}{p}}\|(u,b)(t)\|_{L^{p, \infty}\left(B_{r}\left(x_{0}\right)\right)}>δ^{*} \text { for } \frac{1}μ+\frac{1}ν=\frac{1}{2},\,2 \leq ν\leq \frac{2 p}{3},\, 3 δ^{*} \text { for } \frac{1}μ+\frac{1}ν=\frac{1}{2},\, ν\in\left\{\begin{array}{ll} {[2, \infty],} & p\geq 3 {[2, \frac{2p}{3-p}],} & \frac{3}{2}\leq p<3 \end{array}\right. $$ where $δ^{*}$ is a positive constant independent on $ν$ and $p$.

math.AP↗
arXiv2015

Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows with vacuum

This paper concerns the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space $\mathbb{R}^{2}$ with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the initial orientation satisfies a geometric condition (see \eqref{eq1.3}). In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support. As a byproduct, the large time behavior of the solution is also obtained.

math.AP↗
arXiv2011

Global solution to liquid crystal flows in three dimensions

In this paper, we mainly study a hydrodynamic system modeling the flow of nematic liquid crystals. In three dimensions, we first establish local well-posedness of the initial-boundary value problem of the system. Then, we prove the existence of global strong solution to the system with small initial-boundary condition.

math.AP↗