偏微分方程建模及其理论分析研讨会

本次会议的总参加人数约为45人。近年来，偏微分方程建模及其理论分析的研究发展十分迅速，及时加强国内外专家的交流和合作十分有必要。2023年正值厦门大学数学科学学院学科成立100周年，偏微分方程方向也是厦门大学数学科学学院传统的研究方向之一，为增强偏微分方程团队与其他高校专家的交流合作，促进共同进步与发展，厦门大学数学科学学院及国家天元数学东南中心将于2023年3月3日至4日举办“偏微分方程建模及其理论分析”研讨会，邀请一些资深专家和青年学者参与本次会议并作报告。

一、本次会议的邀请报告人如下：

二、会议日程安排

2023年3月3日报到，4日作报告。

三、会议地址

厦门市软件园一期元汇楼310

四、会议组织者

谭忠、张剑文、王焰金、徐新英、罗珍

五、本次会议得到国家自然科学基金项目“高维可压流体中的若干数学理论研究”、数学建模经费以及国家基金委天元东南中心经费的资助。

“偏微分方程建模及其理论分析”组委会

2023年3月3日

会议报告日程安排：3月4日（周六）

报告题目和摘要

Stability of the viscoelastic Rayleigh–Benard problem with an upper free boundary

江飞，福州大学

Motivated by the stability result of the Rayleigh–Benard problem in a fixed slab domain in Jiang and Liu (Nonlinearity 33:1677–1704, 2020) and the global-in-time well-posedness of an incompressible viscoelastic fluid system with an upper free boundary in Xu et al. (Arch Ration Mech Anal 208:753–803, 2013), we further investigate the Rayleigh–Benard problem for an incompressible viscoelastic fluid in a three-dimensional horizontally periodic domain with the lower fixed boundary and with the upper free boundary. By a careful energy method, we establish an explicit stability condition, under which the viscoelastic Rayleigh–Benard problem has a unique global-in-time solution with exponential time-decay. Our result presents that the elasticity can inhibit the thermal instability for sufficiently large elasticity coefficient.

Optimal decay rates of the solution for generalized Poisson–Nernst–Planck–Navier–Stokes equations in R^3

童雷雷，重庆邮电大学

The Cauchy problem of compressible quantum Navier-Stokes-Poisson equations in three-dimensional space is considered in this paper. Under some smallness conditions on the initial data, we derive the existence of the global classical solution near the non-constant steady state by using the energy method. Combining the linear decay rate and the energy method, we also prove the algebraic decay rate of the solution toward the non-constant steady state with a small doping profile.

Analysis on the viscoelastic electrically conducting fluid equations

王勇，华南师范大学

We investigate the initial-boundary value problem of the three-dimensional compressible viscoelastic fluids with the electrostatic effect, in which the Dirichlet-Neumann mixed boundary condition for the electrostatic potential is imposed. We prove that there exists a unique global-in-time small strong solution in Sobolev space. Moreover, we show that such a solution converges to the constant equilibrium state with an exponential decay rate as time tends to infinity.

Stability and instability for a generic non-conservative compressible two-fluid model

吴国春，华侨大学

We are concerned with stability and instability of the steady state  for a generic non-conservative compressible two-fluid model in whole space. Under the assumption that the initial fraction densities are close to the constant state $(1,1)$ in $H^3\cap\dot B^{s}_{1,\infty}$ and the initial velocities are small in $H^2\cap\dot B^{s}_{1,\infty}$ with $s\in [0,1]$, it is shown that $\frac{1}{2}$ is the critical value of $s$ on the stability of the model in question. More precisely, when $0\leq s< \frac{1}{2}$, the steady state $(1,\overrightarrow{0}, 1, \overrightarrow{0})$ is nonlinearly globally stable; and conversely, the steady state $(1,\overrightarrow{0}, 1, \overrightarrow{0})$ is nonlinearly unstable in the sense of hadamard when $\frac{1}{2}

Exponential decay for Lions–Feireisl's weak solutions to the barotropic compressible Navier–Stokes equations in 3D bounded domains

吴云顺，贵州师范大学

For barotropic compressible Navier-Stokes equations in three-dimensional (3D) bounded domains, we prove that any finite energy weak solution obtained by Lions and Feireisl-Novotny-Petzeltova decays exponentially to the equilibrium state. This result is established by both using the extra integrability of the density due to Lions and constructing a suitable Lyapunov functional just under the framework of Lions-Feireisl’s weak solutions.

Global Stability and Non-Vanishing Vacuum States of 3D

Compressible Navier-Stokes Equations

张映辉，广西师范大学

We investigate the global stability and non-vanishing vacuum states of large solutions to the compressible Navier-Stokes equations on the torus T3, and the main novelty of this work is three-fold: First, under the assumption that the density verifies some condition, it is shown that the solutions converge to equilibrium state exponentially. Second, by employing some new thoughts, we also show that the density converges to its equilibrium state exponentially. Finally, we prove that the vacuum states will not vanish for any time provided that the vacuum states are present initially. This phenomenon is totally new and somewhat surprising, and particularly is in contrast to the previous work of [H. L. Li et al., Commun. Math. Phys., 281 (2008), 401-444], where the authors showed that the vacuum states must vanish within finite time for the 1D compressible Navier-Stokes equations with some density-dependent viscosity.

Global small solutions of MHD boundary layer equations in Gevrey function space

吴忠二，厦门大学

We obtain the existence and decay estimates of the global small solutions for the MHD boundary layer equations in Gevrey $\frac{3}{2}$ space. By using the ``good function'' proposed by M. Paicu and P. Zhang (Arch. Ration. Mech. Anal. 241:403-446,2021), Littlewood-Paley theory, new auxiliary functions and symmetry of the MHD boundary layer equations, we obtain sufficiently fast decay estimates of the solutions and thus the global existence. Here we do not need the monotonicity assumption of the tangential velocity ($\py u_0>0$) and the assumption that the initial tangential magnetic field does not degenerate ($b_0\geq C>0$).

Stability of Boussinesq equations with partial dissipation around the hydrostatic balance

徐赛国，厦门大学

This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system, in order to gain insight into certain atmospheric and oceanographic phenomena. The Boussinesq system focused here is anisotropic and involves only horizontal dissipation and thermal damping. In 2D case $\mathbb{R}^2$, due to the lack of the vertical dissipation, the stability and large-time behavior problem remains open in a Sobolev setting. When the spatial domain is $\mathbb{T}\times\mathbb{R}$, this paper solves the stability problem and gives the precise large-time behavior of the perturbation. By decomposing the velocity $u$ and temperature $\theta$ into the horizontal average $(\bar{u},\bar{\theta})$ and the corresponding oscillation $(\tf{u},\tf{\theta})$, we can derive the global stability in $H^3$ and the exponential decay of $(\tf{u},\tf{\theta})$ to zero in $H^2$. Moreover, we also obtain that $(\bar{u}_2,\bar{\theta})$ decays exponentially to zero in $H^2$, and $\bar{u}_1$ decays exponentially to $\bar{u}_1(\infty)$ in $H^2$ as well, which reflect a strongly stratified phenomenon of buoyancy-driven fluids. In addition, we still establish the global stability in $H^3$ for the 3D case $\mathbb{R}^3$.

Blowup dynamics for equivariant critical Landau-Lifshitz flow

韩芳宇，厦门大学

For the energy critical Landau-Lifshitz flow, we show the existence of 1-equivariant Krieger-Schlag-Tataru type blowup solutions near the lowest energy steady state. More precisely, we prove that for any $\nu>1$, there exists a 1-equivariant finite-time blowup solution of the form $u(x,t)= \phi(\lambda(t)x) +\zeta(x,t), \lambda(t)=t^{-1/2-\nu}$, where $\phi$ is a lowest energy steady state and $\zeta(t)$ is arbitrary small in $\dot{H}^1 \cap\dot{H}^2$.

Study on the well-posedness of three-dimensional inviscid magneto-micropolar fluids

李心亮，厦门大学

Firstly, we consider the well-posedness and decay rates of the solutions to three-dimensional incompressible magneto-micropolar equations with damping term. It is found that the damping term has the effect of increasing the velocity decay rate and weakening the initial conditions. Then we consider the non-uniqueness of the low regularity weak solution for the 3D inviscid magneto-micropolar without damping term by the convex integration.