Special Session 8: Recent Progress on Mathematical Analysis of PDEs Arising in Fluid Dynamics

Optimal decay rates of compressible Navier-Stokes equations and the related model

Guangyi HONG South China University of Technology Peoples Rep of China Co-Author(s):    Changjiang Zhu

Abstract: In this talk, we will introduce our recent results on the large-time behaviors of solutions to the one-dimensional compressible Navier-Stokes equations and the related compressible two-phase model with free boundaries. Specifically, for both the case of constant viscosity and the case of degenerate viscosity, we prove the decay estimates for the solutions of the models in the vacuum free boundary setting. In particular, we derive the optimal pointwise decay estimates of the density function and the mass function for the compressible Navier-Stokes equations and the two-phase model, respectively. This talk is mainly based on some joint work with Prof. Changjiang Zhu.

Some recent progress on mathematical analysis of nematic liquid crystals

Jinrui Huang Wuyi University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will present some results on mathematical analysis of nematic liquid crystals in recent years, mainly about the unique continuation, orientability and singular limit for the Q-tensor system, and Freedericksz Transition for the Ericksen-Leslie system. This talk is based on the joint works with Shijin Ding and Junyu Lin.

Nonlinear stability of traveling waves to a parabolic-hyperbolic system modeling chemotaxis with periodic perturbations

Haiyang Jin South China University of Technology Peoples Rep of China Co-Author(s):    Feifei Zou

Abstract: We prove the nonlinear stability of large-amplitude traveling waves under space-periodic perturbations to a parabolic-hyperbolic system derived from a singular chemotaxis model describing the initiation of tumor angiogenesis. Compared with the previous results, we are able to prove the nonlinear stability of traveling waves even though the perturbation oscillates at the far fields.

Existence of solutions to Dirichlet boundary value problems of the relativistic Boltzmann equation

Li Li Ningbo University Peoples Rep of China Co-Author(s):    Yi Wang, Zaihong Jiang

Abstract: We study the boundary value problem of the steady-state relativistic Boltzmann equation in the half-space, assigning the Dirichlet data for outgoing particles at the boundary and a Maxwellian at the far field, under the hard potential model. It is proved that the solvability condition varies with the Mach number of the problem.

Stability of hyperbolic wave for the viscous vasculogenesis model

Qingqing Liu South China University of Technology Peoples Rep of China Co-Author(s):    Xiaoli Wu, Qian Yan, Wenwen Fu

Abstract: Experiments of in vitro vasculogenesis show that endothelial cells randomly distributed on the gel matrix will organize themselves into a connected capillary network. As a kind of taxis, this network aggregation phenomenon of endothelial cells can not be simulated by the classical Keller-Segel model. The viscous vasculogenesis model proposed by the biologist Gamba et al. can model the experimental phenomenon very well. This talk will present a series of studies on the existence and long-time behavior of the solution for viscous vasculogenesis model, including: the stability of rarefaction wave for the Cauchy problem of the one-dimensional viscous vasculogenesis model, the stability of rarefaction wave and the boundary layer for the initial boundary value problem of the one-dimensional viscous vasculogenesis model over $\mathbb{R}^{1}_{+}$, and the stability of planar staionary solution for the initial boundary value problem of the three-dimensional viscous vasculogenesis model over $\mathbb{R}^3_{+}$.

Recent progress on the 3D kinetic shear flow via the Boltzmann equation in the diffusive limit

Shuangqian Liu Central China Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will present our recent study on the Boltzmann equation in the diffusive limit for 3D kinetic shear flow. Our results show that the first-order approximation of the solutions is governed by the perturbed incompressible Navier-Stokes-Fourier system around the fluid shear flow. The proof is based on: (i) applying the Fourier transform on $\T^2$ to effectively reduce the 3D problem to a one-dimensional one; (ii) using anisotropic Chemin-Lerner type function spaces, incorporating the Wiener algebra, to control nonlinear terms and address the singularities arising from the small Knudsen number in the diffusive limit; and (iii) employing Caflisch`s decomposition, together with the $L^2 \cap L^\infty$ interplay technique, to manage the growth of large velocities.

The Cauchy problem for an inviscid Oldroyd-B model in three dimensions: global well posedness and optimal decay rates

Sili Liu Changsha University of Science and Technology Peoples Rep of China Co-Author(s):    Wenjun Wang; Huanyao Wen.

Abstract: In this talk I will introduce our recent work on the Cauchy problem for an inviscid compressible Oldroyd-B model in three dimensions. The global well posedness of strong solutions and the associated time-decay estimates in Sobolev spaces are established near an equilibrium state. The vanishing of viscosity is the main challenge compared with [Wang-Wen, Sci.China Math., 2021] where the viscosity coefficients are included and the decay rates for the highest-order derivatives of the solutions seem not optimal. One of the main objectives of this paper is to develop some new dissipative estimates such that the smallness of the initial data and decay rates are independent of the viscosity. Moreover, we prove that the decay rates for the highest-order derivatives of the solutions are optimal, which is of independent interest. Our proof relies on Fourier theory and delicate energy method. This talk is based on joint works with Prof. Wenjun Wang and Prof. Huanyao Wen.

On a hyperbolic-parabolic chemotaxis system

Hongyun Peng Guangdong University of Technology Peoples Rep of China Co-Author(s):    Kun Zhao

Abstract: Stability of steady state solutions associated with initial and boundary value problems of a coupled fluid-reaction-diffusion system in one space dimension is analyzed. It is shown that under Dirichlet-Dirichlet type boundary conditions, non-trivial steady state solutions exist and are locally stable when the system parameters satisfy certain constraints.

Some recent progress on blowup criteria for compressible Navier-Stokes equations

Huanyao Wen South China University of Technology Peoples Rep of China Co-Author(s):    Eduard Feireisl and Changjiang Zhu

Abstract: In this talk, we will introduce some recent progress on blowup criteria in terms of concentration of density and absolute temperature for viscous, compressible, and heat-conducting flow with vaccum.

Nonlinear stability of viscous contact wave for the isentropic MHD equations with free boundary

Huancheng Yao South China Agricultural University Peoples Rep of China Co-Author(s):    Changjiang Zhu

Abstract: In this talk, we will introduce our recent results on the nonlinear stability of viscous contact wave for the isentropic compressible magnetohydrodynamics (MHD) equations with a free boundary. It is shown that when time tends to infinity, the fluid part of solutions will asymptotically converge to the viscous contact waves, and more importantly, the magnetic field will converge to a nontrivial wave pattern. This wave phenomenon is different from those of isentropic compressible Navier-Stokes equations, which means the magnetic field truly makes an essential effect on the fluid behavior. The main result is proved by using elaborate wave pattern analysis and elementary energy methods, provided the initial perturbations and wave strength are suitably small.

Initial boundary value problem for 3D non--conservative compressible two--fluid model

Lei Yao Northwestern Polytechnical University Peoples Rep of China Co-Author(s):    Guochun Wu; Lei Yao; Yinghui Zhang;

Abstract: We study the initial boundary value problem of the 3D non--conservative compressible two--fluid model with common pressure (P^+=P^-) in a bounded domain with no--slip boundary conditions. The global existence and uniqueness of classical solution are established when the initial data is near its equilibrium in H^4(\Omega) by delicate energy methods. By a product, the exponential convergence rates of the pressure and velocities in H^3(\Omega) are obtained. To overcome the difficulties arising from boundary effects, on the one hand, we separate the energy estimates for the spatial derivatives into that over the region away from the boundary and near the boundary by using cutoff functions and localizations of \Omega. On the other hand, by exploiting the dissipation structure of the system, we employ regularity theory of the stationary Stokes equations and elliptic equations to get higher--order spatial derivatives of pressure and velocities, which is very different from the Cauchy problem in [Wu--Yao--Zhang, Math. Ann., 2024] where the effective viscous flux played an important role in their analysis.

A new blowup criterion for the 3D barotropic compressible Navier-Stokes equations with vacuum

Yinghui Zhang Guangxi Normal University Peoples Rep of China Co-Author(s):    Saiguo Xu

Abstract: We investigate the blowup criterion of the barotropic compressible viscous fluids for the Cauchy problem, Dirichlet problem and Navier-slip boundary condition. The main novelty of this paper is two-fold: First, for the Cauchy problem and Dirichlet problem, we prove that a strong or smooth solution exists globally, provided that the vorticity of velocity satisfies Serrin`s condition and the maximum norm of the divergence of the velocity is bounded. Second, for the Navier-slip boundary condition, we show that if both the maximum norm of the vorticity of velocity and the maximum norm of the divergence of velocity are bounded, then the solution exists globally. In particular, this criterion extends the well-known Beale-Kato-Majda`s blowup criterion for the 3D incompressible Euler equations (Comm. Math. Phys. 94(1984):61-66) to the 3D barotropic compressible Navier-Stokes equations, and can be regarded as a complement for the work by Huang-Li-Xin (Comm. Math. Phys. 301(2011):23-35). The vacuum is allowed to exist here.

Vanishing shear viscosity limit for the compressible planar MHD system with boundary layer

Xinhua Zhao Guangdong Polytechnic Normal University Peoples Rep of China Co-Author(s):    Huanyao Wen

Abstract: This paper is devoted to the study of the vanishing shear viscosity limit and strong boundary layer problem for the compressible, viscous, and heat-conducting planar MHD equations. The main aim is to obtain a sharp convergence rate which is usually connected to the boundary layer thickness. However, The convergence rate would be possibly slowed down due to the presence of the strong boundary layer effect and the interactions among the magnetic field, temperature, and fluids through not only the velocity equations but also the strongly nonlinear terms in the temperature equation. Our main strategy is to construct some new functions via asymptotic matching method which can cancel some quantities decaying in a lower speed. It leads to a sharp $L^\infty$ convergence rate as the shear viscosity vanishes for global-in-time solution with arbitrarily large initial data.

Stability threshold of Couette flow for the 3D MHD equations

Ruizhao Zi Central China Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider the stability of 3D Couette flow $(y,0,0)^\top$ in a uniform background magnetic field $\al(\sig, 0,1)^\top$. It is shown that if the background magnetic field $\al(\sig, 0,1)^\top$ with $\sig\in\mathbb{R}\backslash\mathbb{Q}$ satisfying a generic Diophantine condition is so strong that $|\al|\gg \fr{\nu+\mu}{\sqrt{\nu\mu}}$, and the initial perturbations $u_{\mathrm{in}}$ and $b_{\mathrm{in}}$ satisfy $ \left\|(u_{\mathrm{in}},b_{\mathrm{in}})\right\|_{H^{N+2}}\ll\min\left\{\nu, \mu\right\}$ for sufficiently large $N$, then the resulting solution remains close to the steady state in $L^2$ at the same order for all time. Compared with the result of Liss [Comm. Math. Phys., 377(2020), 859--908], we use a more general energy method to address the physically relevant case $\nu\ne\mu$ based on some new observations. This is based on a joint work with Yulin Rao and Zhifei Zhang.