# 第三届偏微分方程青年学术论坛

## Program

5月6号全天在君宜王朝大酒店注册报到，5月7-8号两天为学术报告，地点在武汉大学数学与统计学院， 每个报告时间为40分钟（包含30分钟的学术报告和10分钟讨论时间）

• ## May 7, Saturday

8:30-9:00    Opening

• The isometric immersion of two-dimensional Riemannian manifold with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss-Codazzi equations for the first and second fundamental forms. The large $L^\infty$ solution is obtained which leads to a $C^{1,1}$ isometric immersion. The approximate solutions are constructed by the Lax-Friedrichs finite-difference scheme with the fractional step. The uniform estimate is established by studying the equations satisfied by the Riemann invariants and using the sign of the nonlinear part. The $H^{-1}$ compactness is also derived. A compensated compactness framework is applied to obtain the existence of large $L^\infty$ solution to the Gauss-Codazzi equations for the surfaces more general than those in literature.

• In this talk, the unique existence of subsonic Euler-Poisson flow in a finitely long nozzle will be discussed. Moreover, under physically acceptable boundary conditions, we prove that the flow is stable in sense of small perturbation of data at the entrance and exit. Also, some related recent work on supersonic and transonic shock problem will be introduced.

• 10:20-10:40    Tea break

• This talk is devoted to the well poshness issue for the self-gravitating Hookean elastodynamics in dimension three. The solution is constructed near a constant equilibrium. In the linear level, the system contains wave equations and Klein-Gordon equations with different speeds.

• We establish a unified blow-up criterion for strong solutions of various compressible models, including baratropic, fully compressible and heat-conducting magneto hydrodynamic flows, which is analogous to the Serrin's criterion of Navier-Stokes equations. It gives an affirmative answer to a problem proposed by J. Nash in 1958. As an application, we prove global-in-time classical solutions of compressible flows allowing vacuum and large fluctunation of initial data provided the initial energy is small. In the end, we prove a strong version of Nash’s problem based on new observations.

• 12:00-14:00
Lunch

• We investigate the stability and instability of Parker problem of a compressible viscous magnetohydrodynamic (MHD) fluid of zero resistivity in the presence of an modified gravitational force in a vertical strip domain, in which the equilibrium magnetic field is horizontal and vertically stratified, and the velocity of the fluid is non-slip on the boundary of the strip domain. We establish a discriminant value $\mm{C_r}$ for the stability and instability of Parker problem. More precisely, if $\mm{C_r} <0,$ then the Parker problem is unstable, that is the Parker instability occurs; if $\mm{C_r}>0$ and the initial perturbation quantities around a Parker equilibrium state satisfy some relations, then the Parker problem is stable, where the solution enjoys the algebraic decay in time. Our stability result presents that the sufficiently large horizontal equilibrium magnetic field with vertical stratification has stabilizing effect so that the heavier part of gas can persistently float on the lighter one under any small perturbation, and thus prevents the Parker instability from occurrence.

• In this talk, we consider the existence of weak steady solution and periodic solution for the compressible flows of nematic liquid crystals in a 3-D bounded domain with no-slip boundary condition. By an approximation scheme, we establish the existence of weak steady solution under the hypothesis $\gamma>1$ for the adiabatic constant, and the existence of weak periodic solution under the hypothesis $\gamma>3/2$.

• Recently there has been some progress on weak solutions of the Euler systems, such as the uniqueness question of the admissible weak solutions. We will review some of the recent results.

• 16:00-16:20
Tea Break

• In this talk, we consider a class of semilinear pseudo-parabolic equations. By introducing a family of potential wells, we prove the invariance of some sets, global existence, nonexistence and asymptotic behavior of solutions with sub-critical and critical initial energy respectively. Moreover, we obtain finite time blow-up with sup-critical initial energy by comparison principle.

• Convexity of shocks is frequently observed in many experimental results and provides better understanding of mathematical problems with the nonlinear wave, the uniqueness for instance. We consider the pseudo-transonic shock governed by the potential flow equation in the self-similar coordinates, and give a framework to show the strict and uniform convexity by a nonlinear and global argument. Finally, several applications are given.

• This talk is concerned with the large-time behavior of solutions for the one-dimensional compressible Navier-Stokes system. We show that the combination of viscous contact wave with rarefaction waves for the non-isentropic polytropic gas is stable under large initial perturbation without the condition that the adiabatic exponent $\gamma$ is close to 1, provided the strength of the combination waves is suitably small.

• 18:30
Banquet

• ## May 8, Sunday

Consider the dynamics of two layers of immiscible, compressible, viscous fluid lying atop one another. We establish a sharp nonlinear stability criterion of the equilibrium. This is a joint work with Juhi Jang and Ian Tice

• This talk examines the initial-value problem for the two-dimensional magnetohydrodynamic equation with only magnetic diffusion (without velocity dissipation). There establishes two main results. The first result features a regularity criterion in terms of themagnetic field. This criterion comes naturally from our approach to obtain a global bound for the vorticity. Due to the lack of velocity dissipation, it is difficult to conclude the boundedness of the vorticity from the vorticity equation itself. Instead we derive and involve a new equation for the combined quantity of the vorticity and a singular integral operator on the tensor product of the magnetic field. This criterion may be verifiable. Our second main result is a weaker version of the small data global existence result, which is shown by the bootstrap argument.

• Entropy weak solutions with bounded periodic initial data are considered for the system of weakly nonlinear gas dynamics. Through a modified Glimm scheme, the approximate solution sequence is constructed, and then a priori estimates are provided with the methods of approximate characteristics and approximate conservation laws, which gives the existence and the uniform bounds for the entropy solutions.

• 10:00-10:20
Tea Break

• I will report our recent result on the bipolar hydrodynamic model of semiconductors with insulating boundary conditions. Different from the unipolar case, the corresponding time-independent model is an elliptic equation with nonlocal terms. By the calculus of variations, the unique steady state is gained for N=1,2, without any additional assumption on the doping profile and the initial densities. Furthermore, the large time behavior framework of weak solutions and the asymptotic behavior of smooth solutions are considered. The solutions are shown to converge to the unique stationary solution with exponential decay rate.

• In this talk, we consider the global existence and uniqueness of the solutions to the viscous, non-resistive MHD system. First, we provide a much simplified proof of the main result in [Lin, Zhang, Comm. Pure Appl. Math. 2014] concerning the global existence and uniqueness of smooth solutions to the Cauchy problem for a 3D incompressible complex fluid model under the assumption that the initial data are close to some equilibrium states. Beside the classical energy method, the interpolating inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments. We combine the energy estimates with the $L^\infty$ estimates for time slices to deduce the key $L^1$ in time estimates. The latter is responsible for the global in time existence. Second, we can obtain the similar result for 2D the incompressible viscous, non-resistive MHD system under the assumption that the initial data are close to some equilibrium states. When the background magnetic field is sufficiently large, using the idea from Bardos, Sulem and Sulem [Trans. Amer. Math. Soc. 1988], we can obtain the global strong solutions for any initial data. Furthermore, we consider the global existence and uniqueness of the solution for the system with the time-varying background magnetic field. At last, we remark some wellposedness results for the 3D viscous and resistive MHD system.

• This paper is dedicated to the global well-posedness issue of the compressible Oldroyd-B model in the whole space $\mathbb R^d$ with $d\ge2$. By exploiting the intrinsic structure of the system, it is shown that this set of equations admits a unique global solution in a certain critical Besov space provided the initial data, but not necessarily the coupling parameter, is small enough. This result extends the previous work by Fang and the author [{J. Differential Equations}, {256}(2014), 2559--2602] to the non-small coupling parameter case.

• 12:20-14:30
Lunch

• 14:30-17:30
Free Discussion

• 18:00
Dinner