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Well-posedness and wave-breaking for the stochastic rotation-two-component Camassa-Holm system

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¶«ÄÏ´óѧ

We study the global well-posedness and wave-breaking phenomenon for the stochastic rotation-two-component Camassa-Holm (R2CH) system. First, we find a Hamiltonian structure of the R2CH system and use the stochastic Hamiltonian to derive the stochastic R2CH system. Then, we establish the local well-posedness of the stochastic R2CH system by the dispersion-dissipation approximation system and the regularization method. We also show a precise blow-up criterion for the stochastic R2CH system. Moreover, we prove the global existence of the stochastic R2CH system occurs with high probability. At last, we consider transport noise case and establish the local well-posedness and another blow-up criterion.

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Global existence of spherically symmetric solutions of compressible Euler-Poisson equations for white draft

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Öйú¿Æ±¼³Û±¦Âí3555APPÊýѧÓëϵͳ¿ÆѧÑо¿Ôº

In this talk, the three-dimensional Euler-Poisson equations with gravitational potential for general pressure law is considered. It is shown that there exists a global finite-energy solution with spherical symmetry for Cauchy problem by the theory of compensated compactness, and no concentration (delta measure) is formed in the vanishing viscosity limit. Moreover, the constitutive equation of white dwarf stars is included. The results can be extended to the three-dimensional compressible Euler equation with far field vacuum in the same way.

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Some recent results on compressible Navier-Stokes equations

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Äϲý´óѧ&Öйú¿Æ±¼³Û±¦Âí3555APPÊýѧÓëϵͳ¿ÆѧÑо¿Ôº

We investigate the barotropic compressible Navier-Stokes equations with slip boundary conditions in a three-dimensional (3D) simply connected bounded domain, whose smooth boundary has a finite number of two-dimensional connected components. For any adiabatic exponent bigger than one, after obtaining some new estimates on boundary integrals related to the slip boundary conditions, we prove that both the weak and classical solutions to the initial boundary value problem of this system exist globally in time provided the initial energy is suitably small. Moreover, the density has large oscillations and contains vacuum states. Finally, it is also shown that for the classical solutions, the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially.

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Some results on nonlocal dispersal SIS models in heterogeneous environments

ÀîÍòͬ

À¼ÖÝ´óѧ

In this talk we consider a nonlocal dispersal SIS epidemic model, where the spatial movement of individuals is described by a nonlocal diffusion operator, the transmission rate and recovery rate are spatially heterogeneous. We first define the basic reproduction number and discuss the existence, uniqueness and stability of steady states of the nonlocal dispersal SIS epidemic model in terms of . Then we study the asymptotic profiles of the endemic steady states for large and small diffusion rates to illustrate the persistence or extinction of the infectious disease. We also observe the concentration phenomenon which occurs when the diffusion rate of the infected individuals tends to zero. The obtained results indicate that the nonlocal movement of the susceptible or infectious individuals will enhance the persistence of the infectious disease. In particular, our analytical results suggest that the spatial heterogeneity tends to boost the spread of the infectious disease. This talk is based on joint works with Yan-Xia Feng, Shigui Ruan and Fei-Ying Yang.

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Öйú¿Æѧ¼¼Êõ´óѧ

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Propagation of mean curvature flows in cylinders

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ÉϺ£Ê¦·¶´óѧ

I will talk about mean curvature flows in cylinders, which arise in singular limit problems of Allen-Cahn equations. For the problems with periodic/almost periodic/recurrent boundary conditions, I will state the existence, uniqueness and the stability of corresponding traveling wave solutions (translating solutions). For the problem with unbounded boundary slopes, I will give the asymptotic profiles of the curvature flows.

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Global existence and non-existence analyses for a semilinear edge

degenerate parabolic equation with singular potential term

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ÖØÇì´óѧ

This talk discusses initial boundary value problem for a semilinear edge degenerate parabolic equation and corresponding stationary problem. We first find some initial conditions with different energy levels such that the solution exists globally and blows up in finite time, respectively. We also study the asymptotic behaviors like exponential decay and exponential growth for solution and energy function. Especially, we show the solution of evolution problem will converge to the steady state solution. Additionally, we find that there are two explicit vacuum regions which are ball and annulus respectively, that is to say, there is no solution belongs to them and all solutions are isolated by them. Finally, we discuss the existence of ground state solution to the stationary problem. The instability of the ground state solution is considered and we prove that there exists initial value such that the instability occurs as a blow-up in finite time. This is a joint work with Guangyu Xu and Yafeng Li.

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Toward a nonlocal nutrient taxis model

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ÉϺ£½»Í¨´óѧ

As a simplified version of a three-component taxis cascade model accounting for different migration strategies of two population groups in search of food, a two-component nonlocal nutrient taxis system is considered in a two-dimensional bounded convex domain with smooth boundary. We shall discuss the global existence, boundedness as well as stabilization of a global classical solution to the corresponding no-flux initial-boundary value problem. This is a joint work with Michael Winkler (Paderborn).

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Lipschitz continuity of two-dimensional subsonic-sonic flows

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Steady subsonic-sonic potential flows are governed by a nonlinear degenerate elliptic equation. By a Moser iteration, it is shown that a two-dimensional subsonic-sonic flow is locally Lipschitz continuous. The flow is also Lipschitz continuous on a given smooth streamline.

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Global solutions and regularity of active hydrodynamics

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University of Pittsburgh

Active liquid crystals describe fluids with active constituent particles that have elongated shapes arising in wide applications. In this talk, I will present some mathematical results on the existence and regularity of global solutions to the equations governing the active hydrodynamics and discuss some open problems.

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Global asymptotic behaviors of the diffusive prey-predator model

with variable coefficients

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The global asymptotic behaviors is an important topic in the study of reaction diffusion equations. It is of interest to understand the affect of variable coefficients on the global asymptotic behaviors of solutions of reaction diffusion equations. In this talk, we shall show that variable coefficients satisfying certain conditions will not affect the global asymptotic behaviors of solutions for the diffusive prey-predator model.

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Principal spectral theory and variational characterizations

for cooperative systems with nonlocal diffusion

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Ïã¸ÛÖÐÎÄ´óѧ(ÉîÛÚ)

We study a general class of cooperative systems with nonlocal diffusion operators that may or may not be coupled. These systems are either ¡°strong¡± in cooperation or ¡°strong¡± in the coupling of the nonlocal diffusion operators, and in the former case, diffusion may not occur in some of the components of the system at all. We prove results concerning the existence, uniqueness, multiplicity, variational characterizations of the principal eigenvalue of the system, the spectral bound, the essential spectrum, and the relationship between the sign of principal eigenvalue and the validity of the maximum principle. We do so using an elementary method, without resorting to Krein-Rutmen theorem. This is a joint work with Yuanhang Su and Ting Zhang.

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Existence of multi-dimensional MHD contact discontinuities

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ÏÃÃÅ´óѧ

Contact discontinuities of the ideal compressible magnetohydrodynamics (MHD) are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic conservation laws. We prove the local existence and uniqueness of MHD contact discontinuities in both 2D and 3D in Sobolev spaces without any additional conditions, which in particular gives a complete answer to the two open questions raised by Morando, Trakhinin and Trebeschi, and there is no loss of derivatives in our well-posedness theory. The solution is constructed as the inviscid limit of solutions to suitably-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD. This is a joint work with Professor Zhouping Xin (CUHK).

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Local and global stability of cylinder waves for a nonlocal reaction-diffusion model with bounded phenotypic traits

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Ê׶¼Ê¦·¶´óѧ

Consider the following Fisher type equation with nonlocal reaction term under zero Neumann boundary condition in higher dimensional cylinder

In this talk we shall talk about our recent work on the local and global asymptotic stability of the traveling waves with noncritical speeds by applying spectral analysis, sub-supper solution methods and decomposition techniques. It is a joint work with Xinfu Chen (Pittsburgh University) and Qing Li (ÉϺ£º£Ê´óѧ) .

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Global existence and decay rates to self-consistent chemotaxis-fluid system

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In this talk, we investigate the Cauchy problem of a chemotaxis-fluid system involving both the effect of potential force on cells and the effect of chemotactic force on fluid. One of the novelties and difficulties here is that the coupling in this model is stronger and more nonlinear than the most-studied chemotaxis-fluid model. We will first establish several extensibility criteria of classical solutions, which ensure us to extend the local solutions to global ones in the three-dimensional chemotaxis-Stokes case and in the two-dimensional chemotaxis-Navier-Stokes version under suitable smallness assumption on the initial chemical concentration with the help of a new entropy functional inequality. Some further decay estimates are also obtained under some suitable growth restriction on the potential at infinity. As a byproduct of the entropy functional inequality, we also establish the global-in-time existence of weak solutions to the three-dimensional chemotaxis-Navier-Stokes system. To the best of our knowledge, this seems to be the first work addressing the global well-posedness and decay property of solutions to the Cauchy problem of self-consistent chemotaxis-fluid system. This is a joint work with Prof Jose A Carrillo and Dr Yingping Peng.

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On the critical exponent of the 3D quasilinear wave equation

with the short pulse initial data

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ÄϾ©Ê¦·¶´óѧ

Consider the 3D quasilinear wave equation with the short pulse initial data , where , and is sufficiently small. Under the outgoing constraint condition for , we will establish the global existence of smooth large data solution when with being the critical exponent, and meanwhile show the formation of the outgoing shock before the time under the suitable assumption of when . This is joint work with Prof. Ding Bingbing and Lu Yu.

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Global well-posedness and regularity of weak solutions to the Prandtl's system

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Öйú¿Æ±¼³Û±¦Âí3555APPÊýѧÓëϵͳ¿ÆѧÑо¿Ôº

We shall talk about the study on the global solution to the two-dimensional Prandtl's system for unsteady boundary layers in the class considered by Oleinik provided that the pressure is favorable. First, by using a different method from our earlier works, we gave a direct proof of existence of a global weak solution by a direct BV estimate. Then we prove the uniqueness and continuous dependence on data of such a weak solution to the initial boundary value problem. Finally, we show the smoothness of the weak solutions and then the global existence of smooth solutions. This is a jointed work with Zhou Ping Xin and Jun Ning Zhao.

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Hilbert expansion for Landau type equations with non-relativistic Coulomb collision

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Î人´óѧ

This talk is concerned with the hydrodynamic limits of both the Landau equation and the Vlasov-Maxwell-Landau system in the whole space. Our main purpose is two-fold: the first one is to give a rigorous derivation of the compressible Euler equations from the Landau equation via the Hilbert expansion; while the second one is to prove, still in the setting of Hilbert expansion, that the unique classical solution of the Vlasov-Maxwell-Landau system converges, which is shown to be globally in time, to the resulting global smooth solution of the Euler-Maxwell system, as the Knudsen number goes to zero. The main ingredient of our analysis is to derive some novel interplay energy estimates on the solutions of the Landau equation and the Vlasov-Maxwell-Landau system which are small perturbations of both a local Maxwellian and a global Maxwellian, respectively.