Ronghua Pan

Global Dynamics of a Hyperbolic-Parabolic Model Arising from Chemotaxis

We prove global existence and qualitative behavior of classical solutions for a hyperbolic-parabolic system describing chemotaxis on bounded domains. It is shown that classical solutions to the initial-boundary value problem of the one-dimensional model exist globally in time for large initial data, and the solutions converge to constant equilibrium states exponentially in time, which rigorously demonstrates the collapsing of cell populations in chemotaxis. Moreover, similar results are established for the multidimensional model when the initial data are small.

On the Diffusive Profiles for the System of Compressible Adiabatic Flow through Porous Media

We study the Cauchy problem for the system of one dimensional compressible adiabatic flow through porous media and the related diffusive problem. We introduce a new approach which combines the usual energy methods with special L1 -estimates and use the weighted Sobolev norms to prove the global existence and large time behavior for the solutions of the problems. The asymptotic states for the solutions are given by either stationary solutions or similarity solutions depending on the behavior of the initial data when

GLOBAL BV SOLUTIONS FOR THE P-SYSTEM WITH FRICTIONAL DAMPING

|x|\rightarrow \infty

NONLINEAR STABILITY OF RAREFaCTION WAVES FOR A RATE-TYPE VISCOELASTIC SYSTEM

. Our estimates provide asymptotic time decay rates.

Singularity Formation for the Compressible Euler Equations

We construct global

NONLINEAR STABILITY OF TWO-MODE SHOCK PROFILES FOR A RATE-TYPE VISCOELASTIC SYSTEM WITH RELAXATION

BV

THE NONLINEAR STABILITY OF TRAVELLING WAVE SOLUTIONS FOR A REACTING FLOW MODEL WITH SOURCE TERM

solutions to the Cauchy problem for the damped p-system, under initial data with distinct end-states. The solution will be realized as a perturbation of its asymptotic profile, in which the specific volume satisfies the porous media equation and the velocity obeys the classical Darcy law for gas flow through a porous medium.

Global Classical Solutions of Three Dimensional Viscous MHD System Without Magnetic Diffusion on Periodic Boxes

The authors study a 3 × 3 rate-type viscoelastic system, which is a relaxation approximation to a 2 × 2 quasi-linear hyperbolic system, including the well-known p-system. It is shown that the rarefaction waves are nonlinear asymptotically stable in this relaxation approximation.

The zero relaxation behavior of piecewise smooth solutions to the reacting flow model in the presence of shocks

It is well-known that shock will form in finite time for hyperbolic conservation laws from initial nonlinear compression no matter how small and smooth the data are. Classical results, including Lax [14], Liu [22], Li-Zhou-Kong [16], confirm that when initial data are small smooth perturbations near constant states, blowup in gradient of solutions occurs in finite time if and only if intial data contain any compression in some truly nonlinear characteristic field. A natural puzzle is that: Will this picture keep true for large data problem of physical systems such as compressible Euler equations? One of the key issues is how to find an effective way to obtain sharp enough control on density lower bound. For isentropic flow, we offer a complete picture on the finite time shock formation from smooth initial data away from vacuum, which is consistent with small data theory. For adiabatic flow, we show a striking observation that initial weak compressions do not necessarily develop singularity in finite time, in a sharp contrast to the small data theory. Furthermore, we find the critical strength of nonlinear compression, and prove that if the compression is stronger than this critical value, then singularity develops in finite time, and otherwise there are a class of initial data admitting global smooth solutions with maximum strength of compression equals to this critical value.

Global Smooth Solutions in

The authors study a 3 × 3 rate-type viscoelastic system, which is a relaxation approximation to a 2 × 2 quasi-linear hyperbolic system, including the well-known p-system. The nonlinear stability of two-mode shock waves in this relaxation approximation is proved.