Nonlinear Balance Laws in Applied Sciences(1/2)

In this two-years project, we study the existence, uniqueness and behavior ofsolutions to several models of nonlinear systems of balance laws arise in areas of AppliedSciences. The models are governed by the following PDE systems:(1)Compressible Euler equations with gravitational and global heating source inhydrodynamic escape problem (HEP in short) of astrophysics.(2) Keller-Segel systems with small parameters in the chemotaxis of Biology.(3)Compressible Euler-Poisson equations in gravitation theory.In the first year, we focus on the stability of solutions to the HEP model by a newversion of finite difference scheme that involves an iteration of global heating source.The multiple-phases model of HEP is also studied on the steady states and the globalwell-posedness of time-evolutionary solutions. Numerical simulations are also provided.In the second year, we focus on the stability issue of traveling pulses for Keller-Segelsystems with small parameters. We wish to show that the only asymptotically stabletraveling pulse is the steady state. For the compressible Euler-Poisson equations ingravitations, we find the condition of boundary momentum to prevent the gravitationalcollapse of the gas-like stars. We wish to obtain both analytic and numerical results.