In this paper we consider the existence and stability of traveling wave solutions to Cauchy problem of diagonalizable quasilinear hyperbolic systems. Under the appropriate small oscillation assumptions on the initial traveling waves, we derive the stability result of the traveling wave solutions, especially for intermediate traveling waves. As the important examples, we will apply the results to some systems arising in fluid dynamics and elementary particle physics.

In this paper, we consider the formation of singularities of solutions to the compressible Euler equations with damping and pressureless Euler-Poisson equations with damping. Under certain assumptions of the initial data, we can get the blow up result of the solutions in the finite time.

Whether or not classical solutions to the surface quasi-geostrophic (SQG) equation can develop finite time singularities remains an outstanding open problem. This paper constructs a class of large global-in-time classical solutions to the SQG equation with supercritical dissipation. The construction process presented here implies that any solution of the supercritical SQG equation must be globally regular if its initial data is sufficiently close to a function (measured in a Sobolev norm) whose Fourier transform is supported in a suitable region away from the origin.