Qingtian Zhang

Unique Conservative Solutions to a Variational Wave Equation

Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation

No BV bounds for approximate solutions to p-system with general pressure law

{u_{tt} - c (u) (c(u)u_{x}) x = 0}

Local Well-Posedness of an Approximate Equation for SQG Fronts

utt-c(u)(c(u)ux)x=0. Given a solution u(t, x), even if the wave speed c(u) is only Hör continuous in the t – x plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables X, Y, constant along characteristics, we prove that t, x, u, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains the uniqueness of conservative solutions to the Cauchy problem for the wave equation with general initial data

Singularities of solutions to compressible Euler equations with vacuum

{u(0, \cdot) \in H^{1}(I\!R), u_{t} (0, \cdot) \in L^{2}(I\!R).}

Blow-up criterion for compressible visco-elasticity equations in three dimensional space

u(0,·)∈H1(IR),ut(0,·)∈L2(IR).

Lack of BV bounds for approximate solutions to the p-system with large data

We prove shock formation results for the compressible Euler equations and related systems of conservation laws in one space dimension, or three dimensions with spherical symmetry. We establish an L∞ bound for C1 solutions of the one-dimensional (1D) Euler equations, and use this to improve recent shock formation results of the authors. We prove analogous shock formation results for 1D magnetohydrodynamics (MHD) with orthogonal magnetic field, and for compressible flow in a variable area duct, which has as a special case spherically symmetric three-dimensional (3D) flow on the exterior of a ball.