Shengguo Zhu

Singularity Formation for the Compressible Euler Equations

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.

ON CLASSICAL SOLUTIONS OF THE COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITH VACUUM

In this paper, we consider the 3-D compressible isentropic MHD equations with infinity electric conductivity. The existence of unique local classical solutions is firstly established when the initial data is arbitrarily large, contains vacuum and satisfies some initial layer compatibility condition. The initial mass density needs not be bounded away from zero and may vanish in some open set. Moreover, we prove that the

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

L^infty

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

norm of the deformation tensor of velocity gradients controls the possible blow-up (see cite{olga}cite{zx}) for classical (or strong) solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by the losing the bound of the deformtion tensor as the critical time approches. Our result (see (1.12) is the same as Ponces criterion for

On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities

3

On regular solutions of the

-D incompressible Euler equations cite{pc} and Huang-Li-Xins blow-up criterion for the

3

3

D compressible isentropic Euler-Boltzmann equations with vacuum

-D compressible Navier-stokes equations cite{hup}.

Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum

For the p-system with large BV initial data, an assumption introduced in [N. S. Bakhvalov, Ž. Vycisl. Mat. i Mat. Fiz. (Russian) 10 (1970) 969–980] by Bakhvalov guarantees the global existence of entropy weak solutions with uniformly bounded total variation. The present paper provides a partial converse to this result. Whenever Bakhvalov’s condition does not hold, we show that there exist front tracking approximate solutions, with uniformly positive density, whose total variation becomes arbitrarily large. The construction extends the arguments in [A. Bressan, G. Chen and Q. Zhang, J. Diff. Eqs. 256(8) (2014) 3067–3085] to a general class of pressure laws.