Gui-Qiang Chen

Formation of

The phenomena of concentration and cavitation and the formation of

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-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to the Euler Equations for Isentropic Fluids

-shocks and vacuum states in solutions to the Euler equations for isentropic fluids are identified and analyzed as the pre...

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following second-order nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential φ : Ω ⊂ R → R: (1.1) div (ρ(|Dφ|)Dφ) = 0, where the density function ρ(q) is

GLOBAL SOLUTIONS TO THE COMPRESSIBLE EULER EQUATIONS WITH GEOMETRICAL STRUCTURE

AbstractWe prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow and spherically symmetric flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especially as they pertain to radial flow in an unbounded region,

Global solutions to the compressible Navier-Stokes equations for a reacting mixture

|\vec x| \geqq 1,

THE CAUCHY PROBLEM FOR THE EULER EQUATIONS FOR COMPRESSIBLE FLUIDS

, and to transonic nozzle flow. Arbitrary data withL∞ bounds are allowed in these results. A shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework.

Spectral Viscosity Approximations to Multidimensional Scalar Conservation Laws

We prove the global existence of weak solutions to the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data, and we obtain apriori estimates for these solutions which are independent of time, sufficient to determine their asymptotic behavior. In particular, we show that, as time goes to infinity, the solution tends to a constant state determined by the initial mass and the initial energy. and that the magnitudes of singularities in the solution decay to zero.