We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N x N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 x 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 X 2 case. 0 1994 John Wiley & Sons, Inc.
The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the Euler equations for isentropic fluids are identified and analyzed as the pressure vanishes. It is shown that, as the pressure vanishes, any two-shock Riemann solution to the Euler equations for isentropic fluids tends to a δ-shock solution to the Euler equations for pressureless fluids, and the intermediate density between the two shocks tends to a weighted δmeasure that forms the δ-shock. By contrast, any two-rarefaction-wave Riemann solution of the Euler equations for isentropic fluids is shown to tend to a two-contact-discontinuity solution to the Euler equations for pressureless fluids, whose intermediate state between the two contact discontinuities is a vacuum state, even when the initial data stays away from the vacuum. Some numerical results exhibiting the formation process of δ-shocks are also presented.
We develop a well-posedness theory for solutions in L 1 to the Cauchy problem of general degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; and its advantage is that the kinetic equations in the kinetic formulation are well defined even when the macroscopic fluxes are not locally integrable, so that L 1 is a natural space on which the kinetic solutions are posed. Based on this notion, we develop a new, simpler, more effective approach to prove the contraction property of kinetic solutions in L 1 , especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it different from the isotropic case.
We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures.To overcome these difficulties, we first develop uniform energy-type estimates with respect to the viscosity coefficient for solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the NavierStokes equations for weak entropy-entropy flux pairs, generated by compactly supported C 2 test functions, are confined in a compact set in H 1 , which leads to the existence of measure-valued solutions that are confined by the TartarMurat commutator relation.A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations with finite-energy initial data, relative to the different end-states at infinity.
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