Well-Posedness of Nonlinear Hyperbolic Problems and the Dynamics of Compressible Fluids (original) (raw)
Structural stability and data dependencefor fully nonlinear hyperbolic problems
1992
where "sup" denotes the essential supremum and p = 2. The above notation will be used both for scalar and vector fields. This convention applies to all notations used in the sequel. In particular, we write v, g E X even if v is a vector, and g, a scalar. Given an arbitrary function f(t, x), we denote by f(t), for each fixed t, the function f(t,.).
Structural stability and data dependence for fully nonlinear hyperbolic mixed problems
Arch Ration Mech Anal, 1992
where "sup" denotes the essential supremum and p = 2. The above notation will be used both for scalar and vector fields. This convention applies to all notations used in the sequel. In particular, we write v, g E X even if v is a vector, and g, a scalar. Given an arbitrary function f(t, x), we denote by f(t), for each fixed t, the function f(t,.).
Indiana University Mathematics Journal, 2002
We consider hyperbolic-parabolic systems of nonlinear partial differential equations. While such systems are difficult to analyze in general, physical applications often display interesting singular limits for which strong solutions may be constructed. These are viewed as approximate solutions of the original problem for parameter values near the singular limit. Restricting ourselves to problems posed in all space, we show that if: (i) The full system linearized about a constant state is dissipative in a special inner product; (ii) The approximate solution and the defect resulting from its substitution into the full system satisfy mild decay conditions; then for all initial data sufficiently close to the approximate solution, a classical solution of the full system exists for all time which remains close to it. We also show how the inner product required in (i) can be constructed when the linearized problem involves a large, symmetric, hyperbolic part and when the coupling between the hyperbolic and parabolic subsystems is nonvanishing. This enables us to apply the general result to the equations of compressible, heat-conducting fluids at low Mach number. In this case the approximate solution is simply determined by a solution of the incompressible Navier-Stokes equations.
Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations
Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2003
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.
Nonlinear Differential Equations and Applications NoDEA, 2011
We investigate the well-posedness of the Cauchy problem for a class of nonlinear parabolic equations with variable density in the hyperbolic space. We state sufficient conditions for uniqueness or nonuniqueness of bounded solutions, depending on the behavior of the density at infinity. Nonuniqueness relies on the prescription at infinity of suitable conditions of Dirichlet type, and possibly inhomogeneous.
Quarterly of Applied Mathematics, 1997
Discontinuous solutions with shocks for a family of almost incompressible hyperelastic materials are studied. An almost incompressible material is one whose deformations are not a priori constrained but whose stress response reacts strongly (of order ε − 1 {\varepsilon ^{ - 1}} ) to deformations that change volume. The material class considered is isotropic and admits motions that are self-similar, exhibit cavitation, and are energy minimizing. For the initial-value problem when considering the entire material, the solutions converge (as ε \varepsilon tends to zero) to an isochoric solution of the limit (incompressible) system with the corresponding arbitrary hydrostatic pressure being the singular limit of the pressures in the almost incompressible materials. The shocks, if they exist, disappear: their speed tends to infinity and their strength tends to zero.