Extending to systems of hyperbolic–parabolic conservation laws results of Howard and Zumbrun for ... more Extending to systems of hyperbolic–parabolic conservation laws results of Howard and Zumbrun for strictly parabolic systems, we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral (Evans function) conditions for linearized stability established by Mascia and Zumbrun are also sufficient for linearized and nonlinear phase-asymptotic stability, yielding detailed pointwise estimates and sharp rates of convergence in L p, 1 ≤ p ≤ ∞. 1
Journal of Hyperbolic Differential Equations, 2006
Refining previous work of Zumbrun, Mascia–Zumbrun, Raoofi, Howard–Zumbrun and Howard–Raoofi, we d... more Refining previous work of Zumbrun, Mascia–Zumbrun, Raoofi, Howard–Zumbrun and Howard–Raoofi, we derive sharp pointwise bounds on behavior of perturbed viscous shock profiles for large-amplitude Lax or overcompressive type shocks and physical viscosity. These extend well-known results of Liu obtained by somewhat different techniques for small-amplitude Lax type shocks and artificial viscosity, completing a program initiated by Zumbrun and Howard. As pointed out by Liu, the key to obtaining sharp bounds is to take account of cancellation associated with the property that the solution decays faster along characteristic than in other directions. Thus, we must here estimate characteristic derivatives for the entire nonlinear perturbation, rather than judicially chosen parts as in the work of Raoofi and Howard–Raoofi, a requirement that greatly complicates the analysis.
Journal of Hyperbolic Differential Equations, 2005
We investigate the Lp asymptotic behavior (1 ≤ p ≤ ∞) of a perturbation of a Lax or overcompressi... more We investigate the Lp asymptotic behavior (1 ≤ p ≤ ∞) of a perturbation of a Lax or overcompressive type shock wave solution to a system of conservation law in one dimension. The system of the equations can be strictly parabolic, or partially parabolic (real viscosity case, e.g. compressible Navier–Stokes equations or equations of Magnetohydrodynamics). We use known pointwise Green function bounds for the linearized equation around the shock to show that the perturbation of such a solution can be decomposed into a part corresponding to shift in shock position or shape, a part which is the sum of diffusion waves, i.e. the solutions to a viscous Burger's equation, conserving the initial mass and convecting away from the shock profile in outgoing modes, and another part which is more rapidly decaying in any Lp norm.
Extending to systems of hyperbolic–parabolic conservation laws results of Howard and Zumbrun for ... more Extending to systems of hyperbolic–parabolic conservation laws results of Howard and Zumbrun for strictly parabolic systems, we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral (Evans function) conditions for linearized stability established by Mascia and Zumbrun are also sufficient for linearized and nonlinear phase-asymptotic stability, yielding detailed pointwise estimates and sharp rates of convergence in L p, 1 ≤ p ≤ ∞. 1
Journal of Hyperbolic Differential Equations, 2006
Refining previous work of Zumbrun, Mascia–Zumbrun, Raoofi, Howard–Zumbrun and Howard–Raoofi, we d... more Refining previous work of Zumbrun, Mascia–Zumbrun, Raoofi, Howard–Zumbrun and Howard–Raoofi, we derive sharp pointwise bounds on behavior of perturbed viscous shock profiles for large-amplitude Lax or overcompressive type shocks and physical viscosity. These extend well-known results of Liu obtained by somewhat different techniques for small-amplitude Lax type shocks and artificial viscosity, completing a program initiated by Zumbrun and Howard. As pointed out by Liu, the key to obtaining sharp bounds is to take account of cancellation associated with the property that the solution decays faster along characteristic than in other directions. Thus, we must here estimate characteristic derivatives for the entire nonlinear perturbation, rather than judicially chosen parts as in the work of Raoofi and Howard–Raoofi, a requirement that greatly complicates the analysis.
Journal of Hyperbolic Differential Equations, 2005
We investigate the Lp asymptotic behavior (1 ≤ p ≤ ∞) of a perturbation of a Lax or overcompressi... more We investigate the Lp asymptotic behavior (1 ≤ p ≤ ∞) of a perturbation of a Lax or overcompressive type shock wave solution to a system of conservation law in one dimension. The system of the equations can be strictly parabolic, or partially parabolic (real viscosity case, e.g. compressible Navier–Stokes equations or equations of Magnetohydrodynamics). We use known pointwise Green function bounds for the linearized equation around the shock to show that the perturbation of such a solution can be decomposed into a part corresponding to shift in shock position or shape, a part which is the sum of diffusion waves, i.e. the solutions to a viscous Burger's equation, conserving the initial mass and convecting away from the shock profile in outgoing modes, and another part which is more rapidly decaying in any Lp norm.
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Papers by Mohammadreza Raoofi