arXiv subjects
Publications and source records attributed to Marta Strani.
In this paper we describe the metastable behavior of solutions to a class of parabolic systems. In particular, we improve some results contained in \cite{MS} by using different techniques to describe the slow motion of the internal layers. Numerical simulations illustrate the results.
This paper considers the slow motion of the shock layer exhibited by the solution to the initial-boundary value problem for a scalar hyperbolic system with relaxation. Such behavior, known as metastable dynamics, is related to the presence of a first small eigenvalue for the linearized operator around an equilibrium state; as a consequence, the time-dependent solution approaches its steady state in an asymptotically exponentially long time interval. In this contest, both rigorous and asymptotic approaches are used to analyze such slow motion for the Jin-Xin system. To describe this dynamics we derive an ODE for the the position of the internal transition layer, proving how it drifts towards the equilibrium location with a speed rate that is exponentially slow. These analytical results are also validated by numerical computations.
The aim article is to contribute to the definition of a versatile language for metastability in the context of partial differential equations of evolutive type. A general framework suited for parabolic equations in one dimensional bounded domains is proposed, based on choosing a family of approximate steady states, and on the spectral properties of the linearized operators at such states. The slow motion for solutions belonging to a cylindrical neighborhood of the family of approximate steady states is analyzed by means of a system of an ODE for the parameter $ξ$ that describes the family, coupled with a PDE describing the evolution of the perturbation $v$. We state and prove a general result concerning the reduced system for the couple $(ξ,v)$, called quasi-linearized system, obtained by disregarding the nonlinear term in $v$, and we show how such approach suits to the prototypical example of scalar viscous conservation laws with Dirichlet boundary condition in a bounded one-dimensional interval with convex flux.
We consider the compressible Navier-Stokes equations for isentropic dynamics with real viscosity on a bounded interval. In the case of boundary data defining an admissible shock wave for the corresponding unviscous hyperbolic system, we determine a scalar differential equation describing the motion of the internal transition layer. In particular, for small viscosity, the velocity of the motion is exponentially small. The approach is based on the construction of a one-parameter manifold of approximate solutions and on an appropriate projection of the evolution of the complete Navier-Stokes system towards such manifold.