arXiv subjects

Corrado Mascia

Publications and source records attributed to Corrado Mascia.

30 records · Page 2Linked to original sources

arXiv2015

Asymptotic Expansion for Harmonic Functions in the Half-Space with a Pressurized Cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half-space with a cavity $C$. Zero normal derivative is assumed at the boundary of the half-space; differently, at $\partial C$, the normal derivative of the function is required to be given by an external datum $g$, corresponding to a pressure term exerted on the medium at $\partial C$. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half-space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum $g$ which describes a constant pressure at $\partial C$, we recover a simplified representation based on a polarization tensor.

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arXiv2014

Stability analysis for linear heat conduction with memory kernels described by Gamma functions

This paper analyzes heat equation with memory in the case of kernels that are linear combinations of Gamma distributions. In this case, it is possible to rewrite the non-local equation as a local system of partial differential equations of hyperbolic type. Stability is studied in details by analyzing the corresponding dispersion relation, providing sufficient stability condition for the general case and sharp instability thresholds in the case of linear combination of the first three Gamma functions.

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arXiv2013

Velocity-jump processes with a finite number of speeds and their asymptotically parabolic nature

The paper examines a class of first order linear hyperbolic systems, proposed as a generalization of the Goldstein-Kac model for velocity-jump processes and determined by a finite number of speeds and corresponding transition rates. It is shown that the large-time behavior is described by a corresponding scalar diffusive equation of parabolic type, defined by a diffusion matrix for which an explicit formula is given. Such representation takes advantage of a variant of the Kirchoff's matrix tree Theorem applied to the graph associated to the system and given by considering the velocities as verteces and the transition rates as weights of the arcs.

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arXiv2013

Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws

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.

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arXiv2013

Slow motion for compressible isentropic Navier--Stokes equations

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.

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arXiv2012

Small, medium and large shock waves for non-equilibrium radiation hydrodynamic

We examine the existence of shock profiles for a hyperbolic-elliptic system arising in radiation hydrodynamics. The algebraic-differential system for the wave profile is reduced to a standard two-dimensional form that is analyzed in details showing the existence of heteroclinic connection between the two singular points of the system for any distance between the corresponding asymptotic states of the original model. Depending on the location of these asymptotic states, the profile can be either continuous or possesses at most one point of discontinuity. Moreover, a sharp threshold relative to presence of an internal absolute maximum in the temperature profile --also called {\sf Zel'dovich spike}-- is rigourously derived.

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arXiv2010

Numerical exploration of a forward-backward diffusion equation

We analyze numerically a forward-backward diffusion equation with a cubic-like diffusion function, -emerging in the framework of phase transitions modeling- and its "entropy" formulation determined by considering it as the singular limit of a third-order pseudo-parabolic equation. Precisely, we propose schemes for both the second and the third order equations, we discuss the analytical properties of their semi-discrete counter-parts and we compare the numerical results in the case of initial data of Riemann type, showing strengths and flaws of the two approaches, the main emphasis being given to the propagation of transition interfaces.

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arXiv2009

Stability of scalar radiative shock profiles

This work establishes nonlinear orbital asymptotic stability of scalar radiative shock profiles, namely, traveling wave solutions to the simplified model system of radiating gas \cite{Hm}, consisting of a scalar conservation law coupled with an elliptic equation for the radiation flux. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution operator. A new feature in the present analysis is the construction of the resolvent kernel for the case of an eigenvalue system of equations of degenerate type. Nonlinear stability then follows in standard fashion by linear estimates derived from these pointwise bounds, combined with nonlinear-damping type energy estimates.

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arXiv2008

Spectral stability of weak relaxation shock profiles

Using a combination of Kawashima- and Goodman-type energy estimates, we establish spectral stability of general small-amplitude relaxation shocks of symmetric dissipative systems. This extends previous results obtained by Plaza and Zumbrun by singular perturbation techniques under an additional technical assumption, namely, that the background equation be noncharacteristic with respect to the shock.

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arXiv2006

Shock waves for radiative hyperbolic--elliptic systems

The present paper deals with the following hyperbolic--elliptic coupled system, modelling dynamics of a gas in presence of radiation, $u_{t}+ f(u)_{x} +Lq_{x}=0, -q_{xx} + Rq +G\cdot u_{x}=0,$ where $u\in\R^{n}$, $q\in\R$ and $R>0$, $G$, $L\in\R^{n}$. The flux function $f : \R^n\to\R^n$ is smooth and such that $\nabla f$ has $n$ distinct real eigenvalues for any $u$. The problem of existence of admissible radiative shock wave is considered, i.e. existence of a solution of the form $(u,q)(x,t):=(U,Q)(x-st)$, such that $(U,Q)(\pm\infty)=(u_\pm,0)$, and $u_\pm\in\R^n$, $s\in\R$ define a shock wave for the reduced hyperbolic system, obtained by formally putting L=0. It is proved that, if $u_-$ is such that $\nablaλ_{k}(u_-)\cdot r_{k}(u_-)\neq 0$,(where $λ_k$ denotes the $k$-th eigenvalue of $\nabla f$ and $r_k$ a corresponding right eigenvector) and $(\ell_{k}(u_{-})\cdot L) (G\cdot r_{k}(u_{-})) >0$, then there exists a neighborhood $\mathcal U$ of $u_-$ such that for any $u_+\in{\mathcal U}$, $s\in\R$ such that the triple $(u_{-},u_{+};s)$ defines a shock wave for the reduced hyperbolic system, there exists a (unique up to shift) admissible radiative shock wave for the complete hyperbolic--elliptic system. Additionally, we are able to prove that the profile $(U,Q)$ gains smoothness when the size of the shock $|u_+-u_-|$ is small enough, as previously proved for the Burgers' flux case. Finally, the general case of nonconvex fluxes is also treated, showing similar results of existence and regularity for the profiles.

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arXiv2002

Pointwise Green's function bounds and stability of relaxation shocks

We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD.

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arXiv2001

Stability of viscous shock profiles for dissipative symmetric hyperbolic-parabolic systems

Combining pointwise Green's function bounds obtained in a companion paper [MZ.2] with earlier, spectral stability results obtained in [HuZ], we establish nonlinear orbital stability of small amplitude viscous shock profiles for the class of dissipative symmetric hyperbolic-parabolic systems identified by Kawashima [Kaw], notably including compressible Navier--Stokes equations and the equations of magnetohydrodynamics, obtaining sharp rates of decay in $L^p$ with respect to small $L^1\cap H^3$ perturbations, $2\le p\le \infty$. Our analysis follows the approach introduced in [MZ.1] to treat stability of relaxation profiles.

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