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Hermano Frid

Publications and source records attributed to Hermano Frid.

18 recordsLinked to original sources
arXiv2023

On short wave-long wave interactions\\ in the relativistic context: Application to the Relativistic Euler Equations

In this paper we introduce a model of relativistic short wave-long wave interaction where the short waves are described by the massless $1+3$-dimensional Thirring model of nonlinear Dirac equation and the long waves are described by the $1+3$-dimensional relativistic Euler equations. The interaction coupling terms are modeled by a potential proportional to the relativistic specific volume in the Dirac equation and an external force proportional to the square modulus of the Dirac wave function in the relativistic Euler equation. An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow. In particular, an important contribution of this paper is a clear formulation of the relativistic Lagrangian transformation. This is done by means of the introduction of natural auxiliary dependent variables, rendering the discussion totally similar to the non-relativistic case. As far as the authors know the definition of the Lagrangian transformation given in this paper is new. Finally, we establish the short-time existence and uniqueness of a smooth solution of the Cauchy problem for the regularized model. This follows through the symmetrization of the relativistic Euler equation introduced by Makino and Ukai (1995) and requires a slight extension of a well known theorem of T.~Kato (1975) on quasi-linear symmetric hyperbolic systems.

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arXiv2022

Invariant measures for stochastic parabolic-hyperbolic equations in the space of almost periodic functions: Lipschitz flux case

We study the well-posedness and the long-time behavior of almost periodic solutions to stochastic degenerate parabolic-hyperbolic equations in any space dimension, under the assumption of Lipschitz continuity of the flux and viscosity functions and a non-degeneracy condition. We show the existence and uniqueness of an invariant measure in a separable subspace of the space of Besicovitch almost periodic functions.

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arXiv2022

A Boundary Value Problem for a Class of Anisotropic Stochastic Degenerate Parabolic-Hyperbolic Equations

We establish the well-posedness of an initial-boundary value problem of mixed type for a stochastic nonlinear parabolic-hyperbolic equation on a space domain $\cO=\cO'\X\cO''$ where a Neumann boundary condition is imposed on $\po\cO'\X\cO"$, the hyperbolic boundary, and a Dirichlet condition is imposed on $\cO'\X\po\cO"$, the parabolic boundary. Among other points to be highlighted in our analysis of this problem we mention the new strong trace theorem for the special class of stochastic nonlinear parabolic-hyperbolic equations studied here, which is decisive for the uniqueness of the kinetic solution, and the new averaging lemma for the referred class of equations which is a vital part of the proof of the strong trace property. We also provide a detailed analysis of the approximate nondegenerate problems, which is also made here for the first time, as far as the authors know, whose solutions we prove to converge to the solution of our initial-boundary value problem.

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arXiv2021

On short wave-long wave interactions in the relativistic context

In this paper we introduce models of short wave-long wave interactions in the relativistic setting. In this context the nonlinear Schr\"odinger equation is no longer adequate for describing short waves and is replaced by a nonlinear Dirac equation. Two specific examples are considered: the case where the long waves are governed by a scalar conservation law; and the case where the long waves are governed by the augmented Born-Infeld equations in electromagnetism.

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arXiv2020

On Hodge decomposition, effective viscous flux and compressible Navier-Stokes

It has been known, since the pioneering works by Serre, Hoff, Va\u{i}gant-Kazhikhov, Lions and Feireisl, among others, the regularizing properties of the effective viscous flux and its characterization as the function whose gradient is the gradient part in the Hodge decomposition of the Newtonian force of the fluid, when the shear viscosity of the fluid is constant. In this article, we explore further the connection between the Hodge decomposition of the Newtonian force and the regularizing properties of its gradient part, by addressing the problem of the global existence of weak solutions for compressible Navier-Stokes equations with both viscosities depending on a spatial mollification of the density.

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arXiv2020

Homogenization of Stochastic Conservation Laws with Multiplicative Noise

We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Specially in the second type, these stochastic solutions are crucial elements in the homogenization analysis. Our homogenization method is based on the notion of stochastic two-scale Young measure, whose existence is established here.

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arXiv2019

The Strong Trace Property and the Neumann Problem for Stochastic Conservation Laws

We establish the well-posedness of the Neumann problem for stochastic conservation laws with multiplicative noise. As a major step for establishing the uniqueness of the kinetic solution to the referred problem we establish the new strong trace property for stochastic conservation laws. Existence of kinetic solutions is proved through the vanishing viscosity method and the detailed analysis of the corresponding stochastic parabolic problem is also made here for the first time, as far as the authors know.

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arXiv2019

Global smooth solutions with large data for a system modeling aurora type phenomena in the 2-torus

We prove existence and uniqueness of smooth solutions with large initial data for a system of equations modeling the interaction of short waves, governed by a nonlinear Schr\"odinger equation, and long waves, described by the equations of magnetohydrodynamics. In the model, the short waves propagate along the streamlines of the fluid flow. This is translated in the system by setting up the nonlinear Schr\"odinger equation in the Lagrangian coordinates of the fluid. Besides, the equations are coupled by nonlinear terms accounting for the strong interaction of the dynamics. The system provides a simplified mathematical model for studying aurora type phenomena. We focus on the 2-dimensional case with periodic boundary conditions. This is the first result on existence of smooth solutions with large data for the multidimensional case of the model under consideration.

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arXiv2018

Asymptotic Decay of Besicovitch Almost Periodic Entropy Solutions to Anisotropic Degenerate Parabolic-Hyperbolic Equations

We prove the well-posedness and the asymptotic decay to the mean value of Besicovitch almost periodic entropy solutions to nonlinear aniso\-tropic degenerate parabolic-hyperbolic equations. After setting up the problem and its kinetic formulation on the Bohr compact, the main result, that is, the decay property, ia achieved by devising a suitable adaptation of the technique introduced by Chen and Perthame (2009) in their proof of the decay of periodic entropy solutions to the same equations.

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arXiv2018

Modeling Aurora Type Phenomena by Short Wave-Long Wave Interactions in Multi-Dimensional Large MHD Flows

We establish the convergence of an approximation scheme to a model for aurora type phenomena. The latter, mathematically, means a system describing the short wave-long wave (SW-LW) interactions for compressible magnetohydrodynamic (MHD) flows, introduced in a previous work, which presents short waves, governed by a nonlinear Schr\"odinger (NLS) equation based on the Lagrangian coordinates of the fluid, and long waves, governed by the compressible MHD system. The NLS equation and the compressible MHD system are also explicitly coupled by an interaction potential in the NLS equation and an interaction surface force in the momentum equation of the MHD system, both multiplied by a small coefficient. Since the compressible MHD flow is assumed to have large amplitude data, possibly forming vacuum, the coefficient of the interaction terms may be taken as zero, due to the large difference in scale between the two types of waves. In this case, the whole coupling lies in the Lagrangian coordinates of the compressible MHD fluid upon which the NLS equation is formulated. However, due to the possible occurrence of vacuum, these Lagrangian coordinates are not well defined, and herein lies the importance of the approximation scheme. The latter consists of a system that formally approximates the SW-LW interaction system, including non-zero vanishing interaction coefficients, together with an artificial viscosity in the continuity equation, an artificial energy balance term, an artificial pressure in the momentum equation and approximate Lagrangian coordinates, which circumvent the possible occurrence of vacuum. We prove the convergence of the solutions of the approximation scheme to a solution of a system consisting of a NLS equation based on the coordinate system induced by the scheme, and a compressible MHD system.

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arXiv2016

An extension of Bakhvalov's theorem for systems of conservation laws with damping

For $2\X2$ systems of conservation laws satisfying Bakhvalov conditions, we present a class of damping terms that still yield the existence of global solutions with periodic initial data of possibly large bounded total variation per period. We also address the question of the decay of the periodic solution. As applications we consider the systems of isentropic gas dynamics, with pressure obeying a $\gamma$-law, for the physical range $\gamma\ge1$, and also for the "non-physical" range $0<\gamma<1$, both in the classical Lagrangian and Eulerian formulation, and in the relativistic setting. We give complete details for the case $\gamma=1$, and also analyze the general case when $|\gamma-1|$ is small. Further, our main result also establishes the decay of the periodic solution.

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arXiv2016

A Boundary Value Problem for a Class of Anisotropic Degenerate Parabolic-Hyperbolic Equations

We consider a mixed type boundary value problem for a class of degenerate parabolic-hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one of them we impose a Dirichlet boundary condition; in the other, we impose a Neumann condition. We apply a normal trace formula for $L^2$-divergence-measure fields to prove a new strong trace property in the part of the boundary where the Neumann condition is imposed. We prove existence and uniqueness of the entropy solution. This is a revised corrected version of the paper published in Arch.\ Ration.\ Mech.\ Anal.\ {\bf 226} (2017), no. 3, 975--1008.

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arXiv2014

A note on the stochastic weakly* almost periodic homogenization of fully nonlinear elliptic equations

A function $f\in \BUC(\R^d)$ is said to be weakly* almost periodic, denoted $f\in\APs(\R^d)$, if there is $g\in\AP(\R^d)$, such that, $\oM(|f-g|)=0$, where $\BUC(\R^d)$ and $\AP(\R^d)$ are, respectively, the space of bounded uniformly continuous functions and the space of almost periodic functions, in $\R^d$, and $\oM(h)$ denotes the mean value of $h$, if it exists. We give a very simple direct proof of the stochastic homogenization property of the Dirichlet problem for fully nonlinear uniformly elliptic equations of the form $F(\om,\frac{x}{\ve},D^2u)=0$, $x\in U$, in a bounded domain $U\subset\R^d$, in the case where for almost all $\om\in \Om$, the realization $F(\om,\cdot,M)$ is a weakly* almost periodic function, for all $M\in§^d$, where $§^d$ is the space of $d\X d$ symmetric matrices. Here $(\Om,μ,\FF)$ is a probability space with probability measure $μ$ and $\s$-algebra $\FF$ of $μ$-measurable subsets of $\Om$. For each fixed $M\in§^d$, $F(\om,y,M)$ is a stationary process, that is, $F(\om,y,M)=\tilde F(T(y)\om,M):= F(T(y)\om,0,M)$, where $T(y):\Om\to\Om$ is an ergodic group of measure preserving mappings such that the mapping $(\om,y)\to T(y)\om$ is measurable. Also, $F(\om,y,M)$, $M\in§^d$, is uniformly elliptic, with ellipticity constants $0<ł<\gL$ independent of $(\om,y)\in\Om\X\R^d$. The result presented here is a particular instance of the general theorem of Caffarelli, Souganidis and Wang, in CPAM 2005. Our point here is just to show a straightforward proof for this special case, which serves as a motivation for that general theorem, whose proof involves much more intricate arguments. We remark that any continuous stationary process verifies the property that almost all realizations belong to an ergodic algebra, and that $\APs(\R^d)$ is, so far, the greatest known ergodic algebra on $\R^d$.

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arXiv2013

Homogenization of a generalized Stefan Problem in the context of ergodic algebras

We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem which generalizes the well known Stefan model, and includes the classical porous medium equation. It may be represented by the differential inclusion, for a real-valued function $u(x,t)$, $$ \frac{\partial}{\partial t}\partial_u Ψ(x/\ve,x,u)-\nabla_x\cdot \nabla_ηψ( x/\ve,x,t,u,\nabla u) \ni f(x/\ve,x,t, u), $$ on a bounded domain $\Om\subset \R^n$, $t\in(0,T)$, together with initial-boundary conditions, where $Ψ(z,x,\cdot)$ is strictly convex and $ψ(z,x,t,u,\cdot)$ is a $C^1$ convex function, both with quadratic growth, satisfying some additional technical hypotheses. As functions of the oscillatory variable, $Ψ(\cdot,x,u),ψ(\cdot,x,t,u,η)$ and $f(\cdot,x,t,u)$ belong to the generalized Besicovitch space $\BB^2$ associated with an arbitrary ergodic algebra $Å$. The periodic case was addressed by Visintin (2007), based on the two-scale convergence technique. Visintin's analysis for the periodic case relies heavily on the possibility of reducing two-scale convergence to the usual $L^2$ convergence in the cartesian product $Π\X\R^n$, where $Π$ is the periodic cell. This reduction is no longer possible in the case of a general ergodic algebra. To overcome this difficulty, we make essential use of the concept of two-scale Young measures for algebras with mean value, associated with bounded sequences in $L^2$.

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arXiv2013

Homogenization of Degenerate Porous Medium Type Equations in Ergodic Algebras

We consider the homogenization problem for general porous medium type equations of the form $u_t=\D f(x,\frac{x}{\ve}, u)$. The pressure function $f(x,y,\cdot)$ may be of two different types. In the type~1 case, $f(x,y,\cdot)$ is a general strictly increasing function; this is a mildly degenerate case. In the type~2 case, $f(x,y,\cdot)$ has the form $h(x,y)F(u)+S(x,y)$, where $F(u)$ is just a nondecreasing function; this is a strongly degenerate case. We address the initial-boundary value problem for a general, bounded or unbounded, domain $\Om$, with null (or, more generally, steady) pressure condition on the boundary. The homogenization is carried out in the general context of ergodic algebras. As far as the authors know, homogenization of such degenerate quasilinear parabolic equations is addressed here for the first time. We also review the existence and stability theory for such equations and establish new results needed for the homogenization analysis. Further, we include some new results on algebras with mean value, specially a new criterion establishing the null measure of level sets of elements of the algebra, which is useful in connection with the homogenization of porous medium type equations in the type~2 case.

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arXiv2010

$L^\infty$ solutions for a model of polytropic gas flow with diffusive entropy

We establish the global existence of $L^\infty$ solutions for a model of polytropic gas flow with diffusive entropy. The result is obtained by showing the convergence of a class of finite difference schemes, which includes the Lax-Friedrichs and Godunov schemes. Such convergence is achieved by proving the estimates required for the application of the compensated compactness theory.

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