Scalar conservation laws with stochastic forcing

We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.     

Existence,uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces

Summary A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupR _t are obtained. We show thatR _t is a compactC ₀-semigroup in all Sobolev spacesW ^n,p which are built on its invariant measure . Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inL ^p() spaces and spacesW ^1,p . As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to . It is shown also that the density of this measure with respect to is inL ^p() for allp1.This work was done during the first author's stay at UNSW supported by ARC Grant 150.346 and the second author's stay at ód University supported by KBN Grant 2.1020.91.01     

Convergence of a stochastic particle approximation for fractional scalar conservation laws

We are interested in a probabilistic approximation of the solution to scalar conservation laws with fractional diffusion and nonlinear drift. The probabilistic interpretation of this equation is based on a stochastic differential equation driven by an α-stable Lévy process and involving a nonlinear drift. The approximation is constructed using a system of particles following a time-discretized version of this stochastic differential equation, with nonlinearity replaced by interaction. We prove convergence of the particle approximation to the solution of the conservation law as the number of particles tends to infinity whereas the discretization step tends to 0 in some precise asymptotics.     

Existence and uniqueness of traveling waves and error estimates for Godunov schemes of conservation laws

The existence and uniqueness of the Lipschitz continuous traveling wave of Godunov's scheme for scalar conservation laws are proved. The structure of the traveling waves is studied. The approximation error of Godunov's scheme on single shock solutions is shown to be .

     

Scalar conservation laws with monotone pure-jump Markov initial conditions

In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions \(\rho \) to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe \(\rho (x,t)\) as a stochastic process in x with t fixed. In this article we verify an analogue of the conjecture for initial conditions which are bounded, monotone, and piecewise constant. Our argument uses a particle system representation of \(\rho (x,t)\) over \(0 \le x \le L\) for \(L > 0\), with a suitable random boundary condition at \(x = L\).     

Quasilinear parabolic stochastic partial differential equations: Existence,uniqueness

We present a direct approach to existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone. The proof of uniqueness is very elementary, based on a new method of applying Itô’s formula for the $L^1$-norm. The proof of existence relies on a recent regularity result and is direct in the sense that it does not rely on the stochastic compactness method.     

Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q-Brownian motion

In this paper, we prove the existence and uniqueness of the entropy solution for a first-order stochastic conservation law with a multiplicative source term involving a $Q$-Brownian motion. After having defined a measure-valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure-valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure-valued entropy solution of the stochastic conservation law, which proves the existence of the measure-valued entropy solution.     

Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths

In this note we consider a chain of N$N$ oscillators, whose ends are in contact with two heat baths at different temperatures. Our main result is the exponential convergence to the unique invariant probability measure (the stationary state). We use the Lyapunov’s function technique of Rey-Bellet and coauthors [Luc Rey-Bellet, Statistical mechanics of anharmonic lattices, in: Advances in Differential Equations and Mathematical Physics (Birmingham, AL, 2002), in: Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 283–298. MR MR1991548 (2005a:82068) [11]; Luc Rey-Bellet, Lawrence E. Thomas, Fluctuations of the entropy production in anharmonic chains, Ann. Henri Poincaré 3 (3) (2002) 483–502. MR MR1915300 (2003g:82060); Luc Rey-Bellet, Lawrence E. Thomas, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Comm. Math. Phys. 225 (2) (2002) 305–329. MR MR1889227 (2003f:82052); Luc Rey-Bellet, Lawrence E. Thomas, Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators, Comm. Math. Phys. 215 (1) (2000) 1–24. MR MR1799873 (2001k:82061) [12]; Jean-Pierre Eckmann, Claude-Alain Pillet, Luc Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Comm. Math. Phys. 201 (3) (1999) 657–697. MR MR1685893 (2000d:82025); Jean-Pierre Eckmann, Claude-Alain Pillet, Luc Rey-Bellet, Entropy production in nonlinear, thermally driven Hamiltonian systems, J. Statist. Phys. 95 (1–2) (1999) 305–331. MR MR1705589 (2000h:82075)], with different model of heat baths, and adapt these techniques to two new case recently considered in the literature by Bernardin and Olla [Cédric Bernardin, Stefano Olla, Fourier’s law for a microscopic model of heat conduction, J. Statist. Phys. 121 (3–4) (2005) 271–289. MR MR2185330] and Lefevere and Schenkel [R. Lefevere, A. Schenkel, Normal heat conductivity in a strongly pinned chain of anharmonic oscillators, J. Stat. Mech. Theory Exp. 2006 (02) (2006) L02001].     

Stochastic wave equation of pure jumps: Existence,uniqueness and invariant measures

In this paper, we investigate a class of nonlinear damped stochastic hyperbolic equations with jumps. The jump component considered here is described as a Poisson point process. This paper is divided into two parts. The first part deals with existence and uniqueness of global weak and strong solutions to this type of equations, based on the energy approach. The second part devotes to the existence and support of invariant measures corresponding to the weak solution semi-group, based on Markov property of the solution.     

Convex conservation laws with discontinuous coefficients. existence,uniqueness and asymptotic behavior

Existence and uniqueness is proved, in the class of functions satisfying a wave entropy condition, of weak solutions to a conservation law with a flux function that may depend discontinuously on the space variable. The large time limit is then studied, and explicit formulas for this limit is given in the case where the initial data as well as the x dependency of the flux vary periodically. Throughout the paper, front tracking is used as a method of analysis. A numerical example which illustrates the results and method of proof is also presented.     

Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u₀ on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u₀∈L^∞(Ω), f∈L^∞(Q), a∈L^∞((0,T)×∂Ω). In the L¹-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.     

Existence,uniqueness, and numerical approximations for stochastic Burgers equations

Abstract

In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.     

Existence and uniqueness of unbounded entropy solutions of the Cauchy problem for first-order quasilinear conservation laws

We indicate conditions for the well-posedness of the Cauchy problem for a scalar quasilinear conservation law in the class of locally bounded functions. We construct examples showing that if these conditions are violated, then the Cauchy problem may fail to have a generalized entropy solution.     

Systems of stochastic partial differential equations with reflection: Existence and uniqueness

In this paper, we establish the existence and uniqueness of solutions of systems of stochastic partial differential equations (SPDEs) with reflection in a convex domain. The lack of comparison theorems for systems of SPDEs makes things delicate.     

Existence of an invariant measure for stochastic evolutions driven by an eventually compact semigroup

It is shown that for an SDE in a Hilbert space, eventual compactness of the driving semigroup together with compact perturbations can be used to establish the existence of an invariant measure. The result is applied to stochastic functional differential equations and the heat equation perturbed by delay and noise, which are both shown to be driven by an eventually compact semigroup.     

Duality solutions for pressureless gases,monotone scalar conservation laws,and uniqueness

We introduce for the system of pressureless gases a new notion of solution, which consist in interpreting the system as two nonlinearly coupled linear equations. We prove In this setting existence of solutions for the Cauchy Problem, as well as uniqueness under optimal conditions on initlaffata. The proofs rely on the detailed study of the relations between pressureless gases, tie dynamics of sticky particles and nonlinear scalar conservation laws with monotone initial data. We prove for the latter problem that monotonicit implies uniqueness. and a generalization of Oleinik's entropy condition     

Existence and uniqueness for a stochastic age-structured population system with diffusion

In this paper, a class of stochastic age-dependent population dynamic system with diffusion is introduced. Existence and uniqueness of strong solution for a stochastic age-dependent population dynamic system in Hilbert space are established. The analysis use Barkholder–Davis–Gundy’s inequality, Itô’s formula and some special inequalities for our purposes.