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.     

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.     

Scalar multidimensional conservation laws IBVP in noncylindrical Lipschitz domains

We study the initial-boundary-value problems for multidimensional scalar conservation laws in noncylindrical domains with Lipschitz boundary. We show the existence-uniqueness of this problem for initial-boundary data in L^∞ and the flux-function in the class C¹. In fact, first considering smooth boundary, we obtain the L¹-contraction property, discuss the existence problem and prove it by the Young measures theory. In the end we show how to pass the existence-uniqueness results on to some domains with Lipschitz boundary.     

Solutions to quasilinear hyperbolic conservation laws with initial discontinuities

We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t T~*.Then, when T~*≤ t T′, we get a new shock wave between two rarefactions, and then, when t T′,another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t →∞. The technique does not involve dimensional reduction or coordinate transformation.     

Necessary embedding conditions for state-wise monotone Markov chains

In previous work, the embedding problem is examined within the entire set of discrete-time Markov chains. However, for several phenomena, the states of a Markov model are ordered categories and the transition matrix is state-wise monotone. The present paper investigates the embedding problem for the specific subset of state-wise monotone Markov chains. We prove necessary conditions on the transition matrix of a discrete-time Markov chain with ordered states to be embeddable in a state-wise monotone Markov chain regarding time-intervals with length 0.5: A transition matrix with a square root within the set of state-wise monotone matrices has a trace at least equal to 1.     

Scalar conservation laws with fractional stochastic forcing: Existence,uniqueness and invariant measure

We study a fractional stochastic perturbation of a first-order hyperbolic equation of nonlinear type. The existence and uniqueness of the solution are investigated via a Lax–Ole?nik formula. To construct the invariant measure we use two main ingredients. The first one is the notion of a generalized characteristic in the sense of Dafermos. The second one is the fact that the oscillations of the fractional Brownian motion are arbitrarily small for an infinite number of intervals of arbitrary length.     

Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result

We consider a strongly coupled PDE–ODE system that describes the influence of a slow and large vehicle on road traffic. The model consists of a scalar conservation law accounting for the main traffic evolution, while the trajectory of the slower vehicle is given by an ODE depending on the downstream traffic density. The moving constraint is expressed by an inequality on the flux, which models the bottleneck created in the road by the presence of the slower vehicle. We prove the existence of solutions to the Cauchy problem for initial data of bounded variation.     

A solution formula for a non-convex scalar hyperbolic conservation law with monotone initial data

In this paper we prove an explicit representation formula for the solution of a one-dimensional hyperbolic conservation law with a non-convex flux function but monotone initial data. This representation formula is similar to those of Lax [10] and Kunik [7,8] and enables us to compute the solution pointwise explicitly. This result is a generalization of a theorem given in Kunik [8] where the case of only one inflexion point for the fluxes was considered. Its proof uses the polygonal method of Dafermos [2]. The application of this method leads to a simple explicit construction of the solutions for a Kynch sedimentation process [9] and to an explicit parameter representation for the shock curves evolving during the sedimentation process.     

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     

Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains

Summary. This paper is devoted to the study of the finite volume methods used in the discretization of conservation laws defined on bounded domains. General assumptions are made on the data: the initial condition and the boundary condition are supposed to be measurable bounded functions. Using a generalized notion of solution to the continuous problem (namely the notion of entropy process solution, see [9]) and a uniqueness result on this solution, we prove that the numerical solution converges to the entropy weak solution of the continuous problem in for every . This also yields a new proof of the existence of an entropy weak solution. Received May 18, 2000 / Revised version received November 21, 2000 / Published online June 7, 2001     

Transport‐equilibrium schemes for computing nonclassical shocks. Scalar conservation laws

This paper presents a new numerical strategy for computing the nonclassical weak solutions of scalar conservation laws which fail to be genuinely nonlinear. We concentrate on the typical situation of concave–convex and convex–concave flux functions. In such situations the so‐called nonclassical shocks, violating the classical Oleinik entropy criterion and selected by a prescribed kinetic relation, naturally arise in the resolution of the Riemann problem. Enforcing the kinetic relation from a numerical point of view is known to be a crucial but challenging issue. By means of an algorithm made of two steps, namely an Equilibrium step and a Transport step, we show how to force the validity of the kinetic relation at the discrete level. The proposed strategy is based on the use of a numerical flux function and random numbers. We prove that the resulting scheme enjoys important consistency properties. Numerous numerical evidences illustrate the validity of our approach. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions

We revisit the classical theory of multidimensional scalar conservation laws. We reformulate the notion of the classical Kruzkov entropy solutions and study some new properties as well as the well-posedness of the initial value problem with inhomogeneous fluxes and general initial data. We also consider Dirichlet boundary value problems. We put forward a new and transparent definition for solutions and give a simple proof for their well-posedness in domains with smooth boundaries. Finally, we introduce the notion of saturated solutions and show that it is well-posed.     

Stochastically monotone Markov Chains

Summary A real-valued discrete time Markov Chain {X _n} is defined to be stochastically monotone when its one-step transition probability function pr {X _n+1y¦ X _n=x} is non-increasing in x for every fixed y. This class of Markov Chains arises in a natural way when it is sought to bound (stochastically speaking) the process {X _n} by means of a smaller or larger process with the same transition probabilities; the class includes many simple models of applied probability theory. Further, a given stochastically monotone Markov Chain can readily be bounded by another chain {Y _n}, with possibly different transition probabilities and not necessarily stochastically monotone, and this is of particular value when the latter process leads to simpler algebraic manipulations. A stationary stochastically monotone Markov Chain {X _n} has cov(f(X ₀), f(X _n)) cov(f(X ₀), f(X _n+1))0 (n =1, 2,...) for any monotonic function f(·). The paper also investigates the definition of stochastic monotonicity on a more general state space, and the properties of integer-valued stochastically monotone Markov Chains.     

Markov processes with free-Meixner laws

We study a time-non-homogeneous Markov process which arose from free probability, and which also appeared in the study of stochastic processes with linear regressions and quadratic conditional variances. Our main result is the explicit expression for the generator of the (non-homogeneous) transition operator acting on functions that extend analytically to complex domains.     

Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions

We study the boundary layer effect in the small relaxation limit to the equilibrium scalar conservation laws in one space dimension for the relaxation system proposed in [6]. First, it is shown that for initial and boundary data satisfying a strict version of the subcharacteristic condition, there exists a unique global (in time) solution, (u^ε, v^ε), to the relaxation system (1.4) for each ε > 0. The spatial total variation of (u^ε, v^ε) is shown to be bounded independently of ε, and consequently, a subsequence of (u^ε, v^ε) converges to a limit (u, v) as ε → 0⁺. Furthermore, u(x, t) is a weak solution to the scalar conservation law (1.5) and v = f(u). Next, we prove that for data that are suitably small perturbations of a nontransonic state, the relaxation limit function satisfies the boundary-entropy condition (2.11). Finally, the weak solutions to (1.5) with the boundary-entropy condition (2.11) is shown to be unique. Consequently, the relaxation limit of solutions to (1.4) is unique, and the whole sequence converges to the unique limit. One consequence of our analysis shows that the boundary layer occurs only in the u-component in the sense that v^ε(0, ·) converges strongly to γ ○ v = f(γ ○ u), the trace of f(u) on the t-axis. © 1998 John Wiley & Sons, Inc.     

Monotone matrices and monotone Markov processes

A stochastic matrix is “monotone” [4] if its row-vectors are stochastically increasing. Closure properties, characterizations and the availability of a second maximal eigenvalue are developed. Such monotonicity is present in a variety of processes in discrete and continous time. In particular, birth-death processes are monotone. Conditions for the sequential monotonicity of a process are given and related inequalities presented.     

Continuous dependence on the initial and flux functions for solutions of conservation laws

We prove the continuous dependence on the initial and flux functions for the entropy solutions to the Cauchy problem for conservation laws. Accordingly, we can show that the continuous dependence on the flux function for the entropy solutions depends only on the sup norm, not on the Lipschitz norm.     

Hyperbolic balance laws with dissipative source relaxing to adiabatic conservation laws

The paper discusses the long time behavior of BV solutions to the Cauchy problem for hyperbolic systems of balance laws with partial dissipation, when the relaxed system is adiabatic.