Some recent developments in the analysis of long-time behaviors of stochastic solutions of nonlinear conservation laws driven by stochastic forcing are surveyed. The existence and uniqueness of invariant measures are established for anisotropic degenerate parabolic-hyperbolic conservation laws of second-order driven by white noises. Some further developments, problems, and challenges in this direction are also discussed.
We study the derivation of a scalar conservation law with stochastic forcing starting from a stochastic BGK model with a high-field scaling. We prove the convergence to a new kinetic formulation where appears a modified Maxwellian. We deduce from it the existence of a weak solution to the scalar conservation law with stochastic forcing. We establish that this solution satisfies some Krushkov-like entropy relations.
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.
We are interested in viscous scalar conservation laws with a white-in-time but spatially correlated stochastic forcing. The equation is assumed to be one-dimensional and periodic in the space variable, and its flux function to be locally Lipschitz continuous and have at most polynomial growth. Neither the flux nor the noise need to be non-degenerate. In a first part, we show the existence and uniqueness of a global solution in a strong sense. In a second part, we establish the existence and uniqueness of an invariant measure for this strong solution.
We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results obtained in [Gess, Souganidis; Comm. Pure Appl. Math. (2017)]. In addition, we introduce a new path-by-path scaling property which is shown to be sufficient to imply regularizing effects.
Under an hypothesis of non-degeneracy of the flux, we study the long-time behaviour of periodic scalar first-order conservation laws with stochastic forcing in any space dimension. For sub-cubic fluxes, we show the existence of an invariant measure. Moreover for sub-quadratic fluxes we show uniqueness and ergodicity of the invariant measure. Also, since this invariant measure is supported by Lp for some p small, we are led to generalize to the stochastic case the theory of L1 solutions developed by Chen and Perthame in 2003.
We prove the existence and uniqueness of solutions to a class of stochastic scalar conservation laws with joint space-time transport noise and affine-linear noise driven by a geometric p-rough path. In particular, stability of the solutions with respect to the driving rough path is obtained, leading to a robust approach to stochastic scalar conservation laws. As immediate corollaries we obtain support theorems, large deviation results and the generation of a random dynamical system.
We consider stochastic scalar conservation laws with spatially inhomogeneous flux. The regularity of the flux function with respect to its spatial variable is assumed to be low, so that entropy solutions are not necessarily unique in the corresponding deterministic scalar conservation law. We prove that perturbing the system by noise leads to well-posedness.
In this work we consider companion conservation laws to general systems of conservation laws. We investigate sufficient regularity for weak solutions to satisfy companion laws, assuming the fluxes to be $C^{1,γ}$, $0<γ<1$, regular. We discuss applications of the general framework to nonlinear elasticity and compressible Euler system.
In this paper, we study scalar conservation laws where the flux is driven by a geometric Hölder $p$-rough path for some $p\in (2,3)$ and the forcing is given by an Itô stochastic integral driven by a Brownian motion. In particular, we derive the corresponding kinetic formulation and define an appropriate notion of kinetic solution. In this context, we are able to establish well-posedness, i.e. existence, uniqueness and the $L^1$-contraction property that leads to continuous dependence on initial condition. Our approach combines tools from rough path analysis, stochastic analysis and theory of …
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial co…
We determine the Lie point symmetries of a Gardner type system and establish its nonlinear self-adjointness. We then construct conservation laws via Ibragimov's Theorem.
This paper deals with some qualitative properties of entropy solutions to hyperbolic conservation laws. In [11] the jump set of entropy solution to conservation laws has been introduced. We find an entropy solution to scalar conservation laws for which the jump set is not closed, in particular, it is dense in a space-time domain. In the later part of this article, we obtain a similar result for the hyperbolic system. We give two different approaches for scalar conservation laws and hyperbolic system to obtain the results. For the scalar case, obtained solutions are more explicitly calculated.
We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference bet…
For nonlinear Schroedinger equations with a power nonlinearity, a new approach to derive the conservation law of the momentum and the pseudo conformal conservation law is obtained. Since this approach does not contain approximating procedure, the argument is simplified to derive these conservation laws.
Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux.
We obtain conservation laws at negative regularity for the Benjamin-Ono equation on the line and on the circle. These conserved quantities control the $H^s$ norm of the solution for $-\frac{1}{2} < s < 0$. The conservation laws are obtained from a study of the perturbation determinant associated to the Lax pair of the equation.
We are concerned with a control problem related to the vanishing \emph{fractional} viscosity approximation to scalar conservation laws. We investigate the $Γ$-convergence of the control cost functional, as the viscosity coefficient tends to zero.
We present a general method for obtaining conservation laws for integrable PDE at negative regularity and exhibit its application to KdV, NLS, and mKdV. Our method works uniformly for these problems posed both on the line and on the circle.
A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such case most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws. They are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake a role of a physical admissibility condition for weak s…
We provide an informal overview on the theory of transport equations with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.
Solutions to a class of conservation laws with discontinuous flux are constructed relying on the Crandall-Liggett theory of nonlinear contractive semigroups~\cite{CL}. In particular, the paper studies the existence of backward Euler approximations, and their convergence to a unique entropy-admissible solution to the Cauchy problem. The proofs are achieved through the study of the backward Euler approximations to the viscous conservation laws.
A theorem due to Nail H. Ibragimov (2007) provides a connection between symmetries and conservation laws for arbitrary differential equations. The theorem is valid for any system of differential equations provided that the number of equations is equal to the number of dependent variables. In this paper we use the theorem to determine conservation laws for a nonlinear system of differential equations that represents a mathematical model for HIV transmission.
We establish conservation laws for the second order Kudryashov-Sinelshchikov equation, which models pressure waves in liquid with bubbles. For this purpose we use the method of Nail Ibragimov based on the notion of nonlinear self-adjointness.
Motivated by the work of P.L. Lions and J-C. Rochet [12], concerning multi-time Hamilton-Jacobi equations, we introduce the theory of multi-time systems of conservation laws. We show the existence and uniqueness of solution to the Cauchy problem for a system of multi-time conservation laws with two independent time variables in one space dimension. Our proof relies on a suitable generalization of the Lax-Oleinik formula.
We determine the Lie point symmetries of a class of BBM-KdV systems and establish its nonlinear self-adjointness. We then construct conservation laws via Ibragimov's Theorem.
We propose a new sufficient non-degeneracy condition for the strong precompactness of bounded sequences satisfying the nonlinear first-order differential constraints. This result is applied to establish the decay property for periodic entropy solutions to multidimensional scalar conservation laws.
Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group carried out in a preceding work, we establish the conservation laws for the critical Kohn-Laplace equations via the Noether's Theorem.
We prove the convergence of quasilinear parabolic viscous approximations to the entropy solution (in the sense of Bardos-Leroux-Nedelec) of a scalar conservation law, considered on a bounded domain in $\R^d$.
Following Rivière's study of conservation laws for second order quasilinear systems with critical nonlinearty and Lamm/Rivière's generalization to fourth order, we consider similar systems of order $2m$. Typical examples are $m$-polyharmonic maps. Under natural conditions, we find a conservation law for weak solutions on $2m$-dimensional domains. This implies continuity of weak solutions.
We discuss properties of kinetic solutions of scalar conservation laws in the variational approach developed by Eu. Panov and Y. Brenier. Our main result shows that such solutions can be considered as curves in a suitable Hilbert space with tangents that are unique minimizers of an interaction functional.
This paper is concerned with singular shocks for a system of conservation laws modeling incompressible two-phase fluid flow. We prove the existence of viscous profiles using the Geometric Singular Perturbation Theory. Weak convergence and growth rates of the unbounded family of solutions are also obtained.
For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see Panov [Mat. Zametki, 55(5):116--129, 159, 1994]). This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Quart. Appl. Math.,…
We extend the recent work of Oliveras arXiv:2008.00940 and Oliveras & Calatola-Young arXiv:2105.07580 to develop a new nonlocal formulation of the water-wave problem for a three-dimensional fluid with a two-dimensional free surface for an inviscid and irrotational fluid over a flat bathymetry. Using this new formulation, we show how one can systematically derive Benjamin & Olver's twelve conservation laws without explicitly relying on the underlying Lie symmetries. This allows us to make draw new conclusions about conservation laws and posit the potential existence of additional, nonlocal, con…
In this paper we study large time behaviors toward shock waves and rarefaction waves under periodic perturbations for 1-D convex scalar conservation laws. The asymptotic stabilities and decay rates of shock waves and rarefaction waves under periodic perturbations are proved.
We present a quantitative compensated compactness estimate for stochastic conservation laws, which generalises a previous result of Golse & Perthame (2013) for deterministic equations. With a stochastic modification of Kruzkov's interpolation lemma, this estimate provides bounds on the rate at which a sequence of vanishing viscosity solutions becomes compact. This contribution is for the Proceedings of HYP2022.
We study the BGK approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy Problem for the BGK approximation is well-posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.
We consider nondecreasing entropy solutions to 1-d scalar conservation laws and show that the spatial derivatives of such solutions satisfy a contraction property with respect to the Wasserstein distance of any order. This result extends the L^1-contraction property shown by Kruzkov.
This paper deals with an optimal control problem and describes the reachable set for the scalar 1-D conservation laws with discontinuous flux. Regarding the optimal control problem we first prove the existence of a minimizer and then we prescribe an algorithm to compute it. The same method also applies to compute the initial data control. The proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics.
We describe $δ$ shock wave arising from continuous initial data in the case of triangular conservation law system arising from "generalized pressureless gas dynamics model". We use the weak asymptotic method.
We study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we demonstrate that conservation laws with diffusion are "flux--gradient flows". We next construct variational problems for these flows, for which we derive dual PDE systems for regularized conservation laws. Several examples, including traffic flow and Burgers' equation, are presented. Incorporating both primal-dual algorithms and monotone schemes, we successfully com…
Compared with macroscopic conservation law for the solution of the derivative nonlinear Schrödingger equation (DNLS) with small mass in \cite{KlausS:DNLS}, we show the corresponding microscopic conservation laws for the Schwartz solutions of DNLS with small mass. The new ingredient is to make use of the logarithmic perturbation determinant introduced in \cite{Rybkin:KdV:Cons Law, Simon:Trace} to show one-parameter family of microscopic conservation laws of the $A(κ)$ flow and the DNLS flow, which is motivated by \cite{HKV:NLS,KV:KdV:AnnMath,KVZ:KdV:GAFA}.
In this paper we study small shocks of 1D scalar viscous conservation laws with uniformly convex flux and nonlinear dissipation. We show that such shocks are L2 stable independent of the strength of the dissipation, even with large perturbations. The proof uses the relative entropy method with a spatially-inhomogeneous psuedo-norm.
We are concerned with a control problem related to the vanishing viscosity approximation to scalar conservation laws. We investigate the $Γ$-convergence of the control cost functional, as the viscosity coefficient tends to zero. A first order $Γ$-limit is established, which characterizes the measure-valued solutions to the conservation laws as the zeros of the $Γ$-limit. A second order $Γ$-limit is then investigated, providing a characterization of entropic solutions to conservation laws as the zeros of the $Γ$-limit.
We consider forward-forward Mean Field Game (MFG) models that arise in numerical approximations of stationary MFGs. First, we establish a link between these models and a class of hyperbolic conservation laws as well as certain nonlinear wave equations. Second, we investigate existence and long-time behavior of solutions for such models.
Aim of this paper is to review some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. The main focus will be on the uniqueness and stability of entropy weak solutions and on the convergence of vanishing viscosity approximations.
Two-dimensional gas dynamics equations in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler-Lagrange equations. Using group classification and Noether's theorem, conservation laws are obtained. Their counterparts in Eulerian coordinates are given. Among these counterparts there are new conservation laws.
We show that Kruzhkov's theory of entropy solutions to multidimensional scalar conservation laws can be entirely recast in L2 and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by Giga, Miyakawa, Osher, Tsai and the author.
The article first studies the propagation of well prepared high frequency waves with small amplitude $\eps$ near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, Tadmor, (1994), about the maximal regularizing effect for nonlinear conservation laws. For this purpose, a new definition of nonlinear flux is stated and compared to classical definitions. Then it is proved that the smoothness expected in Sobolev spaces cannot be exceeded.
We explore numerical approximation of multidimensional stochastic balance laws driven by multiplicative Lévy noise via flux- splitting finite volume method. The convergence of the approximations is proved towards the unique entropy solution of the underlying problem.
We investigate the quasi-potential problem for the entropy cost functionals of non-entropic solutions to scalar conservation laws with smooth fluxes. We prove that the quasi-potentials coincide with the integral of a suitable Einstein entropy.
Generalizing results by Bryant and Griffiths [Duke Math. J., 1995, V.78, 531-676], we completely describe local conservation laws of second-order (1+1)-dimensional evolution equations up to contact equivalence. The possible dimensions of spaces of conservation laws prove to be 0, 1, 2 and infinity. The canonical forms of equations with respect to contact equivalence are found for all nonzero dimensions of spaces of conservation laws.
We consider two discrete completely integrable evolutions: the Toda Lattice and the Ablowitz-Ladik system. The principal thrust of the paper is the development of microscopic conservation laws that witness the conservation of the perturbation determinant under these dynamics. In this way, we obtain discrete analogues of objects that we found essential in our recent analyses of KdV, NLS, and mKdV. In concert with this, we revisit the classical topic of microscopic conservation laws attendant to the (renormalized) trace of the Green's function.
We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all their conservation laws are (modulo trivial conserved vectors) at most quadratic in the dependent variable and its derivatives.
We determine, by hierarchy, dependencies between higher order linear symmetries which occur when generating them using recursion operators. Thus, we deduce a formula which gives the number of independent generalized symmetries (basis) of several orders. We construct a basis for conservation laws (with respect to the group admitted by the system of differential equation) and hence generate infinite conservation laws in each equivalence class.
