Large time behaviour of solutions of scalar viscous and nonviscous conservation laws (original) (raw)

Large time behaviour of a class of solutions of second order conservation laws

2000

We study the large time behaviour of entropy solutions of the Cauchy problem for a possibly degenerate nonlinear diusion equation with a nonlinear convection term. The initial function is assumed to have bounded total variation. We prove the convergence of the solution to the entropy solution of a Riemann problem for the corresponding first order conservation law.

Large-time behavior of entropy solutions of conservation laws

1999

We are concerned with the large-time behavior of discontinuous entropy solutions for hyperbolic systems of conservation laws. We present two analytical approaches and explore their applications to the asymptotic problems for discontinuous entropy solutions. These approaches allow the solutions of arbitrarily large oscillation without apriori assumption on the ways from which the solutions come. The relation between the large-time behavior of entropy solutions and the uniqueness of Riemann solutions leads to an extensive study of the uniqueness problem. We use a direct method to show the large-time behavior of large L ∞ solutions for a class of m × m systems including a model in multicomponent chromatography; we employ the uniqueness of Riemann solutions and the convergence of self-similar scaling sequence of solutions to show the asymptotic behavior of large BV solutions for the 3 × 3 system of Euler equations in thermoelasticity. These results indicate that the Riemann solution is the unique attractor of large discontinuous entropy solutions, whose initial data are L ∞ ∩ L 1 or BV ∩ L 1 perturbation of the Riemann data, for these systems. These approaches also work for proving the large-time behavior of approximate solutions to hyperbolic conservation laws.

Conservation laws with vanishing nonlinear diffusion and dispersion

2007

Abstract: We study the limiting behavior of the solutions to a class of conservation laws with vanishing nonlinear diffusion and dispersion terms. We prove the convergence to the entropy solution of the first order problem under a condition on the relative size of the diffusion and the dispersion terms. This work is motivated by the pseudo-viscosity approximation introduced by Von Neumann in the 50's.

On Convergence to Entropy Solutions of a Single Conservation Law

Journal of Differential Equations, 1996

Estimates of the difference between the entropy solutions of the single conservation law u t +div g(u)=0, u(0, } )=u 0 and of the evolutionary integral equation (k V (u&u 0 )) t +div g(u)=0 are given in terms of k, g and u 0 . A corresponding result is obtained for more general evolution equations with an accretive nonlinearity.

On the Structure of {L^\infty}$$ L ∞ -Entropy Solutions to Scalar Conservation Laws in One-Space Dimension

Archive for Rational Mechanics and Analysis, 2017

We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a C 0-sense up to the degeneracy due to the segments where f = 0. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.

On the Structure of L∞-Entropy Solutions to Scalar Conservation Laws in One-Space Dimension

2016

We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a C0-sense up to the degeneracy due to the segments where f ′′ = 0. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp. Preprint SISSA 43/2016/MATE

On the time continuity of entropy solutions

Journal of Evolution Equations, 2011

We show that any entropy solution u of a convection diffusion equation ∂tu+divF (u)−∆φ(u) = b in Ω×(0, T) belongs to C([0, T), L 1 loc (Ω)). The proof does not use the uniqueness of the solution.

On the structure of weak solutions to scalar conservation laws with finite entropy production

Calculus of Variations and Partial Differential Equations

We consider weak solutions with finite entropy production to the scalar conservation law ∂tu + divxF (u) = 0 in (0, T) × R d. Building on the kinetic formulation we prove under suitable nonlinearity assumption on f that the set of non Lebesgue points of u has Hausdorff dimension at most d. A notion of Lagrangian representation for this class of solutions is introduced and this allows for a new interpretation of the entropy dissipation measure.