Global existence for a class of viscous systems of conservation laws

Abstract

We prove existence and boundedness of classical solutions for a family of viscous conservation laws in one space dimension for arbitrarily large time. The result relies on H. Amann's criterion for global existence of solutions and on suitable uniform-in-time estimates for the solution. We also apply Jüngel's boundedness-by-entropy principle in order to obtain global existence for systems with possibly degenerate diffusion terms. This work is motivated by the study of a physical model for the space-time evolution of the strain and velocity of an anharmonic spring of finite length.Mathematics Subject Classification (2010). 35B65, 35K51, 35K65, 35Q70.

Trend to equilibrium for systems with small cross-diffusion

“…We are now going to show that solutions (in the sense of Definition 2.1) of problem (2) are, indeed, classical solutions if the initial data is sufficiently smooth. The arguments in this section follow closely those presented in [4] for a different class of parabolic systems. We start by showing that classical solutions exist for a short time (see Proposition 2.3) and then extend their maximal time interval of definition to the positive half-line using uniform estimates (see Proposition 2.4).…”

“…We assume that D αβ ij ∈ C 1 (Ω) is symmetric in the space indices α and β and that it is "diagonal" in the component indices i and j, i.e., (4) D αβ ij = 0, for all i = j. Furthermore, we suppose that there exist two constants Λ ≥ λ > 0 such that for every x ∈ Ω and ξ ∈ R d , it holds that…”

“…Few analytic results are available, and the little results that exist seem to rely on the assumption that the asymptotic gradient flow system is close to its stationary state (see [10]) or that it is close to being a diagonal, decoupled linear problem; cf. [4].…”

Hydrodynamic limit for a diffusive system with boundary conditions

“…Existence and uniqueness of global solutions of class C 1 in time and C 2 in space for systems (3.4) and (3.9) with our initial-boundary conditions have been proven in Alasio and Marchesani (2019). More precisely, we have…”

“…Thus, we may not assume existence of global classical solutions, and such an existence needs to be proven. This is done in Alasio and Marchesani (2019), where more general systems and boundary conditions are considered.…”