Abstract
We are concerned with the problem of the global (in time) existence of weak solutions to hyperbolic systems of conservation laws, in one spatial dimension. First, we provide a survey of the different facets of a technique that has been used in several papers in the last years: the path decomposition. Then, we report on two very recent results that have been achieved by means of suitable applications of this technique. The first one concerns a system of three equations arising in the dynamic modeling of phase transitions, the second one is the famous Euler system for nonisentropic fluid flow. In both cases, the results concern classes of initial data with possibly large total variation.
Highlights
We consider the following initial-value problem for a 1-d system of conservation lawsUt + F (U )x = 0, (x, t) ∈ R × R+, (1)U (x, 0) = U0(x), x ∈ R, (2)where U = t(u1, u2, . . . , un) is an n-tuple of conserved quantities taking values in a connected region Ω ⊂ Rn and F (U ) is the flux function, which is a smooth map from Ω to Rn
We are concerned with the problem of the global existence of weak solutions to hyperbolic systems of conservation laws, in one spatial dimension
We provide a survey of the different facets of a technique that has been used in several papers in the last years: the path decomposition
Summary
Glimm [14] obtained the following remarkable existence theorem (see Lax [16]) for general initial data with small total variation. At some interaction points we have quadratic waves which generate secondary paths. Consider a Riemann problem with initial data (vL, uL, λL), (vR, uR, λR) and solved by waves εi, i = 1, 2, 3; by [7, (4.4)] it follows that 2 |ε1| + |ε3| ≤ 1 − max{kL, kR} (|rR − rL| + |sR − sL|) .
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