Hyperbolic conservation laws with large initial data. Is the Cauchy problem well-posed?

Abstract

We present an example in which the Glimm estimate for a strictly hyperbolic system of two conservation laws is violated. 1. Introduction. Consider the Cauchy problem for a strictly hyperbolic system of conservation laws in one space dimension:

Sharper Total Variation Bounds for the P-System of Fluid Dynamics

“…The idea of polygonal approximations used by several authors, was first introduced by Dafermos [5]; see also Liu, Xin and Yang [14]. We observe that by successively taking closer approximations, the difficulty encountered in Tsikkou [18], induced by the finite jump of the characteristic speed at a single point, disappears and we are able to reproduce the known estimates, albeit with greater precision. In particular, we fix k sufficiently small.…”

“…We will then estimate the two functionals to derive the fundamental estimate (3.1). In Tsikkou [18], we observed that using the piecewise linear function σ 2 and specific initial data, P 1 leads to violation of the Glimm estimate. In the present paper we show that for any σ n , the sum of the two functionals, P 1 (·) + P 2 (·) is controlled by the total variation T V s(·, 0) + T V r(·, 0), under some assumptions on the initial data.…”

“…In Tsikkou [18], we worked with system (1.7), where σ was given by (1.8), and showed that a combination of points of the set Φ leads to solutions with uncontrollable growth, under specific initial data. We continue on with this paper using σ 2 .…”

“…However, the case of initial data with large oscillation is still open. In order to gain a better understanding of this problem we considered, in Tsikkou [18], the initial value problem…”

Existence and uniqueness of singular solutions for a conservation law arising in magnetohydrodynamics

“…Firstly, we remark that SW1 and RW1 emanating from (u L , q L ) cover the entire q ≥ u 2 /2 domain (see Figure 2(a)). In other words, we have for the curve q R defining the SW1 by (25): lim…”