# IJPAM: Volume 61, No. 4 (2010)

**APPROXIMATION AS ATTEMPT TO INFER NONEXISTENCE**

OF WEAK SOLUTIONS OF RIEMANN PROBLEMS FOR

THE ``TWO-DIMENSIONAL P-SYSTEM"

OF WEAK SOLUTIONS OF RIEMANN PROBLEMS FOR

THE ``TWO-DIMENSIONAL P-SYSTEM"

Department of Mathematics

The Hebrew University

Givat Ram, Jerusalem, 91904, ISRAEL

e-mail: [email protected]

**Abstract.**A class of Riemann problems for the two-dimensional p-system is
considered, for which the existence of a traditional weak solution
is at best uncertain. Regarding the existence of such a solution
as a postulated hypothesis, we attempt to prove the hypothesis
wrong by experiment.

In this case, experiment means the construction and analysis of one-parameter
sequences of ostensibly approximate solutions, obtained by both
vanishing viscosity and discretization methods. In each case,
failure of the method to produce a sequence provably converging to
the desired solution is shown to be readily observable in the
sense of numerical computations. The hypothesis of existence of a solution thus survives to
the extent that no such failure is observed.

**Received: **May 1, 2010

**AMS Subject Classification: **35L65

**Key Words and Phrases: **multidimensional conservation laws, self-similar solutions, approximation methods

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 61

**Issue:** 4