An ill posed Cauchy problem for a hyperbolic system in two space dimensions

By Alberto Bressan. 2003

89 citations recorded by the source corpus.

The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global existence of solutions to the Cauchy problem remains a challenging open question. In this note we construct a conterexample showing that, even for a simple class of hyperbolic systems, in two space dimensions the Cauchy problem can be ill posed.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c115a557bfd1c4c1da820bacf2384bac09e142d9

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In Applied Mathematics

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Citations in the corpus

EXISTENCE OF STRONG TRACES FOR QUASI-SOLUTIONS OF MULTIDIMENSIONAL CONSERVATION LAWS (2007, background): “For instance there are numerous examples ( see [ 5 , 8, 22] ) of linear transport equations div(a(x)u) = 0 with divergence free fields a(x) ( i.e.”
Transport equation and Cauchy problem for BV vector fields (2004, background): “We discuss also some conjectures related to stronger forms of uniqueness, and their relation with a compactness conjecture for Lagrangian flows made by A. Bressan in [ 13 ] (Theorem 6.6 provides only a partial answer to that conjecture).”
Lagrangian flows for vector fields with gradient given by a singular integral (2012, background): “[10] Bressan, A.”
Estimates and regularity results for the DiPerna-Lions flow (2008, result): “Actually, the extension to the case Db ∈ L(1) of our (or of similar) estimates would answer positively to the following conjecture of Bressan (see [6]): Conjecture 1.”
Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields (2009, background): “Even in the presence of this special “triangular” structure, Bressan [48] proved that the Cauchy problem (5.”
The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems (2011, background): “Even in the presence of this special “triangular” structure, Bressan [48] proved that the Cauchy problem (5.”
Transport Equation and Cauchy Problem for Non-Smooth Vector Fields (2008, background): “In a recent paper [ 32 ] Bressan showed that the problem can be ill-posed for”
Chapter 4 – Notes on Hyperbolic Systems of Conservation Laws and Transport Equations (2007, background): “In [17] Bressan showed that in 2 space dimensions renormalized entropy solutions might lead to an ill posed Cauchy problem for bounded initial data.”
Differential equations with singular fields (2010, background): “amples are nevertheless known to test this optimality, from the early [1], to [11] and [21] which indicates that in the general framework BV indeed plays a critical role.”