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(2015.6.12 14:00 N620)Professor Alberto Bressan:Some counterexamples in the theory of conservation laws
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Update time: 2015-06-09
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Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:

Professor Alberto Bressan,Penn state University

Inviter: Feimin Huang
Title:
Some counterexamples in the theory of conservation laws
Time & Venue:
2015.6.12 14:00-15:00 N620
Abstract:
The first part of the talk will be concerned with sticky particle models in two space dimensions. Examples of Cauchy problems will be discussed,admitting two distinct solutions, or no solution at all. Similar results apply to the equations of pressureless gases in two space dimensions, with bounded measurable initial data. The second part of the talk will discuss two long standing open problems for one-dimensional systems of conservation laws: The global existence of BV solutions to the p-system of isentropic gas dynamics, and the possible appearance of vacuum in finite time. For initial data with large total variation, one can consider front tracking approximate solutions, where, at each interaction, the strength of outgoing waves is the same as in the exact solution but some error is present in the wave speeds. In this setting, it is possible to construct approximate solutions, in such a way that the total strength of all wave fronts becomes arbitrarily large, or even blows up in finite time. The result remains valid also if the gas density satisfies a uniform lower bound. Finally, an example will be discussed pointing at the difficulties in proving a lower bound on the density, when the solution contains shocks.
 

 

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