Seminar Analysis: Benjamin Texier (Université Paris-Diderot (Paris 7))

When the Garding inequality is ineffective: examples and remedies

The Garding inequality states that positive pseudo-differential symbols are associated with semi-positive operators. It can be used in particular to show time-exponential growth of solutions to initial value problems for elliptic equations. I will give examples in which Garding fails to give appropriate bounds, and a way to overcome this difficulty. Examples include high-frequency asymptotics of systems based on Maxwell's equations, and compressible Euler systems with a Van der Waals pressure law. In these cases, appropriate bounds are derived via a description of the parametrix of a pseudo-differential system.

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