Interpolation between scaling and energy for some nonlinear Cauchy problem
speaker: Isabelle Gallagher (Institut de Mathématiques de Jussieu, Université Paris Diderot)abstract: It is usual for evolution PDEs arising from a physical problem to be associated with some "energy" conservation, and also to have a "scaling" property. To both aspects one can naturally associate some invariant functional spaces. For data in an energy type space it is natural to hope to prove global existence of "weak" solutions, and for data in a scaling invariant space to prove short time existence and uniqueness. The question we shall address here is to try to understand what can happen when one of the spaces is "embedded" in the other : for instance, can energy-type techniques improve the local in time results in scale invariant spaces, if the energy space is itself scale invariant? We shall discuss three different systems, accounting for the three different possibilities : when the energy space is scale invariant (the Navier-Stokes equation in two space dimensions), when the energy space is "below" scaling (the Navier-Stokes equation in three space dimensions) and when the energy space is "above" scaling (the cubic wave equation in three space dimensions).
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