Instability for Geometric Wave Equations
speaker: Piero D'Ancona (Università di Roma "La Sapienza")abstract: The aim of this course is to review a series of results (P.D'Ancona, V.Georgiev) concerning the equations of wave map type. Wave maps are an exact analogue to harmonic maps, if one replaces the background metric with a Lorentzian metric; of course, the system of quasilinear PDEs thus arising is of hyperbolic type. The natural framework is the evolutionary Cauchy problem in the scale of Sobolev spaces, and in this scale the "critical" space is $Hs$ with $s=n2$, $n$ being the dimension of space. In the first part of the course we will show some quite strong results of ill-posedness (namely, non-uniqueness) in the "subcritical" case, i.e., in Sobolev spaces of order lower than $n2$. The second part of the course will discuss the following result: in the critical space $H{n2}$, the Cauchy problem is ill-posed in the Hadamard (or Bourgain-Kenig-Ponce-Vega) sense, i.e., the solution flow is not uniformly continuous in the corresponding spaces. This result can be interpreted as a form of instability of the wave map system in the critical case.
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