Linear and Nonlinear Hyperbolic Equations

On regularity of the solution map for the cubic 1d periodic half - wave equation

speaker: Vladimir Georgiev (Dipartimento di Matematica, Università di Pisa)

abstract: Making a reduction of the 1D nonlinear wave equation on bounded interval to a system of two nonlinear hyperbolic equations, we consider the Cauchy problems associated with this system. Small modification of the system leads to a systems with explicit family of solutions that enables one to study regularity of the solution map and show that this map is not uniformly continuous when the initial data are in the Sobolev space \(H^s\) with \( 0 < s < 1/2.\) Turning back to the original hyperbolic system and using suitable bilinear and trilinear estimates in Bourgain type spaces we show the same ill posedness result for the system and for the 1D nonlinear wave equation.

timetable:
Wed 3 Jul, 17:00 - 17:50, Aula Dini
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