Prerequisite: A graduate level one semester course in PDEs.
Course Description
The purpose of this reading course is to cover several related topics that deal with a new interpretations on solutions of nonlinear PDEs, primarily, entropy measure values solutions (compressible equations), convex integration of “wild” solutions (incompressible compressible Euler equations) and gradient flows.
• Entropy measure valued solution to hyperbolic conservation laws
• Gradient flows
• Non-uniqueness and anomalous dissipation in Euler equations – recent results of De Lellis & Szekelyhidi and their co-workers
• Moser iterations
The discussion on each topic is not supposed to be comprehensive but should be self-contained.
References:
On Young measures:
L. Tartar, Compensated compactness and applications to PDEs, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212.
On mv solutions to conservation laws:
R.J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational
Mech. Anal., 88 (1985), 223–270.
On non-uniqueness:
De Lellis & Szekelyhidi, The Euler eqs as a differential inclusion, Annal. Math. Vol. 170 2009 and The h-Principle and the equations of fluid dynamics, Bull. AMS, v. 49 2012
On gradient flows:
L. Ambrosio, N. Gigli & G. Savare, Gradient Flows: In Metric Spaces and in the Space of Probability Measures (Lectures in Mathematics. ETH Zürich) 2008.
On Nash-Moser iterations:
J. Moser, "A rapidly convergent iteration method and non-linear partial differential equations. I+II", Ann. Scuola Norm. Sup. Pisa (3) v. 20, (1966a+b)
