Non-uniqueness of weak solutions and existence of dissipative solutions for the Euler equations

DATES: 2nd to 6th May 2022 (5 sessions)
TIME: 9:30-11:30, on Friday it will be from 9:00 to 11:00 (a total of 10 hours)
LOCATION: BCAM Seminar Room and Online


ABSTRACT:
n this course we will present the results of De Lellis and Székelyhidi concerning the existence of "wild" solutions for the incompressible Euler equations.
In 2009, the authors, by reformulating the equations as a differential inclusion and using the method of convex integration, were able to obtain an alternative proof of the results of V. Scheffer and A. Shnirelman, about the non-uniqueness of weak solutions and existence of energy-decreasing solutions. By using similar techniques, the authors were also able to extend the results of non-uniqueness to solutions satisfying additional energy inequalities.
We will then present another result by the same authors, concerning the existence of continuous weak solutions on the 3-dimensional torus which dissipate the kinetic energy.

REFERENCES:
[1] De Lellis, C., and Székelyhidi, Jr., L. The Euler equations as a differential inclusion. Ann. of Math. (2) 170, 3 (2009), 1417-1436.
[2] De Lellis, C., and Székelyhidi, Jr., L. On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195, 1 (2010), 225-260.
[3] C. De Lellis and L. Székelyhidi, Jr. The h-principle and the equations of fluid dynamics. Bull. Amer. Math. Soc. (N.S.), 49, 3 (2012), 347-375.
[4] C. De Lellis and L. Székelyhidi, Jr. Dissipative continuous Euler flows. Invent. Math., 193, 2 (2013), 377-407.


*Registration is free, but mandatory before 27th April 2022. To sign-up go to https://forms.gle/1GyWMwc18WqKqAqK7 and fill the registration form.

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