18 research outputs found
h-Principles for the Incompressible Euler Equations
Recently, De Lellis and Sz\'ekelyhidi constructed H\"older continuous,
dissipative (weak) solutions to the incompressible Euler equations in the torus
. The construction consists in adding fast oscillations to the
trivial solution. We extend this result by establishing optimal h-principles in
two and three space dimensions. Specifically, we identify all subsolutions
(defined in a suitable sense) which can be approximated in the -norm by
exact solutions. Furthermore, we prove that the flows thus constructed on
are genuinely three-dimensional and are not trivially obtained
from solutions on .Comment: 29 pages, no figure
Dissipative continuous Euler flows
We show the existence of continuous periodic solutions of the 3D
incompressible Euler equations which dissipate the total kinetic energy
Direct approach to the problem of strong local minima in Calculus of Variations
The paper introduces a general strategy for identifying strong local
minimizers of variational functionals. It is based on the idea that any
variation of the integral functional can be evaluated directly in terms of the
appropriate parameterized measures. We demonstrate our approach on a problem of
W^{1,infinity} weak-* local minima--a slight weakening of the classical notion
of strong local minima. We obtain the first quasiconvexity-based set of
sufficient conditions for W^{1,infinity} weak-* local minima.Comment: 26 pages, no figure
Lack of uniqueness for weak solutions of the incompressible porous media equation
In this work we consider weak solutions of the incompressible 2-D porous
media equation. By using the approach of De Lellis-Sz\'ekelyhidi we prove
non-uniqueness for solutions in in space and time.Comment: 23 pages, 2 fugure
Non-uniqueness of minimizers for strictly polyconvex functionals
In this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557â611, 1982) about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals,
where Ω is homeomorphic to a ball.
We give several examples of non-uniqueness. The main example is a boundary value problem with at least two different global minimizers, both analytic up to the boundary. All these examples are suggested by the theory of minimal surfaces