Existence of Weak Solutions for the Incompressible Euler Equations

Abstract

Using a recent result of C. De Lellis and L. Székelyhidi Jr. we show that, in the case of periodic boundary conditions and for dimension greater or equal 2, there exist infinitely many global weak solutions to the incompressible Euler equations with initial data $v_0$, where $v_0$ may be any solenoidal $L^2$-vectorfield. In addition, the energy of these solutions is bounded in time.

5 pages

Cited by in corpus (22)

• Dissipative continuous Euler flows
• A counterexample to well-posedness of entropy solutions to the compressible Euler system
• Young Measures Generated by Ideal Incompressible Fluid Flows
• Mathematics and Turbulence: where do we stand?
• Weak-strong uniqueness for measure-valued solutions of some compressible fluid models
• On the vortex filament conjecture for Euler flows
• Non Uniqueness of power-law flows
• On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion
• h-Principles for the incompressible Euler equations
• Piecewise constant subsolutions for the Muskat problem
• The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem
• Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise
• Non-uniqueness for the Euler equations: the effect of the boundary
• Global generalized solutions for Maxwell-alpha and Euler-alpha equations
• Planar limits of three-dimensional incompressible flows with helical symmetry
• The 3D incompressible Euler equations with a passive scalar: a road to blow-up?
• Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions
• Dissipative solutions to the stochastic Euler equations
• Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces
• Probabilistic Descriptions of Fluid Flow: A Survey
• Stress-diffusive regularizations of non-dissipative rate-type materials
• An Eulerian-Lagrangian Form for the Euler Equations in Sobolev Spaces