Energy conservation for weak solutions of incompressible fluid equations: The Hölder case and connections with Onsager's conjecture
Luigi C. Berselli
Abstract
In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way. Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one –in the viscous case– with Hölder type hypotheses, but without additional assumptions on the pressure.
Topics & Concepts
MathematicsEuler equationsMathematical proofConjectureMathematical analysisConservation lawCompressibilityDirichlet problemBoundary (topology)Type (biology)Boundary value problemEnergy (signal processing)Conservation of energyHölder conditionDirichlet distributionDirichlet integralWeak solutionPure mathematicsDirichlet's energyGeometryPhysicsMechanicsStatisticsEcologyThermodynamicsBiologyNavier-Stokes equation solutionsAdvanced Mathematical Physics ProblemsFluid Dynamics and Turbulent Flows