Dissipation in 2D inviscid fluid limits is Lebesgue in time and absolutely continuous w.r.t. defect measures, resulting in trivial or atomic measures under sign or oscillation conditions on initial vorticity.
Vishik,Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incom- pressible fluid
4 Pith papers cite this work. Polarity classification is still indexing.
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abstract
In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is locally integrable with values in the same Lebesgue space.
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Strong compactness in the lowest norm making the nonlinearity well-defined prevents anomalous Hamiltonian dissipation for the generalized SQG equation in the inviscid limit, yielding global conservative weak solutions for critical initial data.
A residual set of L² divergence-free initial data exists for which the 2D Euler equations admit unique global weak solutions that conserve energy and are recovered from Navier-Stokes vanishing-viscosity limits.
A simpler proof of Vishik's nonuniqueness theorem for the forced 2D Euler equation is obtained by constructing an unstable vortex first as piecewise constant and then regularizing it via a fixed-point argument.
citing papers explorer
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Dissipation concentration in two-dimensional fluids
Dissipation in 2D inviscid fluid limits is Lebesgue in time and absolutely continuous w.r.t. defect measures, resulting in trivial or atomic measures under sign or oscillation conditions on initial vorticity.
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Hamiltonian compactness and dissipation for the generalized SQG equation in the inviscid limit
Strong compactness in the lowest norm making the nonlinearity well-defined prevents anomalous Hamiltonian dissipation for the generalized SQG equation in the inviscid limit, yielding global conservative weak solutions for critical initial data.
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The 2D Euler equations are well-posed for generic initial data in $L^2$
A residual set of L² divergence-free initial data exists for which the 2D Euler equations admit unique global weak solutions that conserve energy and are recovered from Navier-Stokes vanishing-viscosity limits.
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A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation
A simpler proof of Vishik's nonuniqueness theorem for the forced 2D Euler equation is obtained by constructing an unstable vortex first as piecewise constant and then regularizing it via a fixed-point argument.