Oscillations and concentrations in weak solutions of the incompressible fluid equations.

*(English)*Zbl 0626.35059The authors introduce a new concept of measure-valued solutions for the 3-D incompressible Euler equations using an extension of the concept of Young measure in order to incorporate phenomena like persistence of oscillations and development of concentrations in solution sequences. As an application they prove that a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of the 3-D Euler equations for all times. In section 1 of the paper a very clear survey is given with hints on forthcoming papers. Section 2 presents an example of an \(L^{\infty}\)- bounded sequence of smooth solutions of the 3-D Euler equations which exhibits persistence of oscillations. In section 3 examples of solution sequences of the 2-D Euler equations are given which have locally uniformly bounded kinetic energy and which develop concentrations. In section 4 a generalization of the Young measure is developed which includes both oscillations and concentrations and it is applied to the problem of representing weak limits of exact and approximate solution sequences of the Euler equations under the single assumption of locally uniform boundedness of the kinetic energy. The generalized Young measures are constructed in the form of vector-valued measures acting on arbitrary continuous functions with at most quadratic growth. In the last section the zero viscosity limit for the Navier-Stokes equations is discussed.

Reviewer: R.Racke

##### MSC:

35L60 | First-order nonlinear hyperbolic equations |

35Q30 | Navier-Stokes equations |

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

35A35 | Theoretical approximation in context of PDEs |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

35B65 | Smoothness and regularity of solutions to PDEs |

##### Keywords:

measure-valued solutions; incompressible Euler equations; Young measure; persistence of oscillations; development of concentrations; Leray-Hopf weak solutions; Navier-Stokes equations; weak limits; solution sequences; zero viscosity limit
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\textit{R. J. DiPerna} and \textit{A. J. Majda}, Commun. Math. Phys. 108, 667--689 (1987; Zbl 0626.35059)

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