ON THE UNIQUENESS OF BOUNDED WEAK SOLUTIONS TO THE NAVIER-STOKES CAUCHY PROBLEM (original) (raw)

On uniqueness of weak solutions of the incompressible Navier-Stokes equations

2020

In this article the question on uniqueness of weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case is studied. Here the investigation is carried out with use of another approach. The uniqueness of velocity for the considered problem is proved for given functions from spaces that posseses some smoothness. Moreover, these spaces are dense in respective spaces of functions, under which were proved existence of the weak solutions. In addition here the solvability and uniqueness of the weak solutions of auxiliary problems associated with the main problem is investigated, and also one conditional result on uniqueness is proved.

Remarks on the uniqueness of weak solutions of the incompressible Navier-Stokes equations

arXiv (Cornell University), 2024

This article studies the uniqueness of the weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case. Here, the investigation is provided using two different approaches. The first (the main) result is obtained for given functions possessing a certain smoothness using the new approach. The second result is without the complementary conditions but is, in some sense, the "local" result investigated by another approach. In addition, here the solvability and uniqueness of the weak solutions to auxiliary problems lead out from the main problem are investigated.

Weak and Strong Solutions of the Navier-Stokes Initial Value Problem

This paper reviews the existence, uniqueness and regularity of weak and strong solutions of the Navier-Stokes system. For this purpose we emphasize semigroup theory and the theory of the Stokes operator. We use dimensional analysis to clarify the meaning of the results for the solutions. § 0. Introduction Let D be a bounded domain in R" (n>2) with smooth boundary S. We consider the initial-boundary value problem for the Navier-Stokes equations (NS) duldt-Au+(u, grad)w + gradp=/, divw = 0 in Dx(0, T), w=0 on Sx(0, T), M(X, 0) = 0(x) in D, where (u, gi"ad)=£" = 1 u j (d/dXj). This system describes the motion of viscous incompressible fluid filling a rigid vessel D. The function M=(M I (X, f), ..., M"(X, 0) represents the velocity of the fluid and p(x, t) is the pressure. The function a = (a 1 (x),..., a"(x)) is a given initial velocity and /=(/ 1 (x, i) 9 ...,f n (x, 0) is a given external force. We discuss the existence, uniqueness and regularity of weak and strong solutions of this problem. There is an extensive literature on this subject since Leray [27-29] introduced many useful and fundamental ideas. In [29] he constructed a global (in time) weak solution and a local strong solution of the initial value problem when D = R 3. Hopf [20] has proved the existence of a global weak solution of the initial-boundary value problem. Such weak solu

Addendum to the Paper Ëxistence of Weak Solutions for the Navier-Stokes Equations with Initial Data in L p

Transactions of the American Mathematical Society, 1990

The existence of weak solutions for the Navier-Stokes equations for the infinite cylinder with initial data in IY is considered in this paper. We study the case of initial data in IY(Rn), 2 < p < n, and n = 3,4. An existence theorem is proved covering these important cases and therefore, the "gap" between the Hopf-Leray theory (p = 2) and that of Fabes-Jones-Riviere (p > n) is bridged. The existence theorem gives a new method of constructing global solutions. The cases p = n are treated at the end of the paper.

Weak solutions for the Stokes system for compressible fluids with general pressure

2021

We prove existence and uniqueness of global in time weak solutions for the Stokes system for compressible fluids with a general, non-monotone pressure. We construct the solution at the level of Lagrangian formulation and then define the transformation to the original Eulerian coordinates. For nonnegative and bounded initial density the solution is also nonnegative for all t and belongs to L∞([0,∞) × T). A key point of our considerations is the uniqueness of such transformation. Since the velocity might not be Lipschitz continuous, we develop a method which relies on the results of Crippa & De Lellis, concerning regular Lagriangian flows. The uniqueness is obtained thanks to the application of a certain weighted flow and detail analysis based on the properties of the BMO space.

A new regularity criterion for weak solutions to the Navier–Stokes equations

Journal de Mathématiques Pures et Appliquées, 2005

In this paper we obtain a new regularity criterion for weak solutions to the 3-D Navier-Stokes equations. We show that if any one component of the velocity field belongs to L α ([0, T); L γ (R 3)) with 2 α + 3 γ ≤ 1 2 , 6 < γ ≤ ∞, then the weak solution actually is regular and unique. Titre. Un nouveau critère de régularité pour les solutions faibles deséquations de Navier-Stokes Resumé. Dans cet article, on obtient un nouveau critère de régularité pour les solutions faibles deséquations de Navier-Stokes en dimension 3. On démontre que si une conposante quelconque du champ de vitesse appartientà L α ([0, T ]; L γ (R 3)) avec 2 α + 3 γ ≤ 1 2 , 6 < γ ≤ ∞, alors la solution faible est régulière et unique.

On uniqueness of weak solutions of the incompressible Navier-Stokes equations in 3-dimensional case

2015

In this article we study the uniqueness of the weak solution of the incompressible Navier-Stokes Equation in the 3-dimensional case with use of different approach. Here the uniqueness of the obtained by Leray of weak solution is proved in the case, when datums from spaces that are densely contained into spaces of datums for which was proved the existence of the weak solution. Moreover we investigate the solvability and uniqueness of the weak solutions of problems associated with investigation of the main problem.28 p