Asymptotics and stability for global solutions to the

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Niveau: Supérieur, Doctorat, Bac+8
Asymptotics and stability for global solutions to the Navier-Stokes equations Isabelle Gallagher a Drago? Iftimie a,b Fabrice Planchon c Abstract We consider an a priori global strong solution to the Navier-Stokes equations. We prove it behaves like a small solution for large time. Combining this asymptotics with uniqueness and averaging in time properties, we obtain the stability of such a global solution. Introduction We consider the incompressible Navier-Stokes equations in R3, (1) ? ?? ?? ∂u ∂t = ∆u?? · (u? u)??pi, ? · u = 0, u(x, 0) = u0(x). There exist essentially two di?erent kinds of results on the Cauchy problem for these equations. In the pioneering work [15], Jean Leray introduced the concept of weak solutions and proved global existence for datum u0 ? L2. However, their uniqueness (or propagation/breakdown of regularity for smooth data) has remained an open problem. In [11], H. Fujita and T. Kato obtained solutions for datum u0 ? H˙ 1 2 by semi-group methods. These solutions are unique ([8]) but only local in time: u ? C([0, T ?), H˙ 1 2 ), unless one is willing to make a smallness assumption on the datum.

  • global solution

  • then

  • global strong

  • l˜rt b˙

  • blow-up time

  • sp def

  • there exists


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