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2004 Global solutions of Navier-Stokes equations with large $L^2$ norms in a new function space
Qi S. Zhang
Adv. Differential Equations 9(5-6): 587-624 (2004).

Abstract

First we prove certain pointwise bounds for the fundamental solutions of the perturbed linearized Navier-Stokes equation (Theorem 1.1). Next, utilizing a new framework with very little $L^p$ theory or Fourier analysis, we prove existence of global classical solutions for the full Navier-Stokes equation when the initial value has a small norm in a new function class of Kato type (Theorem 1.2). The smallness in this function class does not require smallness in $L^2$ norm. Furthermore we prove that a Leray-Hopf solution is regular if it lies in this class, which allows much more singular functions then before (Corollary 1). For instance this includes the well-known result in [25]. A further regularity condition (form boundedness) was given in Section 5. We also give a different proof about the $L^2$ decay of Leray-Hopf solutions and prove pointwise decay of solutions for the three-dimensional Navier-Stokes equations (Corollary 2, Theorem 1.2). Whether such a method exists was asked in a survey paper [2].

Citation

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Qi S. Zhang. "Global solutions of Navier-Stokes equations with large $L^2$ norms in a new function space." Adv. Differential Equations 9 (5-6) 587 - 624, 2004.

Information

Published: 2004
First available in Project Euclid: 18 December 2012

zbMATH: 1100.35083
MathSciNet: MR2099973

Subjects:
Primary: 35Q30
Secondary: 35A08, 76D03, 76D05

Rights: Copyright © 2004 Khayyam Publishing, Inc.

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Vol.9 • No. 5-6 • 2004
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