## Advances in Differential Equations

- Adv. Differential Equations
- Volume 9, Number 5-6 (2004), 587-624.

### Global solutions of Navier-Stokes equations with large $L^2$ norms in a new function space

#### 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].

#### Article information

**Source**

Adv. Differential Equations, Volume 9, Number 5-6 (2004), 587-624.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867937

**Mathematical Reviews number (MathSciNet)**

MR2099973

**Zentralblatt MATH identifier**

1100.35083

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Secondary: 35A08: Fundamental solutions 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D05: Navier-Stokes equations [See also 35Q30]

#### Citation

Zhang, Qi S. Global solutions of Navier-Stokes equations with large $L^2$ norms in a new function space. Adv. Differential Equations 9 (2004), no. 5-6, 587--624. https://projecteuclid.org/euclid.ade/1355867937