## Differential and Integral Equations

- Differential Integral Equations
- Volume 10, Number 3 (1997), 577-586.

### Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations

#### Abstract

The purpose of this note is to present new results on linear transport equations with initial values in $W^{1,r}(\Omega)$ spaces, and coefficients in $W^{s+1,p}(\Omega)^N$, where $\Omega$ is a bounded open subset of $\mathbb{R}^N$ ($N \geq 2$) and $sp=N$. As an application, we also give refined regularity and uniqueness results for weak solutions of the two-dimensional density-dependent incompressible Navier-Stokes equations.

#### Article information

**Source**

Differential Integral Equations, Volume 10, Number 3 (1997), 577-586.

**Dates**

First available in Project Euclid: 2 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367525668

**Mathematical Reviews number (MathSciNet)**

MR1744862

**Zentralblatt MATH identifier**

0902.76028

**Subjects**

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

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

#### Citation

Desjardins, Benoit. Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations. Differential Integral Equations 10 (1997), no. 3, 577--586. https://projecteuclid.org/euclid.die/1367525668