## Osaka Journal of Mathematics

- Osaka J. Math.
- Volume 44, Number 1 (2007), 99-119.

### Asymptotic behavior of solutions to the viscous Burgers equation

#### Abstract

We study the asymptotic behavior of solutions to the viscous Burgers equation by presenting a new asymptotic approximate solution. This approximate solution, called a diffusion wave approximate solution to the viscous Burgers equation of $k$-th order, is expanded in terms of the initial moments up to $k$-th order. Moreover, the spatial and time shifts are introduced into the leading order term to capture precisely the effect of the initial data on the long-time behavior of the actual solution. We also show the optimal convergence order in $L^p$-norm, $1\leq p\leq \infty$, of the diffusion wave approximate solution of $k$-th order. These results allow us to obtain the convergence of any higher order in $L^p$-norm by taking such a diffusion wave approximate solution with order $k$ large enough.

#### Article information

**Source**

Osaka J. Math., Volume 44, Number 1 (2007), 99-119.

**Dates**

First available in Project Euclid: 19 March 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.ojm/1174324325

**Mathematical Reviews number (MathSciNet)**

MR2313029

**Zentralblatt MATH identifier**

1122.35017

**Subjects**

Primary: 35B40: Asymptotic behavior of solutions 35C20: Asymptotic expansions 35Q35: PDEs in connection with fluid mechanics

Secondary: 35K05: Heat equation

#### Citation

Yanagisawa, Taku. Asymptotic behavior of solutions to the viscous Burgers equation. Osaka J. Math. 44 (2007), no. 1, 99--119. https://projecteuclid.org/euclid.ojm/1174324325