Osaka Journal of Mathematics
- Osaka J. Math.
- Volume 44, Number 1 (2007), 99-119.
Asymptotic behavior of solutions to the viscous Burgers equation
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.
Osaka J. Math., Volume 44, Number 1 (2007), 99-119.
First available in Project Euclid: 19 March 2007
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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