Control of moments at $t = \infin$ and higher order asymptotics in the burgers equation

Abstract

In this paper, an asymptotic approximate solution to the Burgers equation is constructed. The heat equation with nonnegative initial value has a linear combination of $n$ heat kernels as an approximate solution with convergence order $O(t^{\frac{1}{2r} - \frac{2n+1}{2}})$ in $L^r$ -norm, $1 \leq r \leq \infin$, as $t \rightarrow \infin$. The Burgers equation is obtained by adding a nonlinear convective term to the heat equation, and in turn one may question whether a similar form of the approximate solution exists. Note that, the Cole-Hopf transformation transforms a solution of nonlinear Burgers equation into the solution of the linear heat equation. Using Cole-Hopf transformation, we obtain an approximate solution to the Burgers equation by inverse-transforming a linear combination of $n$ heat kernels. Surprisingly, the algebraic convergence order of the Burgers equation is preserved, having $O(t^{\frac{1}{2r} - \frac{2n+1}{2}})$ in $L^r$ -norm, $1 \leq r \leq \infty$, as $t \rightarrow \infty$. Furthermore, we expect that the moments of the solution to the Burgers equation and those of approximate solution coincide at $t = \infty$. Some numerical results are included to show the convergence order.