Initial boundary value problems for scalar and vector Burgers equations

Abstract

In this article we study Burgers equation and vector Burgers equation with initial and boundary conditions. First we consider the Burgers equation in the quarter plane x>0, t>0 with Riemann type of initial and boundary conditions and use the Hopf-Cole transformation to linearize the problems and explicitly solve them. We study two limits, the small viscosity limit and the large time behavior of solutions. Next, we study the vector Burgers equation and solve the initial value problem for it when the initial data are gradient of a scalar function. We investigate the asymptotic behavior of this solution as time tends to infinity and generalize a result of Hopf to the vector case. Then we construct the exact N-wave solution as an asymptote of solution of an initial value problem extending the previous work of Sachdev et al. (1994). We also study the limit as viscosity parameter goes to 0.Finally, we get an explicit solution for a boundary value problem in a cylinder.