Evolution of a Multiscale Singularity of the Solution of the Burgers Equation in the 4-Dimensional Space-Time

Abstract

The solution of the Cauchy problem for the vector Burgers equation with a small parameter of dissipation ε in the 4-dimensional space-time is studied: ut+(u∇)u=ε△u,uν(x,−1,ε)=−xν+4−ν(ν+1)x2ν+1ν. With the help of the Cole--Hopf transform u=−2ε∇lnH, the exact solution and its leading asymptotic approximation, depending on six space-time scales, near a singular point are found. A formula for the growth of partial derivatives of the components of the vector field u on the time interval from the initial moment to the singular point, called the formula of the gradient catastrophe, is established: ∂uν(0,t,ε)∂xν=1t[1+O(ε|t|−1−1/ν)],tεν/(ν+1)→−∞,t→−0. The asymptotics of the solution far from the singular point, involving a multistep reconstruction of the space-time scales, is also obtained: uν(x,t,ε)≈−2(tν+1)1/2νtanh[xνε(tν+1)1/2ν],tεν/(ν+1)→+∞.