Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation

Abstract

We study the one-dimensional viscous Burgers equation on $(0,L)$ with the conservative boundary conditions $u_x=-u^2$. On the half-line, solutions approach a nonlinear self-similar profile, whereas on a bounded interval they converge to a nonconstant equilibrium. We describe explicitly how the dynamics passes between these states at the critical diffusive scale $t=cL^2$. For compactly supported initial data of mass $M$, the rescaled solution converges as $L\to\infty$ to an explicit crossover profile $Φ_c$. In similarity variables, $Φ_c$ converges to the half-line profile $f_M$ as $c\downarrow0$; after rescaling to domain variables, it converges to the interval equilibrium as $c\to\infty$. We also determine the sharp onset of confinement: \[ \lim_{c\downarrow0}-c\log|Φ_c(0)-f_M(0)|=1\qquad(M\ne0). \] Moreover, on compact sets in similarity variables, the interval and half-line solutions differ in $C^k$ by at most $C_{k,\varepsilon}\exp(-(1-\varepsilon)L^2/t)$, uniformly in $L$, and the exponential constant is optimal. Thus we identify not only the transition scale $L^2$, but also the profile governing the crossover and the sharp rate at which the remote boundary becomes visible. We finally discuss the conclusions that persist for space-dependent diffusivity and illustrate the three asymptotic regimes numerically.