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 ux=−u2. 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=cL2. For compactly supported initial data of mass M, the rescaled solution converges as L→∞ to an explicit crossover profile Φc. In similarity variables, Φc converges to the half-line profile fM as c↓0; after rescaling to domain variables, it converges to the interval equilibrium as c→∞. We also determine the sharp onset of confinement: limc↓0−clog|Φc(0)−fM(0)|=1 (M≠0). Moreover, on compact sets in similarity variables, the interval and half-line solutions differ in Ck by at most Ck,εexp(−(1−ε)L2/t), uniformly in L, and the exponential constant is optimal. Thus we identify not only the transition scale L2, 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.
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BibTeX
@misc{sonego2026crossover,
title = {{Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation}},
author = {Maicon Sonego and Enrique Zuazua},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13414},
eprint = {2609.13414},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13414},
}
RIS
TY - UNPB
TI - Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation
AU - Maicon Sonego
AU - Enrique Zuazua
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13414
UR - https://arxiv.org/abs/2609.13414
AB - 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 an explicit crossover profile Φ_c. In similarity variables, Φ_c converges to the half-line profile f_M as c↓0; after rescaling to domain variables, it converges to the interval equilibrium as c→∞. We also determine the sharp onset of confinement: lim_c↓0−clog|Φ_c(0)−f_M(0)|=1 (M≠0). Moreover, on compact sets in similarity variables, the interval and half-line solutions differ in C^k by at most C_k,εexp(−(1−ε)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.
ER -