Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation
Sonego M, Zuazua E (2026)
Publication Language: English
Publication Status: Submitted
Publication Type: Unpublished / Preprint
Future Publication Type: Article in Edited Volumes
Publication year: 2026
DOI: 10.48550/arXiv.2609.13414
Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/crossover_mSonego_eZuazua_2026.pdf
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 non constant 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 −c log |Φc(0) − fM (0)| = 1 (M = 0).
Moreover, on compact sets in similarity variables, the interval and half-line solutions differ in C k by at most Ck,ε exp(−(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
Involved external institutions
Brazil (BR)How to cite
APA:
Sonego, M., & Zuazua, E. (2026). Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation. (Unpublished, Submitted).
MLA:
Sonego, Maicon, and Enrique Zuazua. Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation. Unpublished, Submitted. 2026.
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