The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

In this work, we examine the solution properties of the Burgers’ equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine–Hugoniot condition that the shocks satisfy. Next, we establish the local existence a...

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Bibliographic Details
Published in:Nonlinear differential equations and applications 2019-12, Vol.26 (6), p.1-33, Article 57
Main Authors: Alonso-Orán, Diego, Bethencourt de León, Aythami, Takao, So
Format: Article
Language:English
Subjects:
Analysis
Burgers equation
Mathematics
Mathematics and Statistics
Transport
Uniqueness
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Summary:In this work, we examine the solution properties of the Burgers’ equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine–Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.
ISSN:1021-9722
1420-9004
DOI:10.1007/s00030-019-0602-6