The Hunter–Saxton equation with noise

Helge Holden; Kenneth H. Karlsen; Peter H.C. Pang

doi:10.1016/j.jde.2020.07.031

The Hunter–Saxton equation with noise

Helge Holden, Kenneth H. Karlsen, Peter H.C. Pang

Research output: Journal Publication › Article › peer-review

17 Citations (Scopus)

Abstract

In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter–Saxton equation (1.1), and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.

Original language	English
Pages (from-to)	725-786
Number of pages	62
Journal	Journal of Differential Equations
Volume	270
DOIs	https://doi.org/10.1016/j.jde.2020.07.031
Publication status	Published - 5 Jan 2021
Externally published	Yes

Keywords

Characteristics
Hunter–Saxton equation
Nonlocal wave equations
Stochastic solutions
Wave-breaking
Well-posedness

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2020.07.031

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title = "The Hunter–Saxton equation with noise",

abstract = "In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter–Saxton equation (1.1), and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.",

keywords = "Characteristics, Hunter–Saxton equation, Nonlocal wave equations, Stochastic solutions, Wave-breaking, Well-posedness",

author = "Helge Holden and Karlsen, \{Kenneth H.\} and Pang, \{Peter H.C.\}",

note = "Publisher Copyright: {\textcopyright} 2020 The Authors",

year = "2021",

month = jan,

day = "5",

doi = "10.1016/j.jde.2020.07.031",

language = "English",

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journal = "Journal of Differential Equations",

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T1 - The Hunter–Saxton equation with noise

AU - Holden, Helge

AU - Karlsen, Kenneth H.

AU - Pang, Peter H.C.

PY - 2021/1/5

Y1 - 2021/1/5

N2 - In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter–Saxton equation (1.1), and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.

AB - In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter–Saxton equation (1.1), and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.

KW - Characteristics

KW - Hunter–Saxton equation

KW - Nonlocal wave equations

KW - Stochastic solutions

KW - Wave-breaking

KW - Well-posedness

UR - https://www.scopus.com/pages/publications/85091076803

U2 - 10.1016/j.jde.2020.07.031

DO - 10.1016/j.jde.2020.07.031

M3 - Article

AN - SCOPUS:85091076803

SN - 0022-0396

VL - 270

SP - 725

EP - 786

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

The Hunter–Saxton equation with noise

Abstract

Keywords

ASJC Scopus subject areas

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