Global convergence of a sticky particle method for the modified Camassa-Holm equation

Yu Gao; Jian Guo Liu

doi:10.1137/16M1102069

Global convergence of a sticky particle method for the modified Camassa-Holm equation

Yu Gao, Jian Guo Liu

Research output: Journal article publication › Journal article › Academic research › peer-review

5 Citations (Scopus)

Abstract

In this paper, we prove convergence of a sticky particle method for the modified Camassa-Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV functions. The total variation of m(•, t) = u(•, t) - u_xx(•, t) is bounded by the total variation of the initial data m₀. We also obtain W^1,1(ℝ)-stability of weak solutions when solutions are in L^∞ (0, ∞; W^1,2(ℝ)). (Notice that peakon weak solutions are not in W^1,2(ℝ).) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.

Original language	English
Pages (from-to)	1267-1294
Number of pages	28
Journal	SIAM Journal on Mathematical Analysis
Volume	49
Issue number	2
DOIs	https://doi.org/10.1137/16M1102069
Publication status	E-pub ahead of print - 6 Apr 2017
Externally published	Yes

Keywords

Global existence
N-peakon solutions
Nonuniqueness
Space-time BV estimates
Sticky collisions

ASJC Scopus subject areas

Analysis
Computational Mathematics
Applied Mathematics

More information

10.1137/16M1102069

Cite this

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title = "Global convergence of a sticky particle method for the modified Camassa-Holm equation",

abstract = "In this paper, we prove convergence of a sticky particle method for the modified Camassa-Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV functions. The total variation of m(•, t) = u(•, t) - uxx(•, t) is bounded by the total variation of the initial data m0. We also obtain W1,1(ℝ)-stability of weak solutions when solutions are in L∞ (0, ∞; W1,2(ℝ)). (Notice that peakon weak solutions are not in W1,2(ℝ).) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.",

keywords = "Global existence, N-peakon solutions, Nonuniqueness, Space-time BV estimates, Sticky collisions",

author = "Yu Gao and Liu, {Jian Guo}",

note = "Funding Information: The work of the authors was supported by the National Science Foundation through the research network KI-Net RNMS11-07444. The work of the second author was supported by the National Science Foundation under grant DMS-1514826. Publisher Copyright: {\textcopyright} 2017 Society for Industrial and Applied Mathematics.",

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AU - Gao, Yu

AU - Liu, Jian Guo

N1 - Funding Information: The work of the authors was supported by the National Science Foundation through the research network KI-Net RNMS11-07444. The work of the second author was supported by the National Science Foundation under grant DMS-1514826. Publisher Copyright: © 2017 Society for Industrial and Applied Mathematics.

PY - 2017/4/6

Y1 - 2017/4/6

N2 - In this paper, we prove convergence of a sticky particle method for the modified Camassa-Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV functions. The total variation of m(•, t) = u(•, t) - uxx(•, t) is bounded by the total variation of the initial data m0. We also obtain W1,1(ℝ)-stability of weak solutions when solutions are in L∞ (0, ∞; W1,2(ℝ)). (Notice that peakon weak solutions are not in W1,2(ℝ).) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.

AB - In this paper, we prove convergence of a sticky particle method for the modified Camassa-Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV functions. The total variation of m(•, t) = u(•, t) - uxx(•, t) is bounded by the total variation of the initial data m0. We also obtain W1,1(ℝ)-stability of weak solutions when solutions are in L∞ (0, ∞; W1,2(ℝ)). (Notice that peakon weak solutions are not in W1,2(ℝ).) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.

KW - Global existence

KW - N-peakon solutions

KW - Nonuniqueness

KW - Space-time BV estimates

KW - Sticky collisions

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U2 - 10.1137/16M1102069

DO - 10.1137/16M1102069

M3 - Journal article

AN - SCOPUS:85018748953

SN - 0036-1410

VL - 49

SP - 1267

EP - 1294

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 2

ER -

Global convergence of a sticky particle method for the modified Camassa-Holm equation

Abstract

Keywords

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