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.
Original language | English |
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Pages (from-to) | 1267-1294 |
Number of pages | 28 |
Journal | SIAM Journal on Mathematical Analysis |
Volume | 49 |
Issue number | 2 |
DOIs | |
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