zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global dissipative solutions of the Camassa-Holm equation. (English) Zbl 1139.35378
Summary: This paper is devoted to the continuation of solutions to the Camassa-Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an $\bold L^{\infty}$ space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data $\overline u \in H^1(\mathbb R)$, and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.

35L65Conservation laws
35L67Shocks and singularities
35Q58Other completely integrable PDE (MSC2000)
35B60Continuation of solutions of PDE
35L45First order hyperbolic systems, initial value problems
35L60Nonlinear first-order hyperbolic equations
Full Text: DOI