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 existence and blow-up for a shallow water equation. (English) Zbl 0918.35005

An interesting phenomenon in water channels is the appearance of waves with length much greater than the depth of the water. Recently, R. Camassa and D. Holm proposed a new model for the same phenomenon:

u t -u xxt +3uu x =2u x u xx +uu xxx ,t>0,x,u(0,x)=u 0 (x),x·(1)

The variable u(t,x) in (1) represents the fluid velocity at time t in the x direction in appropriate nondimensional units (or, equivalently, the height of the water’s free surface above a flat bottom).

The aim of this paper is to prove local well-posedness of strong solutions to (1) for a large class of initial data, and to analyze global existence and blow-up phenomena. In addition, we introduce the notion of weak solutions to (1) suitable for soliton interaction.

35A05General existence and uniqueness theorems (PDE) (MSC2000)
35Q35PDEs in connection with fluid mechanics
35B40Asymptotic behavior of solutions of PDE
35Q53KdV-like (Korteweg-de Vries) equations