zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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 conservative solutions of the generalized hyperelastic-rod wave equation. (English) Zbl 1116.35115

Summary: We prove existence of global and conservative solutions of the Cauchy problem for the nonlinear partial differential equation

u t -u xxt +f(u) x -f(u) xxx +(g(u)+1 2f '' (u)(u x ) 2 ) x =0

where f is strictly convex or concave and g is locally uniformly Lipschitz. This includes the Camassa-Holm equation (f(u)=u 2 /2 and g(u)=κu+u 2 ) as well as the hyperelastic-rod wave equation (f(u)=γu 2 /2 and g(u)=(3-γ)u 2 /2) as special cases. It is shown that the problem is well-posed for initial data in H 1 () if one includes a Radon measure that corresponds to the energy of the system with the initial data. The solution is energy preserving. Stability is proved both with respect to initial data and the functions f and g. The proof uses an equivalent reformulation of the equation in terms of Lagrangian coordinates.

MSC:
35Q72Other PDE from mechanics (MSC2000)
35B35Stability of solutions of PDE
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35PDEs in connection with fluid mechanics