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)
A long wave approximation for capillary-gravity waves and an effect of the bottom. (English) Zbl 1136.35081
The paper aims to generalize the classical derivation of the Korteweg-de Vries equation for small-amplitude long waves on the surface of a shallow layer of water (in the absence of viscosity and for irrotational flows), taking into regard the surface tension and possible non-uniformity of the bottom. The analysis gives rise to a coupled system of two equations for the velocity at the surface, $v(x,t)$, and local elevation of the surface, $\eta(x,t)$: $$\align u_t +\eta_x +\varepsilon u_1u_x - \varepsilon \mu\eta_{xxxx}&=0,\\ \eta_x + u_x+\varepsilon[(\eta - b)]_x + (\varepsilon/3)u_{xxx}&=0, \endalign$$ where $b(x)$ is the local depth, and small parameter $\varepsilon$ is the measure for the smallness of ratio of the local elevation of the free surface to the depth of the layer. It is proven that the full dynamics of the water waves is described by the above system for a long time interval. In fact, a priori estimates for the duration of the validity interval are essentially the same as obtained in earlier works, but estimates for the solution, and especially for its derivatives, are much stronger. The analysis is performed in the Eulerian (rather than Lagrangian) coordinates.

35Q53KdV-like (Korteweg-de Vries) equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B45Capillarity (surface tension)
35B45A priori estimates for solutions of PDE
Full Text: DOI