# 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 weak solutions to a generalized hyperelastic-rod wave equation. (English) Zbl 1100.35106

The Camassa-Holm equation is a nonlinear dispersive wave equation that takes the form

$\frac{\partial u}{\partial t}-\frac{{\partial }^{3}u}{\partial t\partial {x}^{2}}+2\kappa \frac{\partial u}{\partial x}+3u\frac{\partial u}{\partial x}=2\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial {x}^{2}}+u\frac{{\partial }^{3}u}{\partial {x}^{3}},\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{4pt}{0ex}}x\in ℝ·\phantom{\rule{2.em}{0ex}}\left(1\right)$

When $\kappa >0$, this equation models the propagation of unidirectional shallow water waves on a flat bottom, and $u\left(t,x\right)$ represents the fluid velocity at time $t$ in the horizontal direction $x$. The Camassa-Holm equation possesses a bi-Hamiltonian structure (and thus an infinite number of conservation laws) and is completely integrable. Moreover, when $\kappa =0$ it has an infinite number of solitary wave solutions, called peakons due to the discontinuity of their first derivatives at the wave peak. The solitary waves with $\kappa >0$ are smooth, while they become peaked when $\kappa \to 0$. From a mathematical point of view the Camassa-Holm equation is well studied. Local well-posedness results have been proved. It is also known that there exist global solutions for a particular class of initial data and also solutions that blow up in finite time for a large class of initial data.

The authors of the paper are interested in the Cauchy problem for the nonlinear equation

$\frac{\partial u}{\partial t}-\frac{{\partial }^{3}u}{\partial t\partial {x}^{2}}+\frac{\partial }{\partial x}\left(\frac{g\left(u\right)}{2}\right)=\gamma \left(\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial {x}^{2}}+u\frac{{\partial }^{3}u}{\partial {x}^{3}}\right),\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{4pt}{0ex}}x\in ℝ·\phantom{\rule{2.em}{0ex}}\left(2\right)$

where the function $g:ℝ\to ℝ$ and the constant $\gamma \in ℝ$ are given. If $g\left(u\right)=2\kappa u+3{u}^{2}$ and $\gamma =1,$ then (2) is the classical Camassa-Holm equation. With $g\left(u\right)=3{u}^{2}$, this equation describes finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods, and is often referred to as the hyperelastic-rod wave equation. The constant $\gamma$ is given in terms of the material constants and the prestress of the rod. The authors coin equation (2) the generalized hyperelastic-rod wave equation. From a mathematical point of view the generalized hyperelastic-rod wave equation (2) is much less studied than (1). The existence of a global weak solution to (2) for any initial function ${u}_{0}$ belonging to ${H}^{1}\left(ℝ\right)$ is established in the paper. Furthermore, the authors prove the existence of a strongly continuous semigroup, which in particular implies stability of the solution with respect to perturbations of data in the equation as well as perturbation in the initial data. The approach is based on a vanishing viscosity argument, showing stability of the solution when a regularizing term vanishes. This stability result is new even for the Camassa-Holm equation. Finally, the authors prove a “weak equals strong” uniqueness result.

##### MSC:
 35Q72 Other PDE from mechanics (MSC2000) 35D05 Existence of generalized solutions of PDE (MSC2000) 35G25 Initial value problems for nonlinear higher-order PDE 74D10 Nonlinear constitutive equations (materials with memory) 74H20 Existence of solutions for dynamical problems in solid mechanics 74K10 Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics