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Communications on Pure & Applied Analysis

2012 , Volume 11 , Issue 4

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Introduction to the special issue on hydrodynamic model equations
Adrian Constantin and Joachim Escher
2012, 11(4): i-iii doi: 10.3934/cpaa.2012.11.4i +[Abstract](214) +[PDF](164.0KB)
Abstract:
The increased interest in water wave theory over the last decade has been motivated, arguably, by two themes: rst, noticeable progress in the investigation of the governing equations for water waves (well-posedness issues, as well as in-depth qualitative studies of regular wave patterns{see the discussion and the list of references in [1, 17], respectively in [9]), and secondly, by the derivation and study of various model equations that, although simpler, capture with accuracy the prominent features of the governing equations in a certain physical regime. The two themes are intertwined with one another.
Global weak solutions to the generalized Proudman-Johnson equation
Chien-Hong Cho and Marcus Wunsch
2012, 11(4): 1387-1396 doi: 10.3934/cpaa.2012.11.1387 +[Abstract](207) +[PDF](381.0KB)
Abstract:
We consider the generalized Proudman-Johnson equation, in which an artificial parameter $a$ controlling the impact of convection was introduced to the Proudman-Johnson equation ([33]). In the present paper, we are going to show that there are global weak solutions to the generalized Proudman-Johnson equation for certain parameter $a$'s.
Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity
Adrian Constantin
2012, 11(4): 1397-1406 doi: 10.3934/cpaa.2012.11.1397 +[Abstract](175) +[PDF](365.6KB)
Abstract:
In the absence of stagnation points, we derive the dispersion relation for periodic travelling waves of small amplitude propagating at the surface of water with a layer of constant vorticity adjacent to the flat bed within an irrotational flow.
The geometry of a vorticity model equation
Joachim Escher, Boris Kolev and Marcus Wunsch
2012, 11(4): 1407-1419 doi: 10.3934/cpaa.2012.11.1407 +[Abstract](211) +[PDF](417.9KB)
Abstract:
We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$ of orientation-preserving diffeomorphisms $\varphi \in \mathrm{Diff}^{\infty}(\mathbb{S})$ such that $\varphi(1) = 1$ equipped with the right-invariant metric induced by the homogeneous Sobolev norm $\dot H^{1/2}$. On the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$, this induces a weak Riemannian structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.
Integrating factors and conservation laws for some Camassa-Holm type equations
Marianna Euler and Norbert Euler
2012, 11(4): 1421-1430 doi: 10.3934/cpaa.2012.11.1421 +[Abstract](198) +[PDF](310.3KB)
Abstract:
We classify all first-order integrating factors and the corresponding conservation laws for a class of Camassa-Holm type equations.
A note on uniqueness and compact support of solutions in a recent model for tsunami background flows
Anna Geyer
2012, 11(4): 1431-1438 doi: 10.3934/cpaa.2012.11.1431 +[Abstract](159) +[PDF](299.4KB)
Abstract:
We present an elementary proof of uniqueness for solutions of an initial value problem which is not Lipschitz continuous, generalizing a technique employed in [20]. This approach can be applied for a wide class of vorticity functions in the context of [6], where, departing from a recent model for the evolution of tsunami waves developed in [10], the possibility of modelling background ows with isolated regions of vorticity is rigorously established.
On the integrability of KdV hierarchy with self-consistent sources
Vladimir S. Gerdjikov, Georgi Grahovski and Rossen Ivanov
2012, 11(4): 1439-1452 doi: 10.3934/cpaa.2012.11.1439 +[Abstract](149) +[PDF](409.9KB)
Abstract:
Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called ``squared solutions'' (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. Such equations, however in general, are not completely integrable.
On the regularity of steady periodic stratified water waves
David Henry and Bogdan--Vasile Matioc
2012, 11(4): 1453-1464 doi: 10.3934/cpaa.2012.11.1453 +[Abstract](163) +[PDF](348.4KB)
Abstract:
In this paper we prove regularity results for steady periodic stratified water waves, where we allow for the effects of surface tension. Our results concern stratified water waves, without stagnation points, which exist in three distinct physical regimes, namely: capillary, capillary-gravity, and gravity water waves. We prove, for all three types of waves, that, when the Bernoulli function is Hölder continuous and the variable density function has a first derivative which is Hölder continuous, then the free-surface profile is the graph of a smooth function. Furthermore, we show that the streamlines are analytic a priori for capillary stratified waves, whereas for gravity and capillary-gravity stratified waves the streamlines are smooth in general, and analytic in an unstable regime. Moreover, if the Bernoulli function and the streamline density function are both real analytic functions then all of the streamlines, including the wave profile, are real analytic for all gravity, capillary, and capillary-gravity stratified waves.
On the formation of singularities for surface water waves
Vera Mikyoung Hur
2012, 11(4): 1465-1474 doi: 10.3934/cpaa.2012.11.1465 +[Abstract](215) +[PDF](317.5KB)
Abstract:
A Burgers equation with fractional dispersion is proposed to model waves on the moving surface of a two-dimensional, infinitely deep water under the influence of gravity. For a certain class of initial data, the solution is shown to blow up in finite time.
Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity
Delia Ionescu-Kruse
2012, 11(4): 1475-1496 doi: 10.3934/cpaa.2012.11.1475 +[Abstract](151) +[PDF](546.1KB)
Abstract:
We provide analytic solutions of the nonlinear differential equation system describing the particle paths below small-amplitude periodic gravity waves travelling on a constant vorticity current. We show that these paths are not closed curves. Some solutions can be expressed in terms of Jacobi elliptic functions, others in terms of hyperelliptic functions. We obtain new kinds of particle paths. We make some remarks on the stagnation points which could appear in the fluid due to the vorticity.
A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques
R. S. Johnson
2012, 11(4): 1497-1522 doi: 10.3934/cpaa.2012.11.1497 +[Abstract](140) +[PDF](455.8KB)
Abstract:
The methods of analysis based on asymptotic expansions, with a small parameter, are briefly outlined. These techniques are then applied to three examples in the theory of water waves, the aim being to demonstrate the effectiveness of this approach. Throughout, we relate this procedure to more rigorous methods.
The classical problem of water waves is formulated, and then various parameter choices are incorporated. The first problem examines the properties of small-amplitude, periodic waves over constant vorticity and, invoking the ideas of breakdown, scaling and matching, the possibility of stagnation is investigated. In the second case, a Camassa-Holm equation is derived; this is found to be relevant only for the horizontal velocity component in the flow at a specific depth. However, this special, integrable equation requires the retention of a number of terms of different asymptotic order--which is the least satisfactory way to use these methods. The final example, which shows how some new aspects of a familiar problem can be obtained by these methods, develops an asymptotic description of edge waves. This leads to a single equation that captures all aspects of the run-up pattern at a shoreline.
On isolated vorticity regions beneath the water surface
Octavian G. Mustafa
2012, 11(4): 1523-1535 doi: 10.3934/cpaa.2012.11.1523 +[Abstract](163) +[PDF](404.3KB)
Abstract:
We present a class of vorticity functions that will allow for isolated, circular vorticity regions in the background of still water, preceding the arrival of waves at the shoreline.
On particle trajectories in linear deep-water waves
Anca-Voichita Matioc
2012, 11(4): 1537-1547 doi: 10.3934/cpaa.2012.11.1537 +[Abstract](189) +[PDF](386.9KB)
Abstract:
We determine the phase portrait of a Hamiltonian system of equations describing the motion of the particles in linear deep-water waves. The particles experience in each period a forward drift which decreases with greater depth.
Effects of shear flow on KdV balance - applications to tsunami
Raphael Stuhlmeier
2012, 11(4): 1549-1561 doi: 10.3934/cpaa.2012.11.1549 +[Abstract](175) +[PDF](358.2KB)
Abstract:
Building upon recent work in the applicability of soliton theory to tsunami propagation, we discuss the effects of shear flow on the KdV balance. This leads in the shallow-water limit to the Burns condition, and we see that for shear which does not yield critical layer solutions, the speeds determined by the Burns condition arise again in the KdV balance. In the event of waves propagating counter to the shear, KdV dynamics arise earlier, while their appearance is delayed in the case of waves propagating with the shear, the magnitude of this effect depending on the surface shear velocity.
The 2-component dispersionless Burgers equation arising in the modelling of blood flow
Tony Lyons
2012, 11(4): 1563-1576 doi: 10.3934/cpaa.2012.11.1563 +[Abstract](190) +[PDF](358.3KB)
Abstract:
This article investigates the properties of the solutions of the dispersionless two-component Burgers (B2) equation, derived as a model for blood-flow in arteries with elastic walls. The phenomenon of wave breaking is investigated as well as applications of the model to clinical conditions.
On the symmetry of steady periodic water waves with stagnation points
Gerhard Tulzer
2012, 11(4): 1577-1586 doi: 10.3934/cpaa.2012.11.1577 +[Abstract](150) +[PDF](377.7KB)
Abstract:
The aim of this paper is to prove the symmetry of small-amplitude steady periodic water waves with monotonic wave profile even if stagnation points occur in the flow beneath the wave.

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