On the traveling waves governed by the Camassa–Holm equation

The Camassa–Holm equation admits undistorted traveling waves that are either smooth or exhibit peaks or cusps. All three wave types can be periodic or solitary. Also waves of different types may be combined. In the present paper it is shown that, apart from peaks and cusps, the traveling waves governed by the Camassa–Holm equation can be found from some simpler equation. In the case of peaked solutions, this reduced equation is even linear. The governing equation of traveling waves in its original form can be interpreted as a nonlinear combination of the reduced equation and its first integral. For a small range of the integration constant, the reduced equation admits bounded solutions, which then are directly inherited by the Camassa–Holm equation. In general, the solutions of the reduced equation are unbounded and cannot be considered to represent traveling waves. The full equation, however, has a nonlinearity in the highest derivative, which is characteristic for the Camassa–Holm and some other equations. This nonlinear term offers the possibility of constructing bounded traveling waves from the unbounded solutions of the reduced equation. These waves necessarily have discontinuities in the slope and are, therefore, solutions only in a generalized sense.     

On the Small-Amplitude Long Waves in Linear Shear Flows and the Camassa–Holm Equation

We compare two linearizations used in the study of small-amplitude long waves on a constant vorticity flow. For the propagation of such waves, we derive, by a variational approach in the Lagrangian formalism, the nonlinear Camassa–Holm equation.     

The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin–Bona–Mahoney and Korteweg–de Vries equations. In particular, they accommodate wave breaking phenomena.     

Stability of the Camassa–Holm Multi-peakons in the Dynamics of a Shallow-Water-Type System

Consideration herein is the stability issue of a variety of superpositions of the Camassa–Holm peakons and antipeakons in the dynamics of the two-component Camassa–Holm system, which is derived in the shallow water theory. These wave configurations accommodate the ordered trains of the Camassa–Holm peakons, the ordered trains of Camassa–Holm antipeakons and peakons as well as the Camassa–Holm multi-peakons. Using the features of conservation laws and the monotonicity properties of the local energy, we prove the orbital stability of these wave profiles in the energy space by the modulation argument.     

The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory

We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokes-alpha) (NS-) equations. We also provide estimates, in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the Kolmogorov theory of turbulence, we define a small spatial scale, , as the scale at which the balance occurs in the mean rates of nonlinear transport of energy and viscous dissipation of energy. Furthermore, we show that the number of degrees of freedom in the long-time behavior of the solutions to these equations is bounded from above by (L/ )³, where L is a typical large spatial scale (e.g., the size of the domain). This estimate suggests that the Landau–Lifshitz classical theory of turbulence is suitable for interpreting the solutions of the NS- equations. Hence, one may consider these equations as a closure model for the Reynolds averaged Navier–Stokes equations (NSE). We study this approach, further, in other related papers. Finally, we discuss the relation of the NS- model to the NSE by proving a convergence theorem, that as the length scale ₁ tends to zero a subsequence of solutions of the NS- equations converges to a weak solution of the three dimensional NSE.     

Several new types of bounded wave solutions for the generalized two-component Camassa–Holm equation

Li and Qiao studied the bifurcations and exact traveling wave solutions for the generalized two-component Camassa–Holm equation $$\begin{aligned} \left\{ \begin{array}{l} m_{t}+\sigma um_{x}-Au_{x}+2m \sigma u_{x}+3(1-\sigma )uu_{x}\\ \quad +\rho \rho _{x}=0, \\ \rho _{t} +(\rho u)_{x}=0, \end{array} \right. \end{aligned}$$ \(m=u-u_{xx}, A>0\) . They showed that there exist solitary wave solutions, cusp wave solutions, and periodic wave solutions for the equation, and their analysis focused on the bifurcations when \(\sigma >0\) . In this paper, we first complement the bifurcations when \(\sigma <0\) by following the same procedure as that of Li, and then show the existence and implicit expressions of several new types of bounded wave solutions, including solitary waves, periodic waves, compacton-like waves, and kink-like waves. In addition, the numerical simulations of the bounded wave solutions are given to show the correctness of our results.     

On the Regularity Set and Angular Integrability for the Navier–Stokes Equation

We investigate the size of the regular set for suitable weak solutions of the Navier–Stokes equation, in the sense of Caffarelli–Kohn–Nirenberg (Commun Pure Appl Math 35:771–831, 1982). We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space \({\{t > 0\}}\) in an appropriate limit. In particular, we obtain that if the \({L^{2}}\) norm with weight \({|x|^{-\frac12}}\) of the data tends to 0, the regular set invades \({\{t > 0\}}\); this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982).     

Bifurcations and exact solutions of an asymptotic rotation-Camassa–Holm equation

This paper investigates the rotation-Camassa–Holm equation, which appears in long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the earth’s rotation. The rotation-Camassa–Holm equation contains the famous Camassa–Holm equation and is a special case of the generalized Camassa–Holm equation. By using the approach of dynamical systems and singular traveling wave theory to its traveling wave system, in different parameter conditions of the five-parameter space, the bifurcations of phase portraits are studied. Some exact explicit parametric representations of the smooth solitary wave solutions, periodic wave solutions, peakons and anti-peakons, periodic peakons as well as compacton solutions are obtained.

     

On the Robin-Transmission Boundary Value Problems for the Nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes Systems

The purpose of this paper is to study a boundary value problem of Robin-transmission type for the nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes systems in two adjacent bounded Lipschitz domains in \({{\mathbb{R}}^{n} (n\in \{2,3\})}\), with linear transmission conditions on the internal Lipschitz interface and a linear Robin condition on the remaining part of the Lipschitz boundary. We also consider a Robin-transmission problem for the same nonlinear systems subject to nonlinear transmission conditions on the internal Lipschitz interface and a nonlinear Robin condition on the remaining part of the boundary. For each of these problems we exploit layer potential theoretic methods combined with fixed point theorems in order to show existence results in Sobolev spaces, when the given data are suitably small in \({L^2}\)-based Sobolev spaces or in some Besov spaces. For the first mentioned problem, which corresponds to linear Robin and transmission conditions, we also show a uniqueness result. Note that the Brinkman–Forchheimer-extended Darcy equation is a nonlinear equation that describes saturated porous media fluid flows.     

On the thermodynamics of the Swift–Hohenberg theory

We present the microbalance including the microforces, the first- and second-order microstresses for the Swift–Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift–Hohenberg theory via a proper constitutive process. Finally, we present one highly resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift–Hohenberg equation.     

On the forward and backward in time problems in the Kelvin–Voigt thermoviscoelastic materials

This paper studies the uniqueness of solutions to the forward and backward in time boundary value problems associated with the Kelvin–Voigt viscoelastic model of the thermoelastic materials. For thermoviscoelastic materials with a center of symmetry, it is shown the uniqueness of solutions to the forward in time boundary value problems without any assumptions upon the thermoviscoelastic constitutive coefficients other than the symmetry properties and those induced by the dissipation inequality. While for the final boundary value problems two uniqueness theorems are presented: the first one is essentially based on the assumption that the specific heat is of negative definite sign, while the second is established in the class of displacement–temperature variation fields whose dissipation energy has a temporal behavior lower than an appropriate growing exponential.     

On the rheology of ethylene–octene copolymers

It has previously been shown that the plateau modulus, G_N^o, and thus the entanglement molecular weight, M_e, of flexible polymers can be correlated to the unperturbed chain dimension, <R²>_o/M, and mass density, , via the use of the packing length, p. For polyolefins, a method was recently proposed whereby knowledge of the average molecular weight per backbone bond, m_b, allows <R²>_o/M and consequently G_N^o and M_e to be estimated. This is particularly valuable for polyolefin copolymers since the melt chain dimensions are often unknown. This work corroborates these theoretical predictions by studying the rheology of a series of carefully synthesized ethylene/octene copolymers with varying octene content (19–92 wt%). Furthermore, the results reported herein also allow the advancement of rheological characterization techniques of polymer melts. For instance, based on the analysis of the linear viscoelastic properties of these copolymers, it has been found that several rheological parameters scale with the copolymer comonomer content. Analysis of the viscoelastic material functions in terms of the evolution of the phase angle, , as a function of the absolute value of the complex modulus, |G*|, (the so-called van Gurp–Palmen plots), provides a fast and reliable rheological means for determining the composition of ethylene/-olefin copolymers. The crossover parameters, G_co(=G=G) and _co(=1/_co) also scale with copolymer composition.Submitted for publication to Rheologica Acta

---

An erratum to this article is available at .