Optimal control of the viscous Camassa–Holm equation

This paper studies the problem of optimal control of the viscous Camassa–Holm equation. The existence and uniqueness of weak solution to the viscous Camassa–Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa–Holm equation is proved.     

Remarks on regularity criteria for an Ericksen–Leslie system and the viscous Camassa–Holm equations

In this paper, regularity criteria for a simplified Ericksen–Leslie system and the viscous Camassa–Holm equations are established in Besov spaces with negative index.     

Optimal control of the viscous generalized Camassa–Holm equation

In this paper, we study the optimal control problem for the viscous generalized Camassa–Holm equation. We deduce the existence and uniqueness of weak solution to the viscous generalized Camassa–Holm equation in a short interval by using Galerkin method. Then, by using optimal control theories and distributed parameter system control theories, the optimal control of the viscous generalized Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous generalized Camassa–Holm equation is proved.     

A note on regularity criteria for the viscous Camassa–Holm equations in multiplier spaces

In this note, regularity criteria for the viscous Camassa–Holm equations are established in multiplier spaces; these improve on previous results.     

Global attractor for the viscous modified two-component Camassa–Holm equation

We obtain the existence of global attractor for the Cauchy problem of a viscous modified two-component Camassa–Holm equation. The existence of global strong solutions is obtained using Kato’s theory. The key elements in our analysis are the uniform Gronwall lemma and some estimates of the solutions.     

On the solutions of the cross-coupled Camassa–Holm system

In this paper, we study the Cauchy problem for a recently derived system of two cross-coupled Camassa–Holm equations. We firstly establish the local well-posedness result of this system in Besov spaces by using Littlewood–Paley decomposition and the transport equation theory, and then present a precise blow-up scenario for strong solutions.     

Bifurcation of travelling wave solutions for the generalized Camassa–Holm–KP equations

By using the bifurcation theory of planar dynamical systems to the generalized Camassa–Holm–KP equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.     

Null controllability of the viscous Camassa–Holm equation with moving control

In this paper, we study the null controllability of the viscous Camassa–Holm equation on the one-dimensional torus. By using a moving distributed control, we obtain that the system is null controllable for a given data with certain regularity.     

Lyapunov-type stability criterion for periodic generalized Camassa–Holm equations

We study the Lyapunov stability of the periodic generalized Camassa–Holm equation in terms of the periodic/anti-periodic eigenvalues and the associated spectral intervals. Moreover, we establish a Lyapunov-type stability criterion based on the Floquet theory and a Lyapunov-type inequality.     

Large time behavior and convergence for the Camassa–Holm equations with fractional Laplacian viscosity

In this paper, we consider the n-dimensional (\(n=2,3\)) Camassa–Holm equations with fractional Laplacian viscosity in the whole space. In contrast to the Camassa–Holm equations without any nonlocal effect, much less has been known on the large time behavior and convergences of solutions. Here we study first the large time behavior of solutions, then consider the relation between the equations under consideration and the imcompressible Navier–Stokes equations with fractional Laplacian viscosity (INSF). By applying the fractional Leibniz chain rule and the fractional Gagliardo–Nirenberg–Sobolev type estimates, the high and low frequency splitting method and the Fourier splitting method, we shall establish the large time non-uniform decays and algebraic rate decays of solutions. In the critical case \(s=\dfrac{n}{4}\), the nonlocal version of Ladyzhenskaya’s inequality along with the smallness of initial data in suitable Sobolev spaces is needed. In addition, by estimates for the fractional heat kernels, we prove that the solutions to the Camassa–Holm equations with nonlocal viscosity converge strongly as the filter parameter \(\alpha \rightarrow ~0\) to solutions of the equations INSF.     

Global existence and blow-up phenomena for a periodic 2-component Camassa–Holm equation

We first establish local well-posedness for a periodic 2-component Camassa?CHolm equation. We then present two global existence results for strong solutions to the equation. We finally obtain several blow-up results and the blow-up rate of strong solutions to the equation.     

Integrability,existence of global solutions,and wave breaking criteria for a generalization of the Camassa–Holm equation

Recent generalizations of the Camassa–Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative of local solutions, for global well-posedness in Sobolev spaces for the family under consideration. Moreover, we prove that wave breaking phenomena occurs under certain mild hypothesis. Based on the machinery developed by Dubrovin [Commun. Math. Phys. 267, 117–139 (2006)] regarding bi-Hamiltonian deformations, we introduce the notion of quasi-integrability and prove that there exists a unique bi-Hamiltonian structure for the equation only when it is reduced to the Dullin–Gotwald–Holm equation. Our results suggest that a recent shallow water model incorporating Coriollis effects is integrable only in specific situations. Finally, to finish the scheme of geometric integrability of the family of equations initiated in a previous work, we prove that the Dullin–Gotwald–Holm equation describes pseudo-spherical surfaces.     

Wave Breaking of the Camassa–Holm Equation

This paper gives a new and direct proof for McKean’s theorem (McKean in Asian J. Math. 2:867–874, 1998) on wave breaking of the Camassa–Holm equation. The blow-up profile is also analyzed.