Global Conservative Solutions Research Articles

Interior inverse problem for global conservative multipeakon solutions of the Camassa-Holm equation

We consider the interior inverse problem associated with the global conservative multipeakon solution of the Camassa-Holm equation. Based on the inverse spectral theory on the half-line and the oscillation property of eigenfunctions, some (non)uniqueness results of the interior inverse problem are obtained. In addition, we give the trace formula, which connects the global conservative multipeakon solution with the corresponding eigenvalues and normalized eigenfunctions.

Existence and uniqueness of the global conservative weak solutions to the cubic Camassa–Holm-Type equation

This paper is devoted to the cubic Camassa–Holm-Type equation with a nonlocal cubic nonlinearity suggested as an asymptotic model for the 2D full water wave dynamics which is nonlocal with higher-order nonlinearities compared to the classical Camassa–Holm equation. We establish the global existence and uniqueness of the energy conservative weak solution in the energy space.

Existence and uniqueness of the global conservative weak solutions for a cubic Camassa-Holm type equation

The extended cubic Camassa-Holm equation can be derived as asymptotic model from shallow-water approximation to the 2D incompressible Euler equations. This model encompasses both quadratic and cubic nonlinearities and the solution of the extended cubic Camassa-Holm equation possible development of singularities in finite time. This paper is devoted to the continuation of solutions to the extended cubic Camassa-Holm equation beyond wave breaking, and the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the extended cubic Camassa-Holm type equation are investigated. An equivalent semilinear system was first introduced by a new set of independent and dependent variables, which can resolve all singularities due to possible wave breaking. Returning to the original variables, depending continuously on the initial data, the existence of the global conservative weak solutions can be obtained. Moreover, by analyzing the evolution of the quantities u and v=2arctan⁡ux along each characteristic, the uniqueness of the global conservative solutions for the Cauchy problem with general initial data u0∈H1(R) was proved.

Uniqueness of global conservative weak solutions for two-component higher-order shallow water system

Uniqueness of global conservative weak solutions for two-component higher-order shallow water system

Existence and regularity for global weak solutions to the 𝜆-family water wave equations

In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as λ \lambda -family equations, where λ \lambda is the power of nonlinear wave speed. The λ \lambda -family equations include Camassa-Holm equation ( λ = 1 \lambda =1 ) and Novikov equation ( λ = 2 \lambda =2 ) modelling water waves, where solutions generically form finite time cusp singularities, or, in other words, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent 1 − 1 2 λ 1- \frac {1}{2\lambda } . The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.

Global conservative solutions of the nonlocal NLS equation beyond blow-up

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \mathrm{i}\partial_t q(x, t)+\partial_{x}^2q(x, t)+2\sigma q^{2}(x, t)\overline{q(-x, t)} = 0 $ with initial data $ q(x, 0)\in H^{1, 1}(\mathbb{R}) $. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $ H^{1, 1} $ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

Uniqueness and generic regularity of global weak conservative solutions to the Constantin-Lannes equation

ABSTRACT This paper is devoted to the uniqueness of global-in-time conservative solutions and generic regularity for the shallow water waves of moderate amplitude equation (Constantin-Lannes equation). The Constantin-Lannes equation possible development of singularities in finite time, and beyond the occurrence of wave breaking, it exists global conservative solutions. In the present paper, we will prove the uniqueness of global-in-time conservative solutions for the Constantin-Lannes equation with general initial data by analyzing the evolution of the quantities u and along each characteristic. Moreover, we consider that piecewise smooth solutions with only generic singularities are dense in the whole solution set by Thom's transversality Lemma.

Global conservative solution for a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities

This paper is devoted to the global conservative solutions of a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solutions of the semilinear system. Returning to the original variables, we get the global conservative solution of the equation. Finally, we show that the peakon solutions of the equation still conserve in .

Global Conservative Solutions of the Two-Component μ -Hunter–Saxton System

In this paper, we establish global conservative solutions of the two-component μ -Hunter–Saxton system by the methods developed in “A. Bressan, A. Constantin, Global Conservative Solutions of the Camassa-Holm Equation, Arch. Ration. Mech. Anal. 183 (2), 215–239 (2007)” and “H. Holden, X. Raynaud, Periodic Conservative Solutions of the Camassa-Holm Equation, Ann. Inst. Fourier (Grenoble) 58(3), 945–988 (2008).”

Global conservative weak solutions for the two-component Novikov equation

In this paper, we study the existence of global conservative solutions to a two-component Novikov system. The system is an integrable multi-component extension of the Novikov integrable equation. We develop the method of constructing global conservative solutions of the Camassa–Holm equation due to Bressan and Constantin to a two-component case with genuine nonlinear interactions. Our approach is based on the formulation of Bressan and Constantin and additional estimates relating to the new conservation laws of the system.

A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa–Holm system

We define a kinetic and a potential energy such that the principle of stationary action from Lagrangian mechanics yields a Camassa–Holm system (2CH) as the governing equations. After discretizing these energies, we use the same variational principle to derive a semi-discrete system of equations as an approximation of the 2CH system. The discretization is only available in Lagrangian coordinates and requires the inversion of a discrete Sturm–Liouville operator with time-varying coefficients. We show the existence of fundamental solutions for this operator at initial time with appropriate decay. By propagating the fundamental solutions in time, we define an equivalent semi-discrete system for which we prove that there exists unique global solutions. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions converging to the conservative solution of the 2CH system.

Global conservative solution for the periodic dispersive Hunter–Saxton equation

This paper is devoted to the unique global admissible conservative solutions of the periodic dispersive Hunter–Saxton equation. Using the standard ordinary differential equation theory, we first get the global admissible conservative solutions of the periodic dispersive Hunter–Saxton equation. Then, given an admissible conservative solution u(t,x), an equation is introduced which singles out a unique characteristic curve through each initial point. Finally, by building an ordinary differential equation system, we get the uniqueness of the global admissible conservative solutions without any additional assumptions.

Global conservative solutions for a modified periodic coupled Camassa-Holm system

<p style='text-indent:20px;'>In present paper, we deal with the behavior of a solution beyond the occurrence of wave breaking for a modified periodic Coupled Camassa-Holm system. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, this evolution system is rewritten as a closed semilinear system. The local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows us to continue the solution after wave breaking, and gives a global conservative solution where the energy is conserved for almost all times. Returning to the original variables. We finally obtain a semigroup of global conservative solutions, which depend continuously on the initial data. Additionally, our results repair some gaps in the pervious work.

Global weak conservative solutions of the \u03bc-Camassa\u2013Holm equation

In this paper we consider global weak conservative solutions of a μ-Camassa–Holm equation. By employing a technique of change of variables on the Lagrangian variable, we successfully construct global weak conservative solutions of the μ-Camassa–Holm equation.

Uniqueness and stability of global conservative solutions for the modified coupled Camassa–Holm system

ABSTRACT In this paper, we investigate the generic regularity of global-in-time conservative solutions for the modified coupled Camassa–Holm (CH) system on real line. It has been shown that the modified coupled CH system possible development of singularities in finite time and beyond the occurrence of wave breaking the global conservative solution is existent. In this paper, we further prove that the Cauchy problem of the modified coupled CH system with the initial data has a unique global conservative weak solution. We also study the stability for the unique global conservative solutions. Therefore, we obtain the generic regularity of the conservative weak solutions.

Geometric approach on the global conservative solutions of the Camassa–Holm equation

In this paper we construct global weak conservative solutions of the Camassa–Holm (CH) equation on the periodic domain. We first express the equation in Lagrangian flow variable η and then transform it using the change of variables ρ=ηx. The new variable removes the singularity of the CH equation, and we obtain both global weak conservative solutions and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter–Saxton equation using the geometric interpretation.

The global conservative solutions for the generalized camassa-holm equation

&lt;abstract&gt;&lt;p&gt;This paper deals with the continuation of solutions to the generalized Camassa-Holm equation with higher-order nonlinearity beyond wave breaking. By introducing new variables, we transform the generalized Camassa-Holm equation to a semi-linear system and establish the global solutions to this semi-linear system, and by returning to the original variables, we obtain the existence of global conservative solutions to the original equation. We introduce a set of auxiliary variables tailored to a given conservative solution, which satisfy a suitable semi-linear system, and show that the solution for the semi-linear system is unique. Furthermore, it is obtained that the original equation has a unique global conservative solution. By Thom's transversality lemma, we prove that piecewise smooth solutions with only generic singularities are dense in the whole solution set, which means the generic regularity.&lt;/p&gt;&lt;/abstract&gt;

Existence and uniqueness of the global conservative weak solutions to the rotation-Camassa–Holm equation

We study herein the Camassa–Holm-type equation, which can be considered as a model in the shallow water for the long-crested waves propagating near the equator with effect of the Coriolis force due to the Earth's rotation. This quasi-linear equation is nonlocal with higher-order nonlinearities compared to the classical Camassa–Holm equation. We establish the global existence and uniqueness of the energy conservative weak solutions in the energy space H1 to this model equation.

Uniqueness of global conservative weak solutions for the modified two-component Camassa–Holm system

In this paper, we consider the uniqueness of global-in-time conservative weak solutions for the modified two-component Camassa–Holm system on real line. The strategy of proof is based on characteristics. Given a conservative weak solution, an equation is introduced to single out a unique characteristic curve through each initial point coordinate transformation into the Lagrangian coordinates. We prove that the Cauchy problem of the modified two-component Camassa–Holm system with initial data $$z_0=(u_0,\gamma _0)\in H^1(\mathbb {R})\times (H^1(\mathbb {R})\cap W^{1,\infty }(\mathbb {R}))$$ has a unique global conservative weak solution.

Lipschitz metric for the modified two-component Camassa–Holm system

This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa–Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well.