Global Conservative Solutions Research Articles

Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics

It was showed that the generalized Camassa-Holm equation possible development of singularities in finite time, and beyond the occurrence of wave breaking which exists either global conservative or dissipative solutions. In present paper, we will further investigate the uniqueness of global conservative solutions to it based on the characteristics. From a given conservative solution \begin{document}$u = u(t,x)$ \end{document} , an equation is introduced to single out a unique characteristic curve through each initial point. By analyzing the evolution of the quantities \begin{document}$u$ \end{document} and \begin{document}$v = 2 \arctan u_x$ \end{document} along each characteristic, it is obtained that the Cauchy problem with general initial data \begin{document}$u_0∈ H^1(\mathbb{R})$ \end{document} has a unique global conservative solution.

Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation

The integrable Novikov equation can be regarded as one of the Camassa-Holm-type equations with cubic nonlinearity. In this paper, we prove the global existence and uniqueness of the Holder continuous energy conservative solutions for the Cauchy problem of the Novikov equation.

An Isospectral Problem for Global Conservative Multi-Peakon Solutions of the Camassa–Holm Equation

We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the conservative Camassa-Holm equation is integrable by the inverse spectral transform in the multi-peakon case.

Global Conservative and Multipeakon Conservative Solutions for the Modified Camassa-Holm System with Coupling Effects

This paper investigates the continuation of solutions to the modified coupled two-component Camassa-Holm system after wave breaking. The underlying problem is rather challenging due to the mutual coupling effect between two components in the system. By introducing a novel transformation that makes use of a skillfully defined characteristic and a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which the global conservative solution is obtained, which further allows for the establishment of the multipeakon conservative solution of the system. The results obtained herein are deemed useful for understanding the inevitable phenomenon near wave breaking.

Global Conservative Solutions of a Generalized Two-Component Camassa-Holm System

The Cauchy problem for a generalized two-component Camassa-Holm system is investigated. Following the idea of fixed points and using new sets of independent and dependent variables, the existence of the global conservative solutions for the system is established.

Global conservative solutions for a model equation for shallow water waves of moderate amplitude

In this paper, we study the continuation of solutions to an equation for surface water waves of moderate amplitude in the shallow water regime beyond wave breaking (in [11], Constantin and Lannes proved that this equation accommodates wave breaking phenomena). Our approach is based on a method proposed by Bressan and Constantin [2]. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, the evolution problem is rewritten as a semilinear system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows one to continue the solution after collision time, giving a global conservative solution where the energy is conserved for almost all times. Finally, returning to the original variables, we obtain a semigroup of global conservative solutions, which depend continuously on the initial data.

Global conservative and multipeakon conservative solutions for the two-component Camassa-Holm system

The continuation of solutions for the two-component Camassa-Holm system after wave breaking is studied in this paper. The global conservative solution is derived first, from which a semigroup and a multipeakon conservative solution are established. In developing the solution, a system transformation based on a skillfully defined characteristic and a set of newly introduced variables is used. It is the transformation, together with the associated properties, that allows for the establishment of the results for continuity of the solution beyond collision time.

Global conservative and dissipative solutions of the generalized Camassa-Holm equation

This paper is devoted to the continuation of solutions to the generalized Camassa-Holm equationbeyond wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear system. This formulation allows one to continue the solution after collision time, giving either a global conservative solution where the energy is conserved for almost all times or a dissipative solution where energy may vanish from the system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global conservative or dissipative solutions, which depend continuously on the initial data.

Global energy conservative solutions to a system of variational wave equations

This paper is concerned with a system of variational wave equations which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. The global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy is established by using the method of energy-dependent coordinates and the Young measure theory.

Global conservative solutions to the Camassa--Holm equation for initial data with nonvanishing asymptotics

We study global conservative solutions of the Cauchy problem for the Camassa-Holm equation $u_t-u_{txx}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with nonvanishing and distinct spatial asymptotics.

Global Solutions for the Two-Component Camassa–Holm System

We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa–Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa–Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.

Global conservative solutions of a modified two-component Camassa–Holm shallow water system

We first prove the existence of global conservative solutions to the Cauchy problem for a modified two-component Camassa–Holm shallow water system. Then, we show that these global solutions, which depend continuously on the initial date, construct a semigroup.

Global conservative and dissipative solutions of a coupled Camassa-Holm equations

This paper develops a new approach in the analysis of a coupled Camassa-Holm equations. A continuous semigroup of global conservative solutions and a continuous semigroup of global dissipative solutions are obtained, respectively. The solutions are conservative, in the sense that the total energy equals to a constant, for almost every time. While the solutions are dissipative, in the sense that energy loss occurs through wave breaking. Compared to the approaches used by Bressan, A. and Constantin, A., “Global conservative solutions of the Camassa-Holm equation,” Arch. Ration. Mech. Anal. 183, 215–239 (2007)10.1007/s00205-006-0010-z and Bressan, A. and Constantin, A., “Global dissipative solutions of the Camassa–Holm equation,” Anal. Appl. (Singapore) 5, 1–27 (2007)10.1142/S0219530507000857 for the Camassa-Holm equation, two characteristics are introduced and a new set of independent and dependent variables are applied to the coupled Camassa-Holm equations.

Global periodic conservative solutions of a periodic modified two-component Camassa–Holm equation

In the paper, we first show the existence of global periodic conservative solutions to the Cauchy problem for a periodic modified two-component Camassa–Holm equation. Then we prove that these solutions, which depend continuously on the initial data, construct a semigroup.

Global Conservative Solutions of the Hyperelastic Rod Equation

Global Conservative Solutions of the Hyperelastic Rod Equation

Global conservative solutions of the Dullin-Gottwald-Holm equation

A new approach to the analysis of wave-breaking solutions to the Dullin-Gottwald-Holm equation is presented in this paper. Introduction of a set of variables allows for solving the singularities. A continuous semigroup of solutions is also built. The solutions have constant $H^{1}$-energy for almost every time with respect to the Lebesgue measure.

Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View

We show that the Camassa–Holm equation ut − uxxt + 3uux − 2uxuxx − uuxxx = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u|t=0 in H1. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure μ with . The total energy is preserved by the solution.

Global conservative solutions of the generalized hyperelastic-rod wave equation

We prove existence of global and conservative solutions of the Cauchy problem for the nonlinear partial differential equation u t − u x x t + f ( u ) x − f ( u ) x x x + ( g ( u ) + 1 2 f ″ ( u ) ( u x ) 2 ) x = 0 where f is strictly convex or concave and g is locally uniformly Lipschitz. This includes the Camassa–Holm equation ( f ( u ) = u 2 / 2 and g ( u ) = κ u + u 2 ) as well as the hyperelastic-rod wave equation ( f ( u ) = γ u 2 / 2 and g ( u ) = ( 3 − γ ) u 2 / 2 ) as special cases. It is shown that the problem is well-posed for initial data in H 1 ( R ) if one includes a Radon measure that corresponds to the energy of the system with the initial data. The solution is energy preserving. Stability is proved both with respect to initial data and the functions f and g. The proof uses an equivalent reformulation of the equation in terms of Lagrangian coordinates.

Global Conservative Solutions of the Camassa–Holm Equation

This paper develops a new approach in the analysis of the Camassa–Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on the initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time.

On the Structure of Solutions to the Periodic Hunter--Saxton Equation

We prove the local existence of strong solutions of the periodic Hunter--Saxton equation, and we show that all strong solutions except space-independent solutions blow up in finite time.