Conservative Weak Solutions Research Articles

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 uniqueness of the globally conservative solutions for a weakly dissipative Camassa–Holm equation in time weighted H1(R) space

In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa–Holm equation in time weighted H1 space. First, we derive an equivalent semi-linear system by introducing some new variables, and present the globally conservative solutions of this equation in time weighted H1 space. Second, given a globally conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables satisfy a particular semilinear system has a unique solution. In turn, we get the uniqueness of the conservative solutions in the original variables. Finally, we show that the peakon solutions are conservative weak solutions in H1.

The Existence and Uniqueness of Global Admissible Conservative Weak Solution for the Periodic Single-Cycle Pulse Equation

This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. 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 solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find a equation to single out a unique characteristic curve through each initial point and prove the uniqueness of global admissible conservative weak solution without any additional assumptions.

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.

Continuous dependence on data under the Lipschitz metric for the rotation-Camassa-Holm equation

In this article, we consider the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation. Based on defining a Finsler-type norm on the tangent space for solutions, we first establish the Lipschitz metric for smooth solutions, then by proving the generic regularity result, we extend this metric to general weak solutions.

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.

A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff

We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution \begin{document}$ (f_t)_{t\geq 0} $\end{document} , once the initial condition \begin{document}$ f_0 $\end{document} with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a \begin{document}$ (0,\infty)\times {\mathbb{R}}^3 $\end{document} random variable \begin{document}$ (M_t,V_t) $\end{document} such that \begin{document}$ \mathbb{E}[M_t {\bf 1}_{\{V_t \in \cdot\}}] = f_t $\end{document} . We also write down a series expansion of \begin{document}$ f_t $\end{document} . Although both the algorithm and the series expansion might be theoretically interesting in that they explicitly express \begin{document}$ f_t $\end{document} in terms of \begin{document}$ f_0 $\end{document} , we believe that the algorithm is not very efficient in practice and that the series expansion is rather intractable. This is a tedious extension to non-Maxwellian molecules of Wild's sum [ 18 ] and of its interpretation by McKean [ 10 , 11 ].

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.

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.

Energy conservative solutions to the system of full variational sine-Gordon equations in a unit sphere

We establish the global existence of a conservative weak solution to the Cauchy problem for a complete system of variational sine-Gordon equations, which models the motion of long waves on a neutral dipole chain in the continuum limit in a unit sphere. Although singularities may develop in finite time, the energy of the solution is conserved across singular times. We also obtain the continuous dependence of solutions on the given initial data.

Conservative weak solutions of the periodic Cauchy problem for μHS equation

We prove the existence and uniqueness of conservative weak solutions of the periodic Cauchy problem for an integrable evolution equation from mathematical physics. Our method is to first prove the existence and uniqueness of solutions for all time in Lagrangian coordinates and construct the maps to carry this result to the Eulerian coordinates.

Energy Conservative Solutions to a One‐Dimensional Full Variational Wave System

AbstractWe establish the existence of a conservative weak solution to the initial value problem for a complete system of variational wave equations modeling liquid crystals in one space dimension, in which the director has two degrees of freedom. The solutions exist globally in time and singularities may develop in finite time, but the energy of the solutions is conserved across singular times. The method for existence also yields continuous dependence of solutions on the initial data. © 2011 Wiley Periodicals, Inc.

Conservative Solutions to a Nonlinear Variational Wave Equation

We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation u tt − c(u)(c(u)u x ) x =0, for initial data of finite energy. Here c(·) is any smooth function with uniformly positive bounded values.

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

We study the nonlinear hyperbolic partial differential equation, (u t+uux)x=1/2u x 2 . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal. Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up. There are no a priori estimates on the second derivatives in any L p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation. We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.