Related papers: Lipschitz metric for the two-component Camassa--Ho…

We consider the strictly hyperbolic Cauchy problem \begin{align*} &D_t^m u - \sum\limits_{j = 0}^{m-1} \sum\limits_{|\gamma|+j = m} a_{m-j,\,\gamma}(t,\,x) D_x^\gamma D_t^j u = 0, \newline &D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots,\,m,…

The paper aims at developing the Riemann-Hilbert problem approach to the modified Camassa-Holm (mCH) equation in the case when the solution is assumed to approach a non-zero constant at the both infinities of the space variable. In this…

In this paper, we investigate the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation $$m_{t}=b u_{x}+\frac{1}{2}\left[m\left(|u|^{2}-\left|u_{x}\right|^{2}\right)\right]_{x}-\frac{1}{2} m\left(u \bar{u}_{x}-u_{x}…

In this paper, we study the long time asymptotic behavior for the initial value problem of the modified Camassa-Holm (mCH) equation in the solitonic region \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}+\kappa u_{x}=0,…

In this paper we mainly investigate the periodic $\mu$-Camassa-Holm equation. We show the existence of global conservative solutions to the Cauchy problem of the periodic $\mu$-Camassa-Holm equation. The result is obtained by introducing a…

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.

We consider the initial-boundary value (IBV) problem for the modified Camassa--Holm (mCH) equation $ \tilde m_t+\left((\tilde u^2-\tilde u_x^2+2\tilde u)\tilde m\right)_x = 0$, $\tilde m:=\tilde u-\tilde u_{xx}+1$ on the half line $x \ge…

We consider the second order Cauchy problem $$u''+\m{u}Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is…

Let $(X,d)$ be a pathwise connected metric space equipped with an Ahlfors $Q$-regular measure $\mu$, $Q\in[1,\infty)$. Suppose that $(X,d,\mu)$ supports a 2-Poincar\'e inequality and a Sobolev-Poincar\'e type inequality for the…

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right.…

In this paper, we study the Cauchy problem for a two-component higher order Camassa-Holm systems with fractional inertia operator $A=(1-\partial_x^2)^r,r\geq1$, which was proposed by Escher and Lyons. By the transport equation theory and…

This work studies the Cauchy problem of a two-component higher order Camassa-Holm system, which is well-posed in Sobolev spaces $H^{s}(\mathbb{R})\times H^{s-2}(\mathbb{R})$, $s>\frac{7}{2}$ and its solution map is continuous. We show that…

We consider the Cauchy problem for the modified Camassa-Holm equation \[ u_t+\left((u^2-u_x^2)m\right)_x=0,\quad m\coloneqq u-u_{xx}, \quad t>0,\ \ -\infty<x<+\infty \] subject to the step-like initial data: $u(x,0)\to A_1$ as $x\to-\infty$…

We introduce a novel solution concept, denoted $\alpha$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa-Holm system on the…

Following conservative solutions of the two-component Camassa-Holm system $u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}+\rho\rho_x=0$, $\rho_t+(u\rho)_x=0$ along characteristics, we determine if wave breaking occurs in the nearby future or not,…

We investigate the long-time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with nonzero boundary conditions in different regions \begin{align*} &m_{t}+\left((u^2-u_x^2)m\right)_{x}=0,~~ m=u-u_{xx},…

In this paper, we address the existence of global solutions to the Cauchy problem of the modified Camassa-Holm (mCH) equation, which is known as a model for the unidirectional propagation of shallow water waves. Based on the spectral…

We consider a Cauchy problem for a (first-order) path-dependent Hamilton--Jacobi equation with coinvariant derivatives and a right-end boundary condition. Such problems arise naturally in the study of properties of the value functional in…

The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in…

We show that the Cauchy problem for the Camassa-Holm equation has a unique, global, weak, and dissipative solution for any initial data $u_0\in H^1(\mathbb{R})$, such that $u_{0,x}$ is bounded from above almost everywhere. In particular, we…