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.     

Global weak solution for a periodic generalized Hunter–Saxton equation

In this paper, we consider a periodic generalized Hunter–Saxton equation. We obtain the existence of global weak solutions to the equation. First, we give the well-posedness result of the viscous approximate equations and establish the basic energy estimates. Then, we show that the limit of the viscous approximation solutions is a global weak solution to the equation.     

Wave breaking and global existence for the generalized periodic two-component Hunter–Saxton system

In this paper, we study the wave-breaking phenomena and global existence for the generalized two-component Hunter–Saxton system in the periodic setting. We first establish local well-posedness for the generalized two-component Hunter–Saxton system. We obtain a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles. We also determine the exact blow-up rate of strong solutions. Finally, we give a sufficient condition for global solutions.     

A Lipschitz metric for the Hunter–Saxton equation

We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.     

Numerical conservative solutions of the Hunter–Saxton equation

BIT Numerical Mathematics - In the article a convergent numerical method for conservative solutions of the Hunter–Saxton equation is derived. The method is based on piecewise linear...     

Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa–Holm system

We study the Cauchy problem of a weakly dissipative 2-component Camassa–Holm system. We first establish local well-posedness for a weakly dissipative 2-component Camassa–Holm system. We then present a global existence result for strong solutions to the system. We finally obtain several blow-up results and the blow-up rate of strong solutions to the system.     

On initial boundary value problems for variants of the Hunter–Saxton equation

The Hunter?CSaxton equation serves as a mathematical model for orientation waves in a nematic liquid crystal. The present paper discusses a modified variant of this equation, coming up in the study of critical points for the speed of orientation waves, as well as a two-component extension. We establish well-posedness and blow-up results for some initial boundary value problems for the modified Hunter?CSaxton equation and the two-component Hunter?CSaxton system.     

Global well-posedness and blow-up for the 2-D Patlak–Keller–Segel equation

In this paper, we prove that for every nonnegative initial data in $L^1({\mathbb{R}}^2)$, the Patlak–Keller–Segel equation is globally well-posed if and only if the total mass $M\le 8\pi$. Our proof is based on some monotonicity formulas of nonnegative mild solutions.     

Algebro-geometric solutions for the Hunter–Saxton hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the Hunter–Saxton (HS) hierarchy through studying an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker–Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire HS hierarchy.     

Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H^2,2(?) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H^2,2(?) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.     

Global existence and blow-up for a degenerate reaction–diffusion system with nonlocal sources

This work investigates a degenerate reaction–diffusion system with homogeneous Dirichlet boundary data. Using the self-similar form of function and super-solution sub-solution techniques, together with results from interactions among the multi-nonlinearities in the system, described using ten exponents, the global existence and blow-up criteria for nonnegative solutions are determined.     

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

We are concerned with the following class of equations with exponential nonlinearities:

$$\begin{aligned} \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mathrm {in}~B_1\subset \mathbb {R}^2, \end{aligned}$$

which is related to the Tzitzéica equation. Here \(h_1, h_2\) are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser–Trudinger inequality related to this problem. Finally, we give a general existence result.

     

Global weak solutions for a periodic two-component?μ-Hunter–Saxton system

This paper is concerned with global existence of weak solutions for a periodic two-component?μ-Hunter–Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component?μ-Hunter–Saxton system.