Riccati-type Differential Inequality Research Articles

Wave-breaking phenomena and persistence properties for a nonlinear dissipative Camassa–Holm equation

In present paper, we mainly consider the wave-breaking phenomena and persistence properties of a nonlinear dissipative Camassa–Holm equation. The precise blow-up criteria of the Cauchy problem are established and wave-breaking phenomenon is investigated based on the method of characteristics and the Riccati-type differential inequality. Due to the presence of high order nonlinear term, the conservation law is losed. This difficulty has been addressed by establishing the energy inequality. Finally, we establish the persistence properties and some unique continuation properties of the solutions in the weighted -spaces .

Curvature blow-up for the periodic CH-mCH-Novikov equation

We study the CH-mCH-Novikov equation with cubic nonlinearity, which is derived byan asymptotic method from the classical shallow water theory. This model can be related to three different important shallow water equations: CH equation, mCH equation and Novikov equation. We show the curvature blow-up of the CH-mCH-Novikov equation by the method of characteristics and conserved quantities to the Riccati-type differential inequality.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/103/abstr.html

A nonlocal shallow-water model with the weak Coriolis and equatorial undercurrent effects

We derive a quasilinear nonlocal shallow-water model equation for moderate-amplitude equatorial waves with effects of the weak Coriolis force and equatorial undercurrent. This mathematical modeling is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and rotational Euler equations in the equatorial region. Moreover, we investigate the influences and interactions of the weak Coriolis force caused by the Earth rotation, vorticity and nonlocal higher nonlinearities on the wave-breaking phenomena. In certain cases, by applying the method of characteristics and extremal property of the solution to the Riccati-type differential inequality, we demonstrate that wave breaking phenomena occur only depending on a shape of wave initially in the local spacial point.

A Nonlocal Shallow-Water Model Arising from the Full Water Waves with the Coriolis Effect

In the present study a mathematical model of long-crested water waves propagating mainly in one direction with the effect of Earth's rotation is derived by following the formal asymptotic procedures. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.

On the modeling of equatorial shallow-water waves with the Coriolis effect

In the present study a simplified phenomenological model of shallow-water wave propagating mainly in the equatorial ocean regions with the Coriolis effect caused by the Earth’s rotation is formally derived. The model equation which is analogous to the Green–Naghdi equations with the second-order approximation of the Camassa–Holm scaling captures stronger nonlinear effects than the classical dispersive integrable equations like the Korteweg–de Vries and two-component Camassa–Holm system. The local well-posedness of the Cauchy problem is then established by the linear transport theory and wave-breaking phenomenon is investigated based on the method of characteristics and the Riccati-type differential inequality. Finally, the condition of permanent waves is demonstrated by analyzing competition between the slope of average of horizontal velocity component and the free surface component.

Well-posedness and blow-up phenomena for an integrable three-component Camassa–Holm system

This paper studies the Cauchy problem for an integrable three-component Camassa–Holm system. We first establish the local well-posedness with initial condition in Besov spaces. Then we prove a blow-up criteria by arguing inductively with respect to the regularity index. Moreover, we derive a Riccati-type differential inequality by using the structure of equations, and also prove a new blow-up criteria with sufficient conditions on initial condition, whose proof is based on the conservative property of potential densities along the characteristic.

Blow-up phenomena and persistence property for the modified b-family of equations

In this paper, we study the blow-up mechanism and persistence property of solutions to the modified b-family of equations. The dynamics of the blow-up quantity along the characteristics is established by the Riccati-type differential inequality with various parameters. The key feature of the method is to refine the analysis on the growth rate of the relative ratio between solution and its gradient by performing a vertical shift. Furthermore, the persistence results for the solution are established in weighted spaces.

Analysis on the blow-up of solutions to a class of integrable peakon equations

We investigate the blow-up mechanism of solutions to a class of quasilinear integrable equations which could possess peakons. The dynamics of the blow-up quantity along the characteristics is established by the Riccati-type differential inequality which involves the interaction among three parts: a local nonlinearity, a nonlocal term, and a term stemming from the weak linear dispersion. To analyse the interplay among these quantities, we provide two different approaches. The first one is designed for the case when the equations do not exhibit a weak linear dispersion and hence focuses on the interplay between the first two parts. The method is based on a refined analysis on either evolution of the solution u and its gradient u x , that is, C u ± u x or the growth rate of the relative ratio u x / u . The second one handles the general situation when all of three parts are present. The idea is to extract the “truly” blow-up component from the Riccati-type differential inequality and utilizes the Morawetz-type identity or higher order conservation laws to show that such a component blows up in finite time before the other component degenerates.

On the rotation-two-component Camassa–Holm system modelling the equatorial water waves

In this paper, a modified two-component Camassa–Holm system with the effect of the Coriolis force in the rotating fluid is derived, which is a model in the equatorial water waves. The effects of the Coriolis force caused by the Earth's rotation and nonlocal nonlinearities on blow-up criteria and wave-breaking phenomena are then investigated. Our refined analysis relies on the method of characteristics and conserved quantities and is proceeded with the Riccati-type differential inequality. Finally, conditions which guarantee the permanent waves are obtained by using a method of the Lyapunov function.