Search

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces B ˙ p , 1 2 p (R 2) × B ˙ p , q 2 p − 1 (R 2). As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in B ˙ p , 1 2 p (R 2) × B ˙ p , 1 2 p − 1 (R 2) for p < 4 and ill-posed in B ˙ p , 1 2 p (R 2) × B ˙ p , 1 2 p − 1 (R 2) for p > 4. In this paper, we prove that this system is ill-posed for the critical case p = 4 in the sense of "norm inflation". Furthermore, we also show that the system is ill-posed in B ˙ 4 , 1 1 2 (R 2) × B ˙ 4 , q − 1 2 (R 2) for any q ≠ 2. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Cauchy problem for the Fornberg–Whitham equation is not locally well-posed in B p , r s (R) with (s , p , r) ∈ (1 , 1 + 1 p) × [ 2 , ∞) × [ 1 , ∞ ] or (s , p , r) ∈ { 1 } × [ 2 , ∞) × [ 1 , 2 ] by showing norm inflation phenomena of the solution for some special initial data. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem for a two-component Camassa-Holm system with high order nonlinearity, which is a multi-component extension of the Fokas-Olver-Rosenau-Qiao equation. Firstly, we state the local well-posedness for the Cauchy problem of the system in the framework of Sobolev-Besov spaces. Then, we establish the precise blow-up mechanism for the strong solutions by means of the transport equation theory. As is well-known, the H 1 -norm conservation law of the velocity component is crucial to study blow-up phenomena of the single Camassa-Holm type equation. However, it is no longer available for our nonlinear coupled system. To overcome this difficulty, we derive some suitable conservation laws by sufficiently exploiting the fine structure of the system. Based on which, we finally construct several new blow-up strong solutions with certain initial profiles in finite time. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the Cauchy problem of the cubic Camassa-Holm-type equations which arise as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves, which admit the single peaked solitons and multi-peakon solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model. Then, by overcoming the difficulties cased by the complicated mixed nonlinear structure, we present a precise blow-up scenario. Moreover, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions for Cauchy problem in the Gevrey-Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. Finally, we investigate the persistence properties of the solution to the Cauchy problem, we prove that the solution maintains the corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension d ≥ 2. The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contains a Navier-Stokes equation with an additional stress tensor which depend on the deformation gradient tensor and the memory function. The deformation gradient tensor satisfies a transport equation and the memory function satisfies a degenerate parabolic equation. We first proved the local well-posedness for the Doi-Edwards polymer model in Besov spaces by using the Littlewood-Paley theory. Moreover, if the initial velocity and the initial memory is small enough, we obtain a global existence result. [ABSTRACT FROM AUTHOR]

In this paper, we first study the local well-posedness for the Cauchy problem of the hyperbolic Keller-Segel equation in Besov spaces B p , r s (R d) with 1 ≤ p , r ≤ + ∞ and s > 1 + d p , i.e., the local existence, unique and continuous dependence on the initial data for the solution of this system are obtained, then we further show that this data-to-solution map is not uniformly continuous in these Besov spaces. [ABSTRACT FROM AUTHOR]

Let H = − Δ + | x | 2 be the harmonic oscillator on R n. In this paper, we prove estimates on Besov spaces associated to the operator H and the end-point maximal regularity estimates for the fractional harmonic oscillator H α , 0 < α ≤ 1 , on the Besov spaces associated to the harmonic oscillator H. These spaces are the appropriate function spaces for the study of estimates on Besov type spaces and the end-point maximal regularity estimates for the fractional power H α in the sense that similar estimates might fail with the classical Besov spaces. [ABSTRACT FROM AUTHOR]

The present paper is dedicated to the Cauchy problem of the two dimensional chemotaxis-shallow water system in the critical Besov spaces. We first obtain the global solutions to this system with small initial data and then investigate the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). Moreover, the bacterial density is allowed to vanish here. [ABSTRACT FROM AUTHOR]

Considered in this paper is the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa-Holm equation (also called Fokas-Olver-Rosenau-Qiao equation) and the Novikov equation as two special cases. The main investigation is the Cauchy problem of the new shallow-water model with qualitative properties of its solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model, and then derive a precise blow-up scenario and the blow-up mechanisms for solutions with certain initial profiles. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we investigate the persistence properties of the solution to the Cauchy problem of the new shallow-water model. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system which includes the Fokas-Olver-Rosenau-Qiao (FORQ) or modified Camassa-Holm (MCH) equation and the two-component MCH system as two special cases. The system is integrable in the sense of possessing a non-isospectral Lax pair with the spectrum depending on time t , and admits multi-peakon solutions in an explicit form. Furthermore, we establish the local well-posedness for the system in the Besov space B 2 , r s (T) with s > 3 / 2 , 1 ≤ r ≤ ∞ , where the key ingredients include the Friedrichs regularization method, the Littlewood-Paley decomposition theory, and the transport theory in Besov spaces. Then we derive a precise blow-up criteria, which is dependent of the parameters α (t) and γ (t). Moreover, by the intrinsic structure of the system, we obtain a new blow-up result for strong solutions with sufficient conditions on the initial data and parameters. The entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force. We first establish the local well-posedness in a range of the Besov spaces, as well as the local well-posedness in the critical space. Then, we study the Gevrey regularity of the shallow-water model by using a generalized Cauchy-Kovalevsky theorem, which implies that the shallow-water model admits analytical solutions locally in time and globally in space. Moreover, we obtain a precise lower bound of the lifespan and the continuity of the solution. Finally, working with moderate weight functions that are commonly used in time-frequency analysis, some persistence results to the shallow-water model are illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear evolution equation modelling the propagation of surface waves in the shallow water regime of large amplitude, which is characterised by some cubical nonlinearities. First, we establish the local well-posedness in Besov space B 2 , 1 3 / 2. Then, we give a blow-up criterion. Finally, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces H s , s > 3 / 2 for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in H r -topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]

In this paper, we prove the solution map of the Cauchy problem of Camassa–Holm type equations depends continuously on the initial data in nonhomogeneous Besov spaces in the sense of Hadamard by using the Littlewood–Paley theory and the method introduced by Kato [37] and Danchin [21] . [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of a three-component Camassa–Holm system. We first prove the local well-posedness of the system in Besov spaces B p , r s with p , r ∈ [ 1 , ∞ ] , s > max ⁡ { 1 p , 1 2 } by using the Littlewood–Paley theory and transport equations theory. Then, we establish two blow-up criteria which along with the conservation laws enable us to study global existence. Moreover, if the initial data satisfies some certain sign conditions, we obtain a global existence result. Finally, we verify that the system possesses peakon solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global well-posedness of the Cauchy problem of the 3D incompressible nematic liquid crystal system with initial data in the critical Besov space B2,11/2 (R³) × B2,13/2 (R³). In particular, we prove that there exists a positive constant C0 such that the nematic liquid crystal system has a unique global solution with initial data (u0, d0) = (u0h, u0³, d0) which satisfies (...} ≤ c0, for some c0 sufficiently small, where d0 is a constant vector with |d0| = 1. Here ν and μ are two positive viscosity constants. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the Cauchy problem for the generalized Camassa–Holm equation in Besov space. First, we prove that the solutions to the Cauchy problem for the generalized Camassa–Holm equation do not depend uniformly continuously on the initial data in with when . Second, combining the real interpolations among inhomogeneous Besov spaces with Lemma 5.2.1 of [6] which is called Osgood Lemma (a substitute for Gronwall inequality), we show that the Cauchy problem for the generalized Camassa–Holm equation is locally well-posed in . Finally, we give a blow-up criterion. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem of a model equation for shallow water waves of moderate amplitude, which was proposed by A. Constantin and D. Lannes [The hydrodynamical relevance of the Camassa–Holmand Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186]. First, the local well-posedness of the model equation is obtained in Besov spaces , , (which generalize the Sobolev spaces ) by using Littlewood–Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with , , ) is considered. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, persistence properties on strong solutions are also investigated. [Copyright &y& Elsevier]

Abstract: Considered in this paper is the modified Camassa–Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa–Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. [Copyright &y& Elsevier]

Abstract: In this paper we consider the Cauchy problem for the integrable Novikov equation. By using the Littlewood–Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for the integrable Novikov equation is locally well-posed in the Besov space with and . In particular, when with and , for all , we have that . We also prove that the local well-posedness of the Cauchy problem for the Novikov equation fails in . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem for a two-component Degasperis–Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong solutions to the system is derived. Finally, two new blow-up criterions and the exact blow-up rate of strong solutions to the system are presented. [Copyright &y& Elsevier]

Abstract: In this paper, we study a multidimensional bipolar hydrodynamic model for semiconductors or plasmas. This system takes the form of the bipolar Euler–Poisson model with electric field and frictional damping added to the momentum equations. In the framework of the Besov space theory, we establish the global existence of smooth solutions for Cauchy problems when the initial data are sufficiently close to the constant equilibrium. Next, based on the special structure of the nonlinear system, we also show the uniform estimate of solutions with respect to the relaxation time by the high- and low-frequency decomposition methods. Finally we discuss the relaxation-time limit by compact arguments. That is, it is shown that the scaled classical solution strongly converges towards that of the corresponding bipolar drift-diffusion model, as the relaxation time tends to zero. [Copyright &y& Elsevier]

Abstract: Recently, Novikov found a new integrable equation (we call it the Novikov equation in this paper), which has nonlinear terms that are cubic, rather than quadratic, and admits peaked soliton solutions (peakons). Firstly, we prove that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces (which generalize the Sobolev spaces ) with the critical index . Then, well-posedness in with , is also established by applying Kato''s semigroup theory. Finally, we present two results on the persistence properties of the strong solution for the Novikov equation. [Copyright &y& Elsevier]