Journal of Evolution Equations最新文献

Abstract

We consider a two dimensional electroconvection model which consists of a nonlinear and nonlocal system coupling the evolutions of a charge distribution and a fluid. We show that the solutions decay in time in (L^2({{mathbb {R}}}^2)) at the same sharp rate as the linear uncoupled system. This is achieved by proving that the difference between the nonlinear and linear evolution decays at a faster rate than the linear evolution. In order to prove the sharp (L^2) decay we establish bounds for decay in (H^2({{mathbb {R}}}^2)) and a logarithmic growth in time of a quadratic moment of the charge density.

In this paper, we investigate the existence of a weak solution and blow-up of strong solutions to a two-component Fornberg–Whitham system. Due to the absence of some useful conservation laws, we establish the existence of a weak solution to the system in lower order Sobolev spaces (H^{s}times H^{s-1}) ((sin (1,3/2])) via a modified pseudo-parabolic regularization method. And then, a blow-up scenario for strong solutions to this system is shown. By the analysis of Riccati-type inequalities recently, we present some sufficient conditions on the initial data that lead to the blow-up for corresponding strong solutions to the system.

In the present paper, we address the asymptotic behavior of a fish population system structured in age and weight, while also incorporating spatial effects. Initially, we develop an abstract perturbation result concerning the essential spectral radius, employing the regular systems approach. Following that, we present the model in the form of a perturbed boundary problem, which involves unbounded operators on the boundary. Using time-invariant regular techniques, we construct the corresponding semigroup solution. Then, we designate an operator characteristic equation of the primary system via the radius of a bounded linear operator defined on the boundary space. Moreover, we provide a characterization of the uniform exponential stability and the asynchronous exponential growth property (AEG) by localizing the essential radius and proving the irreducibility of the perturbed semigroup. Finally, we precise the projection that emerged from the (AEG) property; this depends on the developed characteristic equation.

We introduce a system of equations that models a non-isothermal magnetoviscoelastic fluid. We show that the model is thermodynamically consistent, and that the critical points of the entropy functional with prescribed energy correspond exactly with the equilibria of the system. The system is investigated in the framework of quasilinear parabolic systems and shown to be locally well-posed in an (L_p)-setting. Furthermore, we prove that constant equilibria are normally stable. In particular, we show that solutions that start close to a constant equilibrium exist globally and converge exponentially fast to a (possibly different) constant equilibrium. Finally, we establish that the negative entropy serves as a strict Lyapunov functional and we then show that every solution that is eventually bounded in the topology of the natural state space exists globally and converges to the set of equilibria.

We show that the locations where finite- and infinite-time clustering occur for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time singularity set and asymptotic clusters associated with a weak solution. In many cases, the eventual size of the cluster can be read off directly from the flux associated with a scalar balance law formulation of the system.

This paper develops an effective approach to establishing the optimal decay estimates on solutions of the 3D anisotropic magnetohydrodynamic (MHD) equations with only horizontal dissipation. As our first step, we prove the global existence and stability of solutions to the aforementioned MHD system emanating from any initial data with small (H^1)-norm. Due to the lack of dissipation in the vertical direction, the large-time behavior does not follow from the classical approaches. The analysis of the nonlinear terms are much more difficult than in the case of full dissipation. In particular, we need to represent the MHD equations in an integral form, exploit cancellations and other properties such as the incompressibility in order to control terms involving vertical derivatives.

We studied the pointwise space-time behavior of the classical solution to the Cauchy problem of two-phase fluid model derived by Choi (SIAM J Math Anal 48:3090–3122, 2016) when the initial data is sufficiently small and regular. This model is the compressible damped Euler system coupled with the compressible Naiver–Stokes system via a drag force. As we know, Liu and Wang (Commun Math Phys 196:145–173, 1998) verified that the solution of the compressible Naiver–Stokes system obeys the generalized Huygens’ principle, while Wang and Yang (J Differ Equ 173:410–450, 2001) verified the solution of the compressible Euler system does not obey the generalized Huygens’ principle due to the damped mechanism. In this paper, we proved that both of two densities and two momentums for the two-phase fluid model obey the generalized Huygens’ principle as that in Liu and Wang (Commun Math Phys 196:145–173, 1998). The main contribution is to overcome the difficulty of the non-conservation arising from the damped mechanism of the system. As a byproduct, we also extended (L^2)-estimate in Wu et al. (SIAM J Math Anal 52(6):5748–5774, 2020) to (L^p)-estimate with (p>1).

We consider persistence properties of solutions for a generalised wave equation including vibration in elastic rods and shallow water models, such as the BBM, the Dai’s, the Camassa–Holm, and the Dullin–Gottwald–Holm equations, as well as some recent shallow water equations with Coriolis effect. We establish unique continuation results and exhibit asymptotic profiles for the solutions of the general class considered. From these results we prove the non-existence of non-trivial spatially compactly supported solutions for the equation. As an aftermath, we study the equations earlier mentioned in light of our results for the general class.

The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital (nonlinear) instability in appropriate periodic Sobolev spaces. The analysis is based on the well-posedness theory, the smoothness of the data-solution map, and an abstract result of instability of equilibria under nonlinear iterations. The resulting instability criterion is applied to two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a local Hopf bifurcation around a critical value of the velocity. The second family comprises arbitrarily large period waves which arise from a homoclinic (global) bifurcation and tend to a limiting traveling pulse when their fundamental period tends to infinity. In the case of both families, the criterion is applied to conclude their orbital instability under the flow of the nonlinear viscous balance law in periodic Sobolev spaces with same period as the fundamental period of the wave.

We consider the controllability of a class of systems of n Stokes equations, coupled through terms of order zero and controlled by m distributed controls. Our main result states that such a system is null-controllable if and only if a Kalman type condition is satisfied. This generalizes the case of finite-dimensional systems and the case of systems of coupled linear heat equations. The proof of the main result relies on the use of the Kalman operator introduced in [1] and on a Carleman estimate for a cascade type system of Stokes equations. Using a fixed-point argument, we also obtain that if the Kalman condition is verified, then the corresponding system of Navier–Stokes equations is locally null-controllable.