Journal of Evolution Equations最新文献

Abstract

Noncommutative Euclidean spaces—otherwise known as Moyal spaces or quantum Euclidean spaces—are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the opportunity to highlight some of their peculiar features. For example, the theory of nonlinear partial differential equations has unexpected properties in this noncommutative setting. We develop elementary aspects of paradifferential calculus for noncommutative Euclidean spaces and give some applications to nonlinear evolution equations. We demonstrate how the analysis of some equations radically simplifies in the strictly noncommutative setting.

We consider the initial-boundary value problem for degenerate parabolic and pseudo-parabolic equations associated with Hörmander-type operator. Under the subcritical growth restrictions on the nonlinearity f(u), which are determined by the generalized Métivier index, we establish the global existence of solutions and the corresponding attractors. Finally, we show the upper semicontinuity of the attractors in the topology of (H_{X,0}^1(Omega )).

In this paper, we prove global-in-time (dot{textrm{H}}^{alpha ,q})-maximal regularity for a class of injective, but not invertible, sectorial operators on a UMD Banach space X, provided (qin (1,+infty )), (alpha in (-1+1/q,1/q)). We also prove the corresponding trace estimate, so that the solution to the canonical abstract Cauchy problem is continuous with values in a not necessarily complete trace space. In order to put our result in perspective, we also provide a short review on (textrm{L}^q)-maximal regularity which includes some recent advances such as the revisited homogeneous operator and interpolation theory by Danchin, Hieber, Mucha and Tolksdorf. This theory will be used to build the appropriate trace space, from which we want to choose the initial data, and the solution of our abstract Cauchy problem to fall in.

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).