Journal of Differential Equations最新文献

The Ladyzhenskaya-Prodi-Serrin type $L^{s,r}$ criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. This global $L^{s,r}$ norm is usually large and hence hard to control. Replacing the global $L^{s,r}$ norm with some kind of local norm is interesting. In this article, we introduce a local $L^{s,r}$ space and establish some localized criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local $L^{s,r}$ type norms. These local norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local $L^{s,r}$ type norms is necessary and sufficient to affirmatively answer the millennium problem.

New results on the behaviour of the fast motion in slow-fast systems of ODEs with dependence on the fast time are given in terms of tracking of nonautonomous attractors. Under quite general assumptions, including the uniform ultimate boundedness of the solutions of the layer problems, inflated pullback attractors are considered. In general, one cannot disregard the inflated version of the pullback attractor, but it is possible under the continuity of the fiber projection map of the attractor. The problem of the limit of the solutions of the slow-fast system at each fixed positive value of the slow time is also treated and in this formulation the critical set is given by the union of the fibers of the pullback attractors. The results can be seen as extensions of the classical Tikhonov theorem to the nonautonomous setting.

In this paper, we study the Cauchy problem of the inhomogeneous Landau equation with hard potentials under the perturbation framework to global equilibrium. We prove that the solution to the Cauchy problem enjoys the analytic Gelfand-Shilov regularizing effect with a Sobolev initial datum for positive time.

This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path $(B^H,W)$ with $H\in (1/3,1/2)$ as $\epsilon \to 0$. Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the Itô map restricted on the Cameron-Martin space of the mixed fBm. To this end, we embed the Cameron-Martin space into a larger Hilbert space, then the Hessian is computable. Subsequently, the probability representation for the Hessian is shown. Finally, the Laplace approximation is constructed, which asserts the more precise asymptotics in the exponential scale.

In this paper, we present a methodology for establishing constructive proofs of existence of smooth, stationary, non-radial localized patterns in the planar Swift-Hohenberg equation. Specifically, given an approximate solution $u_0$, we construct an approximate inverse for the linearization around $u_0$, enabling the development of a Newton-Kantorovich approach. Consequently, we derive a sufficient condition for the existence of a unique localized pattern in the vicinity of $u_0$. The verification of this condition is facilitated through a combination of analytic techniques and rigorous numerical computations. Moreover, an additional condition is derived, establishing that the localized pattern serves as the limit of a family of periodic solutions (in space) as the period tends to infinity. The integration of analytical tools and meticulous numerical analysis ensures a comprehensive validation of this condition. To illustrate the efficacy of the proposed methodology, we present computer-assisted proofs for the existence of three distinct unbounded branches of periodic solutions in the planar Swift-Hohenberg equation, all converging towards a localized planar pattern, whose existence is also proven constructively. All computer-assisted proofs, including the requisite codes, are accessible on GitHub at [1].

This paper considers the Keller-Segel model coupled to stochastic Navier-Stokes equations (KS-SNS, for short), which describes the dynamics of oxygen and bacteria densities evolving within a stochastically forced 2D incompressible viscous flow. Our main goal is to investigate the existence and uniqueness of global solutions (strong in the probabilistic sense and weak in the PDE sense) to the KS-SNS system. A novel approximate KS-SNS system with proper regularization and cut-off operators in $H^s\left(R^2\right)$ is introduced, and the existence of approximate solution is proved by some a priori uniform bounds and a careful analysis on the approximation scheme. Under appropriate assumptions, two types of stochastic entropy-energy inequalities that seem to be new in their forms are derived, which together with the Prohorov theorem and Jakubowski-Skorokhod theorem enables us to show that the sequence of approximate solutions converges to a global martingale weak solution. In addition, when $\chi (\cdot )\equiv \text{const.}>0$, we prove that the solution is pathwise unique, and hence by the Yamada-Wantanabe theorem that the KS-SNS system admits a unique global pathwise weak solution.

We consider non-isentropic Euler-Maxwell equations with relaxation times (small physical parameters) arising in the models of magnetized plasma and semiconductors. For smooth periodic initial data sufficiently close to constant steady-states, we prove the uniformly global existence of smooth solutions with respect to the parameter, and the solutions converge global-in-time to the solutions of the energy-transport equations in a slow time scaling as the relaxation time goes to zero. We also establish error estimates between the smooth periodic solutions of the non-isentropic Euler-Maxwell equations and those of energy-transport equations. The proof is based on stream function techniques and the classical energy method but with some new developments.

We study the Liouville problem for the steady p-Stokes system in the half-space. We prove that a bounded weak solution of the p-Stokes system with $p>1$ vanishes in two dimensions. For the three dimensional case, the same result is concluded, provided that $p>\frac{5}{3}$.

In this paper, our objective is to investigate the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we first establish the tightness of the law of ${\left\{X^{\epsilon }\right\}}_{0<\epsilon ⩽1}$ in $C\left(\right[0,T];R^n)$. Subsequently, we demonstrate that any accumulation point of ${\left\{X^{\epsilon }\right\}}_{0<\epsilon ⩽1}$ can be regarded as a solution to the martingale problem or a weak solution of a distribution-dependent stochastic differential equation, which incorporates new drift and diffusion terms compared to the original equation. Our main contribution lies in employing two different methods to explicitly characterize the accumulation point. The diffusion matrices obtained through these two methods have different forms, however we assert their essential equivalence through a comparison.

In this paper, we study the spectrum $\sigma \left(L\right)$ of the Lamé operator$L=\frac{d^2}{d{x}^2}-12\wp (x+z_0;\tau )\text{in}{L}^2(R,C),$ where $\wp (z;\tau )$ is the Weierstrass elliptic function with periods 1 and τ, and $z_0\in C$ is chosen such that L has no singularities on $R$. We prove that a point $\lambda \in \sigma \left(L\right)$ is an intersection point of different spectral arcs but not a zero of the spectral polynomial if and only if λ is a zero of the following cubic polynomial:$\frac{4}{15}{\lambda }^3+\frac{8}{5}{\eta }_1{\lambda }^2-3g_2\lambda +9g_3-6{\eta }_1{g}_2=0.$ We also study the deformation of the spectrum as $\tau =\frac{1}{2}+ib$ with $b>0$ varying. We discover 10 different types of graphs for the spectrum as b varies around the double zeros of the spectral polynomial.