The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits

We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie-Poisson spaces, and the underlying Banach space can be chosen to be even a Hilbert space. Smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established. Several pathological properties of the restricted algebra are pointed out.     

若干Banach空间及有界凸域上的凸映射

本文研究了一类Ｂａｎａｃｈ空间上凸映射的性质。找出了一类Ｂａｎａｃｈ空间上单位球上凸映射的特征，并利用这些结果研究了一类有界凸域上的所有凸映射．     

On the global existence and time decay estimates in critical spaces for the Navier–Stokes–Poisson system

We are concerned with the study of the Cauchy problem for the Navier–Stokes–Poisson system in the critical regularity framework. In the case of a repulsive potential, we first establish the unique global solvability in any dimension for small perturbations of a linearly stable constant state. Next, under a suitable additional condition involving only the low frequencies of the data and in the L²‐critical framework (for simplicity), we exhibit optimal decay estimates for the constructed global solutions, which are similar to those of the barotropic compressible Navier–Stokes system. Our results rely on new a priori estimates for the linearized Navier–Stokes–Poisson system about a stable constant equilibrium, and on a refined time‐weighted energy functional.     

Local existence and uniqueness of the mild solution to the 1D Vlasov–Poisson system with an initial condition of bounded variation

We propose a result of local existence and uniqueness of a mild solution to the one‐dimensional Vlasov–Poisson system. We establish the result for an initial condition lying in the space W^1,1(?²), then we extend it to initial conditions lying in the space BV(?²), without any assumption of continuity, boundedness or compact support. Copyright © 2010 John Wiley & Sons, Ltd.     

Growth estimates and uniform decay for the Vlasov–Poisson system

We consider smooth compactly supported solution to the classical Vlasov–Poisson system in three space dimensions in the electrostatic case. An estimate on velocities is derived, showing a growth rate at most like the power 1/8 of the time variable. As a consequence, a better decay estimate is obtained for the electric field in the norm. Copyright © 2017 John Wiley & Sons, Ltd.     

Local strong solution of Navier–Stokes–Poisson equations with degenerated viscosity coefficient

In this paper, we investigate the vacuum free boundary problem of a compressible Navier–Stokes–Poisson system with density‐dependent viscosity. By introducing Eulerian and Lagrange energy, we obtain a local in time well‐posedness of the strong solution to the Navier–Stokes–Poisson system in a spherically symmetric case. Copyright © 2015 John Wiley & Sons, Ltd.     

Diagonal sequence property in Banach spaces with weaker topologies

We investigate the diagonal sequence property in Banach spaces with weaker topologies. In particular, we present examples of Banach spaces with weaker locally convex topologies which have the diagonal sequence property but are not Fréchet–Urysohn. The examples answer negatively a question of Averbukh and Smolyanov. We give also a very simple proof of the fact that each Banach space contains a subset A whose weak closure includes 0, but 0 is not contained in the weak closure of any bounded subset of A.     

Stability of the superposition of rarefaction wave and contact discontinuity for the non‐isentropic Navier–Stokes–Poisson system

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of a rarefaction wave and a viscous contact discontinuity wave of the non‐isentropic Navier–Stokes–Poisson system with free boundary. We first construct the composite wave through the quasineutral Euler equations and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding initial‐boundary value problem of the non‐isentropic Navier–Stokes–Poisson system. Only the strength of the viscous contact wave is required to be small. However, the strength of the rarefaction wave can be arbitrarily large. In our analysis, the domain decomposition plays an important role in obtaining the zero‐order energy estimates. By introducing this technique, we successfully overcome the difficulty caused by the critical terms involved with the linear term, which does not satisfy the quasineural assumption for the composite wave. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal decay rate of the bipolar Euler–Poisson system with damping in dimension three

By rewriting a bipolar Euler–Poisson equations with damping into a Euler equation with damping coupled with a Euler–Poisson equation with damping and using a new spectral analysis, we obtain the optimal decay results of the solutions in L² norm. More precisely, the velocities u₁ and u₂ decay at the L²?rate , which is faster than the normal L²‐rate for the heat equation and the Navier–Stokes equations. In addition, the decay rates of the disparities of two densities ρ₁?ρ₂ and the disparity of two velocities u₁?u₂ could reach to and in L² norm, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

Blow‐up of smooth solutions to the Navier–Stokes–Poisson equations

The main purpose of this paper is concerned with blow‐up smooth solutions to Navier–Stokes–Poisson (N‐S‐P) equations. First, we present a sufficient condition on the blow up of smooth solutions to the N‐S‐P system. Then we construct a family of analytical solutions that blow up in finite time. Copyright © 2010 John Wiley & Sons, Ltd.     

Decomposition for Morrey type Besov–Triebel spaces

In this paper, the author establishes the decomposition of Morrey type Besov–Triebel spaces in terms of atoms and molecules concentrated on dyadic cubes, which have the same smoothness and cancellation properties as those of the classical Besov–Triebel spaces. The results extend those of M. Frazier, B. Jawerth for Besov–Triebel spaces and those of A. L. Mazzucato for Besov–Morrey spaces (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

On the Hyers–Ulam stabilities of functional equations on ‐Banach spaces

The purpose of this article is to generalize the theory of stability of functional equations to the case of n‐Banach spaces. In this article, we prove the generalized Hyers–Ulam stabilities of the Cauchy functional equations, Jensen functional equations and quadratic functional equations on n‐Banach spaces.     

Initial layers and zero‐relaxation limits of multidimensional Euler–Poisson equations

In this paper, we consider zero‐relaxation limits for periodic smooth solutions of the time‐dependent Euler–Poisson system. For well‐prepared initial data, we construct an approximate solution by an asymptotic expansion up to any order. For ill‐prepared initial data, we construct initial layer corrections in an explicit way. In both cases, the asymptotic expansions are valid in a time interval independent of the relaxation time, and their convergence is justified by establishing uniform energy estimates. Copyright © 2012 John Wiley & Sons, Ltd.     

Local existence for the one‐dimensional Vlasov–Poisson system with infinite mass

A collisionless plasma is modelled by the Vlasov–Poisson system in one dimension. We consider the situation in which mobile negative ions balance a fixed background of positive charge, which is independent of space and time, as ∣x∣ → ∞. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behaviour are shown to exist locally in time, and criteria for the continuation of these solutions are established. Copyright © 2006 John Wiley & Sons, Ltd.     

On the existence of solutions to the Navier–Stokes–Poisson equations of a two‐dimensional compressible flow

In this paper, we consider the Navier–Stokes–Poisson equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(?)=a?log^d (?) for large ?. Here d>1 and a>0. Copyright © 2006 John Wiley & Sons, Ltd.