STABILITY OF GLOBAL MAXWELLIAN FOR NON-LINEAR VLASOV-POISSON-FOKKER-PLANCK EQUATIONS

In this article, we establish the exponential time decay of smooth solutions around a global Maxwellian to the non-linear Vlasov–Poisson–Fokker–Planck equations in the whole space by uniform-in-time energy estimates. The non-linear coupling of macroscopic part and Fokker–Planck operator in the model brings new difficulties for the energy estimates, which is resolved by adding tailored weighted-in-v energy estimates suitable for the Fokker–Planck operator.     

Statistical Estimation of the Shannon Entropy

The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L~2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy–Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.     

Scattering Problem for Klein–Gordon Equation with Cubic Convolution Nonlinearity

The scattering problem for the Klein–Gordon equation with cubic convolution nonlinearity is considered. Based on the Strichartz estimates for the inhomogeneous Klein–Gordon equation, we prove the existence of the scattering operator, which improves the known results in some sense.     

Scattering Theory for the Defocusing Fourth Order NLS with Potentials

Based on the endpoint Strichartz estimates for the fourth order Schr?dinger equation with potentials for n ≥ 5 by [Feng, H., Soffer, A., Yao, X.: Decay estimates and Strichartz estimates of the fourth-order Schr?dinger operator. J. Funct. Anal., 274, 605–658(2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in [Pausader,B.: The cubic fourth-order Schr?dinger equation. J. Funct. Anal., 256, 2473–2517(2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity 1 +8/n p 1 +8/(n-4) in dimensions n ≥ 7.     

规范场论中自对偶磁单极子解的存在唯一性（英文）

Magnetic monopoles stand for the static solution arising from a(1 + 3)–dimensional theory describing the interaction between a real scalar triplet and non–Abelian gauge field. In this paper, we obtain a two–point boundary value problem of a first–order ordinary differential equations from the self–dual monopole model. Then we establish the existence and uniqueness theorem for the problem by using a dynamical shooting method, we also obtain sharp asymptotic estimates for the solutions at infinity...     

Stability of Navier–Stokes System with Singular External Force in Fourier–Herz Space

We prove the asymptotic properties of the solutions to the 3D Navier–Stokes system with singular external force, by making use of Fourier localization method, the Littlewood–Paley theory and some subtle estimates in Fourier–Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339(2022)]. We deal either with the nonstationary problem or with the stationary problem where solution may be singular due to singular external force. In this p...     

Perturbations of Moore–Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper,the perturbations of the Moore–Penrose metric generalized inverses of linear operators in Banach spaces are described.The Moore–Penrose metric generalized inverse is homogeneous and nonlinear in general,and the proofs of our results are different from linear generalized inverses.By using the quasi-additivity of Moore–Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition,we show some error estimates of perturbations for the singlevalued Moore–Penrose metric generalized inverses of bounded linear operators.Furthermore,by means of the continuity of the metric projection operator and the quasi-additivity of Moore–Penrose metric generalized inverse,an expression for Moore–Penrose metric generalized inverse is given.     

Functionals for multilinear fractional embedding

A novel representation is developed as a measure for multilinear fractional embedding.Corresponding extensions are given for the Bourgain–Brezis–Mironescu theorem and Pitt's inequality. New results are obtained for diagonal trace restriction on submanifolds as an application of the Hardy–Littlewood–Sobolev inequality. Smoothing estimates are used to provide new structural understanding for density functional theory, the Coulomb interaction energy and quantum mechanics of phase space. Intriguing connections are drawn that illustrate interplay among classical inequalities in Fourier analysis.     

Two Weighted Estimates for the Commutators of Multilinear Maximal Singular Integral Operators with Dini-type Kernels

The main purpose of this paper is to investigate the properties of the higher order commutators of maximal Calderón–Zygmund operators with Dini-type kernels. Two weighted estimates have been established for these commutators. Similar results have also been extended to multilinear setting.     

Monge–Ampère Type Equations on Almost Hermitian Manifolds

In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds. We derive C^∞ a priori estimates under the existence of an admissible C-subsolution. Finally,we obtain an existence result if there exists an admissible supersolution.     

Gradient estimates for parabolic systems from composite material

In this paper, we derive W~(1,∞) and piecewise C~(1,α) estimates for solutions, and their t-derivatives, of divergence form parabolic systems with coefficients piecewise H¨older continuous in space variables x and smooth in t. This is an extension to parabolic systems of results of Li and Nirenberg [Comm Pure Appl Math, 2003, 56:892–925] on elliptic systems. These estimates depend on the shape and the size of the surfaces of discontinuity of the coefficients, but are independent of the distance between these surfaces.     

Sharp Heat Kernel Estimates in the Fourier–Bessel Setting for a Continuous Range of the Type Parameter

The heat kernel in the setting of classical Fourier–Bessel expansions is defined by an oscillatory series which cannot be computed explicitly.We prove qualitatively sharp estimates of this kernel.Our method relies on establishing a connection with a situation of expansions based on Jacobi polynomials and then transferring known sharp bounds for the related Jacobi heat kernel.Keywords Fourier–Bessel expansions,heat kernel,Bessel process,transition density     

Multilinear singular integral operators with generalized kernels and their multilinear commutators

In this paper, the authors study a class of multilinear singular integral operators with generalized kernels and their multilinear commutators with BMO functions. By establishing the sharp maximal estimates, the boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimate of this class of mutilinear singular integral operators is also established. These results can improve the corresponding known results of classical multilinear Calderón–Zygmund operators and multilinear Calder′on–Zygmund operators with Dini type kernels.     

Global regularity to the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion in a bounded domain

This article considers the global regularity to the initial–boundary value problem for the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion.To overcome the difficulty caused by the vanishing viscosities,we first establish the elliptic system for uxand by,which are estimated by▽×u_x and▽×b_y,respectively.Then,we establish the global estimates for▽×u and▽×b.     

A positive solution for some critical p-Laplacian systems

This paper deals with the existence of a positive solution for two classes of critical quasilinear system. We prove these results by a variant of mountain pass lemma, combining two convergence theorems and two estimate results. Here we avoid the usual compactness arguments(e.g., Palais–Smale condition or Cerami condition) and reveal the potential of some energy level estimates for the existence of nontrivial solutions.     

Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data

In this paper,the global existence of the classical solution to the vacuum free boundary problem of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied.The solutions to the system(1.6)–(1.8) are in the class of radius-dependent solutions,i.e.,independent of the axial variable and the angular variable.In particular,the expanding rate of the moving boundary is obtained.The main difficulty of this problem lies in the strong coupling of the magnetic field,velocity,temperature and the degenerate density near the free boundary.We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates,and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variable r;by weighted estimates,and also the uniform-in-time weighted estimates of the higher-order derivatives of solutions by delicate analysis.     

Variational inequalities for the commutators of rough operators with BMO functions

Starting with the relatively simple observation that the variational estimates of the commutators of the standard Calderón-Zygmund operators with the bounded mean oscillation(BMO) functions can be obtained from their weighted variational estimates, we establish the similar variational estimates for the commutators of the BMO functions with rough singular integrals, which do not admit any weighted variational estimates. The proof involves several Littlewood-Paley-type inequalities with the commutators as well as Bony decomposition and related para-product estimates.     

Global gradient estimate on graph and its applications

Continuing our previous work(ar Xiv:1509.07981v1),we derive another global gradient estimate for positive functions,particularly for positive solutions to the heat equation on finite or locally finite graphs.In general,the gradient estimate in the present paper is independent of our previous one.As applications,it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs.These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al.[J.Differential Geom.,99,359–409(2015)].In many topics,such as eigenvalue estimate and heat kernel estimate(not including the Liouville type theorems),replacing the Li–Yau inequality by the global gradient estimate,we can get similar results.     

Estimates of the Isovariant Borsuk–Ulam Constants of Connected Compact Lie Groups

The isovariant Borsuk–Ulam constant c G of a compact Lie group G is defined to be the supremum of c ∈ R such that the inequality c(dim V-dim V~G) ≤ dim W-dim W~G holds whenever there exists a G-isovariant map f : S(V) → S(W) between G-representation spheres.In this paper,we shall discuss some properties of c G and provide lower estimates of c G of connected compact Lie groups,which leads us to some Borsuk–Ulam type results for isovariant maps.We also introduce and discuss the generalized isovariant Borsuk–Ulam constant G for more general smooth G-actions on spheres.The result is considerably different from the case of linear actions.     

Zero dissipation limit of the compressible heat-conducting navier-stokes equations in the presence of the shock

The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficient ε satisfy κ = O（ε）, κ/ε≥ c 〉 0, as ε→ 0 （see （1.3））, if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier–Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].