Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary

We present some new regularity criteria for “suitable weak solutions” of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are Hölder continuous up to the boundary provided that the scaled mixed norm with 3/p+2/q?2, 2<q?∞, (p,q)≠(3/2,∞) is small near the boundary. Our methods yield new results in the interior case as well. Partial regularity of weak solutions is also analyzed under some additional integral conditions.     

Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type

We consider the quasi-linear Keller-Segel system of singular type, where the principal part Δum represents a fast diffusion like 0<m<1. We first construct a global weak solution with small initial data in the scaling invariant norm for all dimensions N?2 and all exponents q?2. As for the large initial data, we show that there exists a blow-up solution in the case of N=2. In the second part, the decay property in Lr with 1<r<∞ for with the mass conservation is shown. On the other hand, in the case of , the extinction phenomenon of solution is proved. It is clarified that the case of exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across . For the borderline case of , our solution decays in Lr exponentially as t→∞.     

Composition operators on Lizorkin-Triebel spaces

This paper is devoted to the study of the composition operator Tf(g):=f○g on Lizorkin-Triebel spaces . In case s>1+(1/p), 1<p<∞, and 1?q?∞ we will prove the following: the operator Tf takes to itself if and only if f(0)=0 and f belongs locally to .     

Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means

Let s∈R. In this paper, the authors first establish the maximal function characterizations of the Besov-type space with and τ∈[0,∞), the Triebel-Lizorkin-type space with p∈(0,∞), q∈(0,∞] and τ∈[0,∞), the Besov-Hausdorff space with p∈(1,∞), q∈[1,∞) and and the Triebel-Lizorkin-Hausdorff space with and , where t^′ denotes the conjugate index of t∈[1,∞]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0 and are also new even for Q spaces and Hardy-Hausdorff spaces.     

Lifting Galois representations of number fields

Let F be a number field. Given a continuous representation with insoluble image we show, under moderate assumptions at primes dividing ?∞, that for some continuous representation which is unramified outside finitely many primes. We also establish level lowering when F is totally real, is the reduction of a nearly ordinary Hilbert modular form and is distinguished at ?.     

Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition

We first represent the pressure in terms of the velocity in . Using this representation we prove that a solution to the Navier-Stokes equations is in under the critical assumption that , with r?3, while for r=3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L^∞ estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate for L^∞-norm of u.     

Large deviation principles for sequences of logarithmically weighted means

In this paper we consider several examples of sequences of partial sums of triangular arrays of random variables {Xn:n?1}; in each case Xn converges weakly to an infinitely divisible distribution (a Poisson distribution or a centered Normal distribution). For each sequence we prove large deviation results for the logarithmically weighted means with speed function . We also prove a sample path large deviation principle for {Xn:n?1} defined by , where σ²∈(0,∞) and {Un:n?1} is a sequence of independent standard Brownian motions.     

A model containing both the Camassa-Holm and Degasperis-Procesi equations

A nonlinear dispersive partial differential equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases, is investigated. Although the H¹-norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space Hs with is established under the assumptions u₀∈Hs and ‖u0xL^∞_‖<∞. The local well-posedness of solutions for the equation in the Sobolev space Hs(R) with is also developed.     

Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v₀(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.     

Global regularity results for the 2D Boussinesq equations with vertical dissipation

This paper furthers the study of Adhikari et al. (2010) [2] on the global regularity issue concerning the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. It is shown here that the vertical velocity v of any classical solution in the Lebesgue space Lq with 2?q<∞ is bounded by C₁q for C₁ independent of q. This bound significantly improves the previous exponential bound. In addition, we prove that, if v satisfies , then the associated solution of the 2D Boussinesq equations preserve its smoothness on [0,T]. In particular, implies global regularity.     

Asymptotic behavior of coupled dynamical systems with multiscale aspects

We study the asymptotic behavior, as time variable t goes to +∞, of nonautonomous dynamical systems involving multiscale features. As a benchmark case, given H a general Hilbert space, and two closed convex functions, and β a function of t which tends to +∞ as t goes to +∞, we consider the differential inclusion

     

A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space

This paper is devoted to solving globally the boundary value problem for the incompressible inhomogeneous Navier-Stokes equations in the half-space in the case of small data with critical regularity. In dimension n?3, we state that if the initial density ρ₀ is close to a positive constant in and the initial velocity u₀ is small with respect to the viscosity in the homogeneous Besov space then the equations have a unique global solution. The proof strongly relies on new maximal regularity estimates for the Stokes system in the half-space in , interesting for their own sake.     

Embedding of Lipschitz classes into classes of functions of Λ-bounded variation

In this note, sufficient or necessary conditions for embedding of Lipschitz classes , 1?p<∞ into classes ΛBV of functions of Λ-bounded variation are considered. Based on the technique of decomposition of functions, we obtain sufficient and necessary conditions for the embedding , 0<α,β?1.     

The global Cauchy problem for a vibrating beam equation

We consider the global Cauchy problem for an evolution equation which models an Euler-Bernoulli vibrating beam with time dependent elastic modulus under a force linear function of the displacement u, of the slope x_∂u, of and . These two last derivatives are proportional to the bending moment and to the shear respectively. We show results of well-posedness in Sobolev spaces assuming that the coefficient of the shear term has a decay rate |x|^−σ, σ?1, for the position x→±∞ and that all the coefficients of , 1?k?3, satisfy suitable Levi conditions since we allow the elastic modulus to vanish at some time t=t₀.     

Some estimates for Bochner-Riesz operators on the weighted Herz-type Hardy spaces

In this paper, by using the atomic decomposition and molecular characterization of the homogeneous and non-homogeneous weighted Herz-type Hardy spaces , we obtain some weighted boundedness properties of the Bochner-Riesz operator and the maximal Bochner-Riesz operator on these spaces for α=n(1/p−1/q), 0<p?1 and 1<q<∞.     

Blaschke products and the rank of backward shift invariant subspaces over the bidisk

Let H²(D²) be the Hardy space over the bidisk. For sequences of Blaschke products {φn(z):−∞<n<∞} and {ψn(w):−∞<n<∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H²(D²)?M for the pair of operators and .     

Steady compressible Navier-Stokes-Fourier system for monoatomic gas and its generalizations

We consider steady compressible Navier-Stokes-Fourier system for a gas with pressure p and internal energy e related by the constitutive law p=(γ−1)?e, γ>1. We show that for any there exists a variational entropy solution (i.e. solution satisfying the weak formulation of balance of mass and momentum, entropy inequality and global balance of total energy). This result includes the model for monoatomic gas (). If , these solutions also fulfill the weak formulation of the pointwise total energy balance.     

Regularity for a fourth-order critical equation with gradient nonlinearity

Given Ω a smooth bounded domain of Rn, n?3, we consider functions that are weak solutions to the equation

     

Kolmogorov and linear widths of weighted Besov classes

We study the Kolmogorov m-widths and the linear m-widths of the weighted Besov classes on [−1,1], where Lq,μ, 1?q?∞, denotes the Lq space on [−1,1] with respect to the measure , μ>0. Optimal asymptotic orders of and as m→∞ are obtained for all 1?p,τ?∞. It turns out that in many cases, the orders of are significantly smaller than the corresponding orders of the best m-term approximation by ultraspherical polynomials, which is somewhat surprising.