In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equations with fractional dissipative terms (−∆)αu. It is proved that if div(u/|u|) ∈ Lp(0, T ;Lq(R3)) with 2α p + 3 q ≤ 2α− 3 2 , 6 4α− 3 < q ≤ ∞. then any smooth on GNS in [0, T ) remains smooth on [0, T ].

Consider a non-doubling manifold with ends M=Rn♯Rm where Rn=Rn×Sm−n for m>n≥3. We say that an operator L has generalised Poisson kernel if generates semigroup e−tL whose pt(x,y) upper bound similar to the of e−tΔ Δ is Laplace-Beltrami on M. An example operators Gaussian bounds Schrödinger L=Δ+V V arbitrary non-negative locally integrable potential. In this paper, our aim introduce BMO space BMO...

Abstract. Here we utilize operator–valued Lq → Lp Fourier multiplier theorems to establish lower bound estimates for large class of elliptic integrodifferential equations in R. Moreover, we investigate separability properties of parabolic convolution operator equations that arise in heat conduction problems in materials with fading memory. Finally, we give some remarks on optimal regularity of ...

In this paper we give a characterization for the uniform exponential stability of evolution families {Φ(t, t0)}t≥t0 on R+ that do not have an exponential growth, using the hypothesis that the pairs of function spaces (L1(X), L∞(X)) and (Lp(X), Lq(X)), (p, q) 6= (1,∞), are admissible to the evolution families.

An Lq(Lp)-theory of divergence and non-divergence form parabolic equations is presented. The main coefficients are supposed to belong to the class V MOx, which, in particular, contains all measurable functions depending only on t. The method of proving simplifies the methods previously used in the case p = q.

We extend a discrete version of an extension of Carleson’s theorem proved in [5] to a large class of trees T that have certain radial properties. We introduce the geometric notion of s-vanishing Carleson measure on such a tree T (with s ≥ 1) and give several characterizations of such measures. Given a measure σ on T and p ≥ 1, let Lp(σ) denote the space of functions g defined on T such that |g|...

In this paper, a good λ estimate for the multilinear commutator associated to the fractional integral operator on the spaces of homogeneous type is obtained. Under this result, we get the(Lp(X),Lq(X)) -boundedness of the multilinear commutator. Mathematics subject classification (2010): 42B20, 42B25.

Consider the partition function S μ( ) associated in the theory of Rényi dimension to a finite Borel measure μ on Euclidean d-space. This partition function S μ( ) is the sum of the q-th powers of the measure applied to a partition of d-space into d-cubes of width . We further Guérin’s investigation of the relation between this partition function and the Lebesgue Lp norm (Lq norm) of the convol...

We show that b ∈ BMO( n) if and only if the commutator [b, Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from generalized Morrey spaces Lp,φ( n) to Lq,φ q/p ( n), where φ is non-decreasing, and 1 < p < ∞, 0 < α < n and 1/q = 1/p− α/n.

In this note we consider the regularity of solutions to 3-D nematic liquid crystal flows, we prove that if either u ∈ Lq(0, T ;Lp(R3)), 2 q + 3 p ≤ 1, 3 < p ≤ ∞; or u ∈ Lα(0, T ;Lβ(R3)), 2 α + 3 β ≤ 2, 3 2 < β ≤ ∞, then the solution (u, d) is regular on (0, T ].