Sharp embeddings of Besov spaces involving only slowly varying smoothness

AS CRGRICES201/05/2033201/08/0383Grant Agency of the CzechInstitutional Research Plan no. AV0Z10190503 of theFCT05-1000008-815

Existence and equilibration of global weak solutions to Hookean-type bead-spring chain models for dilute polymers

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.Comment: 86 page

Hardy inequalities on metric measure spaces

16 page

Hardy's inequality for functions vanishing on a part of the boundary

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.Comment: 26 pages, 2 figures, includes several improvements in Sections 6-8 allowing to relax the assumptions in the main results. Final version published at http://link.springer.com/article/10.1007%2Fs11118-015-9463-

Sharp embeddings of Besov spaces with logarithmic smoothness

We prove sharp embeddings of Besov spaces B with the classical smoothness a and a logarithmic smoothness a into Lorentz-Zygmund spaces. Our results extend those with a = 0, which have been proved by D. E. Edmunds and H. Triebel. On page 88 of their paper (Math. Nachr. 207 (1999), 79-92) they have written: "Nevertheless a direct proof, avoiding the machinery of function spaces, would be. desirable." In our paper we give such a proof even in a more general context. We cover both the sub-limiting and the limiting cases and we determine growth envelopes of Besov spaces with logarithmic smoothness

A sharp rearrangement inequality for the fractional maximal operator

We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of ⨍, $M_{γ}⨍$, by an expression involving the nonincreasing rearrangement of ⨍. This estimate is used to obtain necessary and sufficient conditions for the boundedness of $M_γ$ between classical Lorentz spaces

Ulyanov-type inequalities between Lorentz-Zygmund spaces

We establish inequalities of Ulyanov-type for moduli of smoothness relating the source Lorentz-Zygmund space $\, L^{p,r}(\log L)^{\alpha -\gamma},\, \gamma >0,$ and the target space $\, L^{p^*,s}(\log L)^\alpha$ over $\, {\mathbb R}^n$ if $\, 1<p<p^*<\infty$ and over $\, \mathbb{T}^n$ if $\, 1<p \le p^*<\infty.$ The stronger logarithmic integrability (corresponding to $\, L^{p^*,s}(\log L)^\alpha$ ) is balanced by an additional logarithmic smoothness