Traces of some weighted function spaces and related non‐standard real interpolation of Besov spaces

We study traces of weighted Triebel–Lizorkin spaces F p , q s ( R n , w ) $F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes R n − k $\mathbb {R}^{n-k}$ , where the weight is of Muckenhoupt type. We concentrate on the example weight w α ( x ) = | x n | α $w_\alpha (x) = {\big\vert x_n\big\vert }^\alpha$ when | x n | ≤ 1 $\big\vert x_n\big\vert \le 1$ , x ∈ R n $x\in \mathbb {R}^n$ , and w α ( x ) = 1 $w_\alpha (x)=1$ otherwise, where α > − 1 $\alpha >-1$ . Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space ( B p 1 , p 1 s 1 ( R n − k ) , B p 2 , p 2 s 2 ( R n − k ) ) θ , r $\big (B^{s_1}_{p_1,p_1}\big (\mathbb {R}^{n-k}\big ), B^{s_2}_{p_2,p_2}{\big (\mathbb {R}^{n-k}\big )\big )}_{\theta ,r}$ , 0 0$sufficiently large, 0 < θ < 1$0<\theta <1$, 0 < r ≤ ∞$

Dynamic Transitions for Quasilinear Systems and Cahn-Hilliard equation with Onsager mobility

The main objectives of this article are two-fold. First, we study the effect of the nonlinear Onsager mobility on the phase transition and on the well-posedness of the Cahn-Hilliard equation modeling a binary system. It is shown in particular that the dynamic transition is essentially independent of the nonlinearity of the Onsager mobility. However, the nonlinearity of the mobility does cause substantial technical difficulty for the well-posedness and for carrying out the dynamic transition analysis. For this reason, as a second objective, we introduce a systematic approach to deal with phase transition problems modeled by quasilinear partial differential equation, following the ideas of the dynamic transition theory developed recently by Ma and Wang

On some iterated weighted spaces

[EN] It is proved that Hormander Bp,kloc (Omega 1 x Omega 2) and B-p,k1(loc) (Omega 1, B-p,k2(loc) (Omega(2))) spaces (Omega(1) subset of R-n, Omega(2) subset of R-m open sets, 1 <= p < infinity, k(i) Beurling-Bjorck weights, k = k(1) circle times k(2)) are isomorphic whereas the iterated spaces B-p,k1(loc) (Omega 1, B-p,k2(loc) (Omega(2))) and B-p,k2(loc) (Omega 1, B-p,k1(loc) (Omega(1))) are not if 1 < p not equal q < infinity. A similar result for weighted L-p-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: B-2,q(s) (R-n, B-2,q(s) (R-m)) and B-2,q(s)(Rn+m) are not isomorphic when 1 < q not equal 2 < infinity. (c) 2007 Elsevier Inc. All rights reserved.The author is partially supported by DGES, Spain, Project MTM2005-08350.Motos Izquierdo, J.; Planells Gilabert, MJ.; Talavera Usano, CF. (2008). On some iterated weighted spaces. Journal of Mathematical Analysis and Applications. 338(1):162-174. https://doi.org/10.1016/j.jmaa.2007.05.009S162174338

Regularity properties of distributions through sequences of functions

We give necessary and sufficient criteria for a distribution to be smooth or uniformly H\"{o}lder continuous in terms of approximation sequences by smooth functions; in particular, in terms of those arising as regularizations $(T\ast\phi_{n})$.Comment: 10 page

Orientability and energy minimization in liquid crystal models

Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory through a unit vector field $n$. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which $n$ should be equivalent to -$n$. This symmetry is preserved in the constrained Landau-de Gennes theory that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class $W^{1,2}$ the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains we completely characterise the instances in which the predictions of the constrained Landau-de Gennes theory differ from those of the Oseen-Frank theory

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order $s\geq0$ on the group \Diff_c(M) of compactly supported diffeomorphisms of a manifold $M$. We show that for the important special case $M=S^1$ the geodesic distance on \Diff_c(S^1) vanishes if and only if $s\leq\frac12$. For other manifolds we obtain a partial characterization: the geodesic distance on \Diff_c(M) vanishes for $M=\R\times N, s<\frac12$ and for $M=S^1\times N, s\leq\frac12$, with $N$ being a compact Riemannian manifold. On the other hand the geodesic distance on \Diff_c(M) is positive for $\dim(M)=1, s>\frac12$ and $\dim(M)\geq2, s\geq1$. For $M=\R^n$ we discuss the geodesic equations for these metrics. For $n=1$ we obtain some well known PDEs of hydrodynamics: Burgers' equation for $s=0$, the modified Constantin-Lax-Majda equation for $s=\frac 12$ and the Camassa-Holm equation for $s=1$.Comment: 16 pages. Final versio

Thermodynamical Consistent Modeling and Analysis of Nematic Liquid Crystal Flows

The general Ericksen-Leslie system for the flow of nematic liquid crystals is reconsidered in the non-isothermal case aiming for thermodynamically consistent models. The non-isothermal model is then investigated analytically. A fairly complete dynamic theory is developed by analyzing these systems as quasilinear parabolic evolution equations in an $L^p-L^q$-setting. First, the existence of a unique, local strong solution is proved. It is then shown that this solution extends to a global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the natural norm of the underlying state space. In these cases, the solution converges exponentially to an equilibrium in the natural state manifold

Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off

Accepted versio

On the Usefulness of Modulation Spaces in Deformation Quantization

We discuss the relevance to deformation quantization of Feichtinger's modulation spaces, especially of the weighted Sjoestrand classes. These function spaces are good classes of symbols of pseudo-differential operators (observables). They have a widespread use in time-frequency analysis and related topics, but are not very well-known in physics. It turns out that they are particularly well adapted to the study of the Moyal star-product and of the star-exponential.Comment: Submitte