Sobolev Interpolation Inequalities Research Articles

Global regularity for the micropolar Rayleigh-Bénard problem with only velocity dissipation

This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in$\mathbb {R}^{d}$with$d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in$\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in$\mathbb {R}^{3}$by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.

Gagliardo–Nirenberg and Sobolev interpolation inequalities on Besov spaces

We establish the Gagliardo–Nirenberg and Sobolev interpolation inequalities on Besov spaces. In particular, we improve the recent results of Miyazaki where he proved the inequalities on BMO spaces (see Yoichi Miyazaki [Proc. Amer. Math. Soc. 148 (2020), pp. 4257–4261]).

Absolute continuity and numerical approximation of stochastic Cahn–Hilliard equation with unbounded noise diffusion

In this article, we consider the absolute continuity and numerical approximation of the solution of the stochastic Cahn–Hilliard equation with unbounded noise diffusion. We first obtain the Hölder continuity and Malliavin differentiability of the solution of the stochastic Cahn–Hilliard equation by using the strong convergence of the spectral Gakerkin approximation. Then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic Cahn–Hilliard equation with sublinear growth diffusion coefficient, which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1/3<α<1. To approximate the density function of the exact solution, we propose a full discretization based on the spatial spectral Galerkin approximation and the temporal drift implicit Euler scheme. Furthermore, a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method. Consequently, we obtain the sharp mean square convergence rates in both time and space via Sobolev interpolation inequalities and semigroup theories. To the best of our knowledge, this is the first result on the convergence rate of full discretizations for the considered equation.

Global regularity for 2D fractional magneto-micropolar equations

The magneto-micropolar equations are important models in fluid mechanics and material sciences. This paper focuses on the global regularity problem on the 2D magneto-micropolar equations with fractional dissipation. We establish the global regularity for three important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ various technics including the regularization of generalized heat operators on the Fourier frequency localized functions, logarithmic Sobolev interpolation inequalities and the maximal regularity property of the heat operator.

The relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space

We are concerned with the Cauchy problem for the relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space. For perturbative initial data with suitable regularity and integrability, we obtain the global classical solutions and the large time decay rates. For the proof, a new structure of the linearized relativistic Fokker–Planck operator is observed so that the coercivity estimates are established, crucial for the optimal large time decay rates is the pivotal Sobolev inequalities and interpolation inequalities as well as the structure of the coupled system. Moreover, it is shown that the energy without any weights decays in time with the polynomial rates varying directly with the derivatives of the solution, and the lower order derivatives of the solution except the magnetic field are further proved to decay faster than the optimal ones which coincide with the linearized equations without electro and magnetic fields, such a new phenomenon occurs only in the case of the relativistic Fokker–Planck model.

Weak perturbations of the p-Laplacian

We consider the p-Laplacian in R^d perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p>d and p=d and discuss the connection with Sobolev interpolation inequalities.

Improved interpolation inequalities, relative entropy and fast diffusion equations

We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolevʼs inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.

Sobolev interpolation inequalities on generalized John domains

We obtain weighted Sobolev interpolation inequalities on generalized John domains that include John domains (bounded or unbounded) for δ-doubling measures satisfying a weighted Poincare inequality. These measures include ones arising from power weights d(x, ∂Ω) α and need not be doubling. As an application, we extend the Sobolev interpolation inequalities obtained by Caffarelli, Kohn and Nirenberg. We extend these inequalities to product spaces and give some applications on products Ω 1 × Ω 2 of John domains for A p (ℝ n x ℝ m ) weights and power weights of the type w(x, y) = dist(x, G 1 ) α dist(y, G 2 ) β , where G 1 C ∂Ω 1 and G 2 C ∂Ω 2 . For certain cases, we obtain sharp conditions.

SMOOTHNESS OF WEAK SOLUTIONS OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION

We prove that any weak solution (in the sense of Desvillettes and Villani [6]) to the Landau homogeneous equation is smooth and, in particular, that it lies in Schwartz space [Formula: see text]. Our result has to be compared to [6], where Desvillettes and Villani have shown existence of smooth weak solutions along with a uniqueness result, under further assumptions on initial data. Our method of proof uses the Littlewood–Paley decomposition of velocity space, and elementary Cauchy–Schwarz or Young's inequalities, thus avoiding any use of complex Sobolev interpolation inequalities.

Extension Theorems on Weighted Sobolev Spaces and Some Applications

AbstractWe extend the extension theorems to weighted Sobolev spaces on (ε, δ) domains with doubling weight w that satisfies a Poincaré inequality and such that w–1/p is locally Lp′. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

Sharp conditions for weighted Sobolev interpolation inequalities

Sharp conditions for weighted Sobolev interpolation inequalities

Weighted Inequalities on John Domains

We study two-weighted inequalities on John domains. We first introduce domains that generalize the C(λ,M) domains defined by HajLasz and Koskela (J. London Math. Soc., to appear) and then show that our domains are actually just John domains. We then extend an interesting and nice result of HajLasz and Koskela by modifying their proof and then obtain some interesting consequences that include weighted Sobolev interpolation inequalities and an almost necessary and sufficient condition for two-weighted Poincaré inequality on cubes.

Polynomials, Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

This paper consists of three main parts. One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups. Despite the extensive research after Jerison's work [3] on Poincare-type inequalities for Hormander's vector fields over the years, our results given here even in the nonweighted case appear to be new. Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE's involving vector fields. The main tools to prove such inqualities are approximating the Sobolev functions by polynomials associated with the left invariant vector fields on ?. Some very usefull properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Finding the existence of such polynomials is the second main part of this paper. Main results of these two parts have been announced in the author's paper in Mathematical Research Letters [38]. The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on (ɛ,δ) domains. Some results of weighted Sobolev spaces are also given here. We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously. In particular, we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions. Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.

Weighted Sobolev interpolation inequalities on product spaces

Weighted Sobolev interpolation inequalities on product spaces

Local and global interpolation inequalities on the Folland-Stein Sobolev spaces and polynomials on stratified groups

We derive both local and global Sobolev interpolation inequalities of anyhigher orders for the Folland-Stein Sobolev on stratified nilpotent Lie groups G and on domains satisfying a certain chain condition. Weighted versions of such inequalities are also included for doubling weights satisfying a weighted Poincare inequality. This paper appears to be the first one to deal with gen- eral Sobolev interpolation inequalities for vector fields on Lie groups; Despite the extensive research for Poincare type inequalities for vector fields over the years, interpolation inequalities given here even in the nonweighted case appear to be new. Such interpolation inequalities have important applications to subelliptic or parabolic pde's involving vector fields. The main tools to prove such inequalities are approximating the functions bypolynomials on G. Some veryuseful proper- ties for projections of polynomials associated with the functions are given here and theyappear to have independent interests in their own rights. Main ideas of the proofs of the theorems are explained here. Detailed proofs of the results pre- sented here are given in the paper Polynomials on stratified groups and Sobolev interpolation inequalities on nonisotropic Folland-Stein spaces bythe author.

Sharp Sobolev interpolation inequalities for the Stokes operator

We prove Sobolev interpolation inequalities for the Stokes operator, giving sharp $L^{\infty}$ estimates for solenoidal vector fields in $\mathbb{R}^2$ and $\mathbb{R}^3$. We use fundamental solutions of a generalized Stokes system in the proofs.

On Weighted Sobolev Interpolation Inequalities

We obtain some weighted Sobolev interpolation inequalities on ${\mathbb {R}^n}$ and domains satisfying the Boman chain condition for doubling weights satisfying a weighted Poincaré inequality.

On weighted Sobolev interpolation inequalities

We obtain some weighted Sobolev interpolation inequalities on R n {\mathbb {R}^n} and domains satisfying the Boman chain condition for doubling weights satisfying a weighted Poincaré inequality.

Sobolev interpolation inequalities with weights

We study weighted local Sobolev interpolation inequalities of the form \[ 1 w 2 ( B ) ∫ B | u ( x ) | p h w 2 ( x ) d x ≤ c ( 1 v ( B ) ∫ B | u ( x ) | p v ( x ) d x ) h − 1 × ( | B | p / n w 1 ( B ) ∫ B | ∇ u ( x ) | p w 1 ( x ) d x + 1 v ( B ) ∫ B | u ( x ) | p v ( x ) d x ) , \begin {gathered} \frac {1} {{{w_2}(B)}}{\int \limits _B {|u(x){|^{ph}}{w_2}(x)dx \leq c\left ( {\frac {1} {{v(B)}}\int \limits _B {|u(x){|^p}v(x)dx} } \right )} ^{h - 1}} \hfill \\ \times \left ( {\frac {{|B{|^{p/n}}}} {{{w_1}(B)}}\int \limits _B {|\nabla u(x){|^p}{w_1}(x)dx + \frac {1} {{v(B)}}} \int \limits _B {|u(x){|^p}v(x)dx} } \right ), \hfill \\ \end {gathered} \] , where 1 &gt; p &gt; ∞ , h &gt; 1 , B 1 &gt; p &gt; \infty ,h &gt; 1, B is a ball in R n {{\mathbf {R}}^n} , and v v , w 1 {w_1} , and w 2 {w_2} are weight functions. The case p = 2 p = 2 is of special importance in deriving regularity results for solutions of degenerate parabolic equations. We also study the analogous inequality without the second summand on the right in the case u u has compact support in B B , and we derive global Landau inequalities ‖ ∇ u ‖ L w q ≤ c ‖ ∇ u ‖ L v p 1 − a ‖ ∇ 2 u ‖ L v p a , 0 &gt; a &gt; 1 , 1 &gt; p ≤ q &gt; ∞ {\left \| {\nabla u} \right \|_{L_w^q}} \leq c\left \| {\nabla u} \right \|_{L_v^p}^{1 - a}\left \| {{\nabla ^2}u} \right \|_{L_v^p}^a,0 &gt; a &gt; 1,1 &gt; p \leq q &gt; \infty , when u u has compact support.

Sobolev Interpolation Inequalities with Weights

Sobolev Interpolation Inequalities with Weights