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1.
We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α<1/2) dissipation α(−Δ). This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical (α=1/2) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from L2 to L∞, from L∞ to Hölder (Cδ, δ>0), and from Hölder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be L∞, but it does not appear that their approach can be easily extended to establish the Hölder continuity of L∞ solutions. In order for their approach to work, we require the velocity to be in the Hölder space C1−2α. Higher regularity starting from Cδ with δ>1−2α can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. 相似文献
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Let P(D) be a nonnegative homogeneous elliptic operator of order 2m with real constant coefficients on Rn and V be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e−tH generated by H=P(D)+V with Kato type perturbing potential V , which naturally generalizes the classical result for Schrödinger semigroup e−t(Δ+V) as V∈K2(Rn), the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e−tP(D) on L1(Rn). As a consequence of the Gaussian upper bound, the Lp-spectral independence of H is concluded. 相似文献
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We exhibit balance conditions between a Young function A and a Young function B for a Korn type inequality to hold between the LB norm of the gradient of vector-valued functions and the LA norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in Lp, with 1<p<∞, and an Orlicz version involving a Young function A satisfying both the Δ2 and the ∇2 condition. 相似文献
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In this paper, we consider the problem (Pε) : Δ2u=un+4/n-4+εu,u>0 in Ω,u=Δu=0 on ∂Ω, where Ω is a bounded and smooth domain in Rn,n>8 and ε>0. We analyze the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev inequality as ε→0 and we prove existence of solutions to (Pε) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε small, (Pε) has at least as many solutions as the Ljusternik–Schnirelman category of Ω. 相似文献
5.
We study boundary value problems for semilinear elliptic equations of the form −Δu+g°u=μ in a smooth bounded domain Ω⊂RN. Let {μn} and {νn} be sequences of measure in Ω and ∂Ω respectively. Assume that there exists a solution un with data (μn,νn), i.e., un satisfies the equation with μ=μn and has boundary trace νn. Further assume that the sequences of measures converge in a weak sense to μ and ν respectively while {un} converges to u in L1(Ω). In general u is not a solution of the boundary value problem with data (μ,ν). However there exists a pair of measures (μ?,ν?) such that u is a solution of the boundary value problem with this data. The pair (μ?,ν?) is called the reduced limit of the sequence {(μn,νn)}. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence. A closely related problem was studied by Marcus and Ponce [3]. 相似文献
6.
In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)up-1, u>0, u∈H1(RN), p∈(2,2N/(N-2)) was proved under assumption b(x)?b∞?lim|x|→∞b(x). In this paper we prove the existence for certain functions b satisfying the reverse inequality b(x)<b∞. For any periodic lattice L in RN and for any b∈C(RN) satisfying b(x)<b∞, b∞>0, there is a finite set Y⊂L and a convex combination bY of b(·-y), y∈Y, such that the problem -Δu+u=bY(x)up-1 has a positive solution u∈H1(RN). 相似文献
7.
A net (xα) in a vector lattice X is said to be unbounded order convergent (or uo-convergent, for short) to x∈X if the net (|xα−x|∧y) converges to 0 in order for all y∈X+. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X be a Banach lattice. We prove that every norm bounded uo-convergent net in X? is w?-convergent iff X has order continuous norm, and that every w?-convergent net in X? is uo-convergent iff X is atomic with order continuous norm. We also characterize among σ -order complete Banach lattices the spaces in whose dual space every simultaneously uo- and w?-convergent sequence converges weakly/in norm. 相似文献
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We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α<1/2) dissipation α(−Δ): If a Leray–Hopf weak solution is Hölder continuous θ∈Cδ(R2) with δ>1−2α on the time interval [t0,t], then it is actually a classical solution on (t0,t]. 相似文献
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We study the existence of weak solutions to (E) (−Δ)αu+g(u)=ν in a bounded regular domain Ω in RN(N≥2) which vanish in RN?Ω, where (−Δ)α denotes the fractional Laplacian with α∈(0,1), ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|k−1r with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved. 相似文献
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Let K be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of L∞(K) and C0(K), the class of left translation invariant w?-subalgebras of L∞(K) and finally the class of non-zero left translation invariant C?-subalgebras of C0(K) in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w?-subalgebras of L∞(K), and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C?-subalgebras of C0(K). By the help of these two characterizations, we extract some results about invariant complemented subspaces of L∞(K) and C0(K). 相似文献
14.
A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G∈X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word wq is indeed concise in the class of residually finite groups. Further, we show that in the case where w=γk the word wq is boundedly concise in the class of residually finite groups. It remains unknown whether the word wq is actually concise in the class of all groups. 相似文献
15.
If U,V are closed subspaces of a Fréchet space, then E is the direct sum of U and V if and only if E′ is the algebraic direct sum of the annihilators U° and V°. We provide a simple proof of this (possibly well-known) result. 相似文献
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We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G . In particular, we show that the operators Tα:f?|⋅|−αL−α/2f, where |⋅| is a homogeneous norm, 0<α<Q/p, and L is the sub-Laplacian, are bounded on the Lebesgue space Lp(G). As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli–Weyl inequality, relating the Lp norm of a function f to the Lq norm of |⋅|βf and the Lr norm of Lδ/2f. 相似文献
18.
Dongwen Qi 《Expositiones Mathematicae》2009,27(1):87-91
A new proof is given for the statement: For an irreducible, infinite Coxeter group (W,S) and w∈W, if wSw-1=S, then w=1 (the identity element of W). 相似文献
19.
João Marcos do Ó Manassés de SouzaEveraldo de Medeiros Uberlandio Severo 《Journal of Differential Equations》2014
In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W1,n(Rn), n?2, into the Orlicz space LΦα determined by the Young function Φα(s) behaving like eα|s|n/(n−1)−1 as |s|→+∞. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rn. 相似文献