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1.
P. Maremonti 《Journal of Mathematical Sciences》2009,159(4):486-523
The Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations are
studied. The existence and uniqueness of classical solutions (u, π) (considered at least C
2 × C
1 smooth with respect to the space variable and C
1 × C
0 smooth with respect to the time variable) without requiring convergence at infinity are proved. A priori the fields u and π are nondecreasing at infinity. In the case of the Stokes problem, the existence, for any t > 0, and the uniqueness of solutions with kinetic field and pressure field are established for some β ∈ (0, 1) and γ ∈ (0, 1 − β). In the case of Navier–Stokes equations, the existence (local in time) and the uniqueness of classical solutions to the
Navier–Stokes equations are shown under the assumption that the initial data are only continuous and bounded, by proving that,
for any t ∈ (0, T), the kinetic field u(x, t) is bounded and, for any γ ∈ (0, 1), the pressure field π(x, t) is O(1 + |x|
γ
). Bibliography: 20 titles.
To V. A. Solonnikov on his 75th birthday
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 176–240. 相似文献
2.
姚奎 《高校应用数学学报(英文版)》2001,16(2):161-170
§ 1 PreliminariesWe considerψ( x)∈ L1 ( Rn) satisfying the mean valuezero,i.e.∫Rnψdx=0 ,and definethe square function g( f) on Rnbyg( f) ( x) =( k|ψk* f|2 ) 1 2 ( x)for f∈ S( Rn) ,the Schwartz space,whereψk( x) =ψ2 k( x) . Whenψ has some smooth property,one can obtain the weak type estimate by viewingthe square function g( f) as the vector-valued singularintegrals,which the readercan referto [1 ,2 ] .As for the results aboutthe Lp-estimates,see [3,4 ] .In this paper,we sha… 相似文献
3.
In this paper we establish results on the existence of nontangential limits for weighted
-harmonic functions in the weighted Sobolev space
, for some q>1 and w in the Muckenhoupt A
q
class, where
is the unit ball in
. These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted
-harmonic functions are weak solutions of the partial differential equation
where
for some fixed q∈(1,∞), where 0<α≤β<∞, and w(x) is a q-admissible weight as in Chap. 1 of Heinonen et al., Nonlinear Potential Theory, 2006.
Later, we apply these results to improve on results of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996 and Martio and Srebro, Math. Scand. 85, 49–70, 1999 on the existence of radial limits for bounded quasiregular mappings in the unit ball of
with some growth restriction on their multiplicity function.
相似文献
4.
Jiang Liya Chen Jiecheng 《高校应用数学学报(英文版)》2006,21(1):69-78
The boundedness on Triebel-Lizorkin spaces of oscillatory singular integral operator T in the form e^i|x|^aΩ(x)|x|^-n is studied,where a∈R,a≠0,1 and Ω∈L^1(S^n-1) is homogeneous of degree zero and satisfies certain cancellation condition. When kernel Ω(x' )∈Llog+L(S^n-1 ), the Fp^a,q(R^n) boundedness of the above operator is obtained. Meanwhile ,when Ω(x) satisfies L^1- Dini condition,the above operator T is bounded on F1^0,1 (R^n). 相似文献
5.
In this paper, we study a Green’s functions G
E
, G
S
for an elasto-static equations and Stokes equations in a three-dimensional bounded Lipschitz domain Ω. We prove that there
is a positive constant c > 0 depending on the Lipschitz constant such that for all . Furthermore, we show that there is a positive constant η ∈ (0,1) depending on the Lipschitz constant such that for all .
The second author is partially supported by Korea Research Foundation Grant KRF C-00005. 相似文献
6.
The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous
to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our
earlier work (Levine and Peres, Indiana Univ Math J 57(1):431–450, 2008). For the shape consisting of sites, where ω
d
is the volume of the unit ball in , we show that the inradius of the set of occupied sites is at least r − O(logr), while the outradius is at most r + O(r
α
) for any α > 1 − 1/d. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total
mass. For the classical abelian sandpile model in two dimensions, with n = πr
2 particles, we show that the inradius is at least , and the outradius is at most . This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and
Redig.
Yuval Peres is partially supported by NSF grant DMS-0605166. 相似文献
7.
GuoEnHU BoLinMA 《数学学报(英文版)》2004,20(5):785-792
The commutators of singular integral operators with homogeneous kernel Ω(x)/|x|^n are studied, where Ω is homogeneous of degree zero, and has mean value zero on the unit sphere. It is proved that Ω∈ L(logL)^(k 1)(S^(n-1)) is a sufficient condition such that the k-th order commutator is bounded on L^2(R^n). 相似文献
8.
In this paper, the boundedness of Toeplitz operator T
b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε
(ℝn) is discussed from L
p(ℝn) to L
q(ℝn),
, and from L
p(ℝn) to Triebel-Lizorkin space
. We also obtain the boundedness of generalized Toeplitz operator Θ
α0
b
from L
p(ℝn) to L
q(ℝn),
. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator
T
b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L
p(ℝn), 1 < p < ∞. 相似文献
9.
V. A. Kofanov 《Ukrainian Mathematical Journal》2008,60(10):1557-1573
We obtain a new sharp inequality for the local norms of functions x ∈ L
∞, ∞
r
(R), namely,
where φ
r
is the perfect Euler spline, on the segment [a, b] of monotonicity of x for q ≥ 1 and for arbitrary q > 0 in the case where r = 2 or r = 3.
As a corollary, we prove the well-known Ligun inequality for periodic functions x ∈ L
∞
r
, namely,
for q ∈ [0, 1) in the case where r = 2 or r = 3.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1338–1349, October, 2008. 相似文献
10.
Riccardo De Arcangelis 《Annali dell'Universita di Ferrara》1989,35(1):135-145
Summary Letf: (x, z)∈R
n×Rn→f(x, z)∈[0, +∞] be measurable inx and convex inz.
It is proved, by an example, that even iff verifies a condition as|z|
p≤f(x, z)≤Λ(a(x)+|z|q) with 1<p<q,a∈L
loc
s
(R
n),s>1, the functional
that isL
1(Ω)-lower semicontinuous onW
1,1(Ω), does not agree onW
1,1(Ω) with its relaxed functional in the topologyL
1(Ω) given by inf
Riassunto Siaf: (x, z)∈R n×Rn→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| p≤f(x, z)≤Λ(a(x)+|z|q) con 1<p<q,a∈L loc s (R n),s>1, il funzionale , che èL 1(Ω)-semicontinuo inferiormente suW 1,1(Ω), non coincide suW 1,1(Ω) con il suo funzionale rilassato nella topologiaL 1(Ω) definito da inf相似文献
11.
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p
1, p
2, and q satisfy 1 < p
2(x) < q(x) < p
1(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any is an eigenvalue, while any is not an eigenvalue of the above problem. 相似文献
12.
We consider the weighted Hardy integral operatorT:L
2(a, b) →L
2(a, b), −∞≤a<b≤∞, defined by
. In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa
n(T) ofT. In this paper, we show that under suitable conditions onu andv,
where ∥w∥p=(∫
a
b
|w(t)|p
dt)1/p.
Research supported by NSERC, grant A4021.
Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic. 相似文献
13.
The commutators of oscillatory singular integral operators with homogeneous kernel
are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphere. It is proved that Ω∈L (logL)K+1(Sn-1) is a sufficient condition under which the k-th order commutator is bounded on L2(Rn).
Supported by NSF of China (1997 1010, 1970 1039) 相似文献
14.
Using measure-capacity inequalities we study new functional inequalities, namely L
q
-Poincaré inequalities
and L
q
-logarithmic Sobolev inequalities
for any q ∈ (0, 1]. As a consequence, we establish the asymptotic behavior of the solutions to the so-called weighted porous media
equation
for m ≥ 1, in terms of L
2-norms and entropies.
相似文献
15.
Karl-Theodor Sturm 《Acta Mathematica》2006,196(1):65-131
We introduce and analyze lower (Ricci) curvature bounds
⩾ K for metric measure spaces
. Our definition is based on convexity properties of the relative entropy
regarded as a function on the L
2-Wasserstein space of probability measures on the metric space
. Among others, we show that
⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds,
⩾ K if and only if
⩾ K
for all
.
The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation.
We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence. 相似文献
16.
In this paper we consider
, in one case that fx
0 (t) is a ΛBMV function on [0, ∞], and in another case thatfεL
1
m-1(Rn) and
when |k|=m−1 and f(x)=0 when |x−x0|<δ for some δ>0. Our theormes improve the results of Pan Wenjie ([1]). 相似文献
17.
Roy Meshulam 《Order》2008,25(2):153-155
Let L be a finite lattice and let . It is shown that if the order complex satisfies then |L| ≥ 2
k
. Equality |L| = 2
k
holds iff L is isomorphic to the Boolean lattice {0,1}
k
.
Research supported by the Israel Science Foundation. 相似文献
18.
B. A. Khudaigulyev 《Ukrainian Mathematical Journal》2011,62(12):1989-1999
We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R
n
, n ≥ 3:
- Du = V(x)u, u| ?B = f(x), - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right. 相似文献
19.
Multilinear Singular Integrals with Rough Kernel 总被引:9,自引:0,他引:9
ShanZhenLU HuoXiongWU PuZHANG 《数学学报(英文版)》2003,19(1):51-62
For a class of multilinear singular integral operators T
A
,
20.
Feng-Yu Wang 《Potential Analysis》2008,28(4):321-334
For μ: = e
V(x)dx a probability measure on a complete connected Riemannian manifold, we establish a correspondence between the Entropy-Information
inequality and the transportation-cost inequality for μ(f
2) = 1, where Φ and Ψ are increasing functions. Moreover, under the curvature–dimension condition, a Sobolev type HWI (entropy-cost-information)
inequality is established. As applications, explicit estimates are obtained for the Sobolev constant and the diameter of a
compact manifold, which either extend or improve some corresponding known results.
Supported in part by NNSFC(10721091) and the 973-project in China. 相似文献
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