共查询到20条相似文献,搜索用时 328 毫秒
1.
Luan Thach Hoang 《Journal of Differential Equations》2008,245(9):2585-2594
We show that ‖Au+Δu‖L2(Ωε)?C(ε‖∇u‖L2(Ωε)+‖u‖L2(Ωε)), where Ωε is a thin domain in R3 of depth ε, the vector field u belongs to the domain of A, which is the Stokes operator for divergence-free vector fields on Ωε satisfying the Navier boundary condition. 相似文献
2.
ABSTRACT The variational problem in L ∞ considered is to minimize F(u) = ‖Du‖ L ∞(Ω) subject to ∈ t Ω |Du|2 dx ≤ E for given E > 0. It is proven that a constrained minimizer exists and satisfies an Aronsson-Euler equation in the viscosity sense which depends on a parameter Λ∞ ≥ 0. This parameter splits Ω into two parts. In one part the minimizer satisfies the infinity laplace equation and in the remaining part the minimizer is the solution of the elasto-plastic torsion problem with constraint ‖Du‖ L ∞ ≤ Λ∞. 相似文献
3.
Justin Holmer 《偏微分方程通讯》2013,38(5):878-905
We consider solutions u(t) to the 3d NLS equation i? t u + Δu + |u|2 u = 0 such that ‖xu(t)‖ L 2 = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to ?Q + ΔQ + |Q|2 Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u 0‖ L 2 ‖?u 0‖ L 2 > ‖Q‖ L 2 ‖?Q‖ L 2 , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t n → + ∞ such that ‖?u(t n )‖ L 2 → ∞. Similar statements hold for negative time. 相似文献
4.
Chen Zhimin 《Arkiv f?r Matematik》1990,28(1-2):371-381
A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rn is obtained under the the assumption thatf is C1+r forr>2/n and ‖u 0‖C2(R n ) +‖u 0‖W 1 2 (R n ) is small. This implies that the assumption thatf is smooth and ‖u 0 ‖W 1 k (R n )+‖u 0‖W 2 k (R n ) is small fork large enough, made in earlier work, is unnecessary. 相似文献
5.
We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u0 on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u0∈L∞(Ω), f∈L∞(Q), a∈L∞((0,T)×∂Ω). In the L1-setting, a corresponding result is proved for the more general notion of renormalised entropy solution. 相似文献
6.
Weilin Zou 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(9):3069-3082
This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L1(Ω) and f∈(Lp′(Ω))N. 相似文献
7.
Ana Portilla Yamilet Quintana Eva Tourís 《Journal of Mathematical Analysis and Applications》2007,334(2):1167-1198
We characterize the set of functions which can be approximated by continuous functions with the norm ‖f‖L∞(w) for every weight w. This fact allows to determine the closure of the space of polynomials in L∞(w) for every weight w with compact support. We characterize as well the set of functions which can be approximated by smooth functions with the norm
‖f‖W1,∞(w0,w1):=‖f‖L∞(w0)+‖f′‖L∞(w1), 相似文献
8.
Tsing-San Hsu 《Journal of Mathematical Analysis and Applications》2007,332(2):814-832
In this paper, assume that h is nonnegative and ‖hL2‖>0, we prove that if ‖hL2‖ is sufficiently small, then there are at least three positive solutions of Eq. (1) in an exterior cylinder domain. 相似文献
9.
For a strong solution u(x,t) of the Navier-Stokes equations in exterior domain Ω in Rn where n=2,3, we study the time decay of ‖α|x|u(t)Lp‖ for α<n. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in [H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier-Stokes solutions in exterior domains, Bull. Korean Math. Soc. 44 (3) (2007) 547-567; H.-O. Bae, B.J. Jin, Asymptotic behavior for the Navier-Stokes solutions in 2D exterior domains, J. Funct. Anal. 240 (2006) 508-529] and we will extend Bae and Jin's results by modifying their methods. 相似文献
10.
The Cheeger problem for a bounded domain Ω⊂RN, N>1 consists in minimizing the quotients |∂E|/|E| among all smooth subdomains E⊂Ω and the Cheeger constant h(Ω) is the minimum of these quotients. Let be the p-torsion function, that is, the solution of torsional creep problem −Δp?p=1 in Ω, ?p=0 on ∂Ω, where Δpu:=div(|∇u|p−2∇u) is the p-Laplacian operator, p>1. The paper emphasizes the connection between these problems. We prove that . Moreover, we deduce the relation limp→1+‖?p‖L1(Ω)?CNlimp→1+‖?p‖L∞(Ω) where CN is a constant depending only of N and h(Ω), explicitely given in the paper. An eigenfunction u∈BV(Ω)∩L∞(Ω) of the Dirichlet 1-Laplacian is obtained as the strong L1 limit, as p→1+, of a subsequence of the family {?p/‖?p‖L1(Ω)}p>1. Almost all t-level sets Et of u are Cheeger sets and our estimates of u on the Cheeger set |E0| yield |B1|hN(B1)?|E0|hN(Ω), where B1 is the unit ball in RN. For Ω convex we obtain u=|E0|−1χE0. 相似文献
11.
Cheng He 《Journal of Functional Analysis》2004,211(1):153-162
In this paper, we study the partial regularity of the general weak solution u∈L∞(0,T;L2(Ω))∩L2(0,T;H1(Ω)) to the Navier-Stokes equations, which include the well-known Leray-Hopf weak solutions. It is shown that there is a absolute constant ε such that for the weak solution u, if either the scaled local Lq(1?q?2) norm of the gradient of the solution, or the scaled local ) norm of u is less than ε, then u is locally bounded. This implies that the one-dimensional Hausdorff measure is zero for the possible singular point set, which extends the corresponding result due to Caffarelli et al. (Comm. Pure Appl. Math. 35 (1982) 717) to more general weak solution. 相似文献
12.
Reinhard Farwig Hermann Sohr 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(6):1459-1465
There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω⊆R3, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the Lq-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution u∈L8(0,T;L4(Ω)) in some interval [0,T), 0<T≤∞, with u(0)=u0, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u0∈D(A1/4) is strictly stronger and therefore not optimal. 相似文献
13.
Guilong Gui 《Advances in Mathematics》2010,225(3):1248-1284
In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution v∈C([0,∞);H0,s0(R3)∩L3(R3)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w0 has a unique global solution in u∈C([0,∞);H0,s0(R3)) with ∇u∈L2(R+,H0,s0(R3)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R3)). 相似文献
14.
A. F. Tedeev 《Ukrainian Mathematical Journal》1996,48(7):1119-1130
We establish exact upper and lower bounds as t ← ∠ for the norm ‖u(·, t)‖ L ∞(Ω) of a solution of the Neumann problem for a second-order quasilinear parabolic equation in the region D=Ω×{>0}, where Ω is a region with noncompact boundary. 相似文献
15.
We consider the generalized Ostrovsky equation utx=u+(up)xx. We show that the equation is locally well posed in Hs, s>3/2 for all integer values of p?2. For p?4, we show that the equation is globally well posed for small data in H5∩W3,1 and moreover, it scatters small data. The latter results are corroborated by numerical computations which confirm the heuristically expected decay of ‖uLr‖∼t−(r−2)/(2r). 相似文献
16.
Guanggui Ding 《中国科学 数学(英文版)》2001,44(3):273-279
In this paper we shall assert that if T is an isomorphism of L∞(Ω1, A, μ) into L∞(Ω2, B, υ) satisfying the condition ‖T‖·‖T ?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\parallel }}$ is close to an isometry with an error less than 6ε in some conditions. 相似文献
17.
Sungwon Cho 《Journal of Mathematical Analysis and Applications》2007,336(1):372-398
Partial regularity is proved for Lipschitzian critical points of polyconvex functionals provided ‖DuL∞‖ is small enough. In particular, the singular set for a Lipschitzian critical point has Hausdorff dimension strictly less than n when ‖DuL∞‖ is small enough. Model problems treated include
18.
We show that an isolated singularity at the origin 0 of a smooth solution (u,p) of the stationary Navier-Stokes equations is removable if the velocity u satisfies u∈Ln or |u(x)|=o(|x|-1) as x→0. Here n?3 denotes the dimension. As a byproduct of the proof, we also obtain a new interior regularity theorem. 相似文献
19.
20.
We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with +(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u0∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u0(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L1-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L∞‖ for (2+mn)/(n+1)<p<2. 相似文献