共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
3.
Arturo de Pablo Fernando Quirós Ana Rodríguez Juan Luis Vázquez 《Advances in Mathematics》2011,226(2):1378
We develop a theory of existence, uniqueness and regularity for the following porous medium equation with fractional diffusion, with m>m?=(N−1)/N, N?1 and f∈L1(RN). An L1-contraction semigroup is constructed and the continuous dependence on data and exponent is established. Nonnegative solutions are proved to be continuous and strictly positive for all x∈RN, t>0. 相似文献
4.
We consider non-local linear Schrödinger-type critical systems of the type(1) where Ω is antisymmetric potential in L2(R,so(m)), v is an Rm valued map and Ωv denotes the matrix multiplication. We show that every solution v∈L2(R,Rm) of (1) is in fact in , for every 2?p<+∞, in other words, we prove that the system (1) which is a-priori only critical in L2 happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the regularity of weak 1/2-harmonic maps into C2 compact sub-manifolds without boundary. 相似文献
5.
In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces and Triebel–Lizorkin spaces for all s∈(0,1) and p,q∈(n/(n+s),∞], both in Rn and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve on Rn for all s∈(0,1) and q∈(n/(n+s),∞]. A metric measure space version of the above morphism property is also established. 相似文献
6.
Fritz Gesztesy Yuri Latushkin Konstantin A. Makarov Fedor Sukochev Yuri Tomilov 《Advances in Mathematics》2011,(1):145
We compute the Fredholm index, index(DA), of the operator DA=(d/dt)+A on L2(R;H) associated with the operator path , where (Af)(t)=A(t)f(t) for a.e. t∈R, and appropriate f∈L2(R;H), via the spectral shift function ξ(⋅;A+,A−) associated with the pair (A+,A−) of asymptotic operators A±=A(±∞) on the separable complex Hilbert space H in the case when A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A−.We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function ξ(⋅;A+,A−) for the pair (A+,A−), and the corresponding spectral shift function ξ(⋅;H2,H1) for the pair of operators in this relative trace class context,This formula is then used to identify the Fredholm index of DA with ξ(0;A+,A−). In addition, we prove that index(DA) coincides with the spectral flow of the family {A(t)}t∈R and also relate it to the (Fredholm) perturbation determinant for the pair (A+,A−): with the choice of the branch of ln(detH(⋅)) on C+ such thatWe also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant. 相似文献
7.
Tomasz Piasecki 《Journal of Differential Equations》2010,248(8):2171-2198
We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω=Ω0×(0,L)∈R3. We show existence of a solution , p>3, where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of . The term u⋅∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (vn,ρn) that is bounded in and satisfies the Cauchy condition in a larger space L∞(0,L;L2(Ω0)) what enables us to deduce that the weak limit of a subsequence of (vn,ρn) is in fact a strong solution to our problem. 相似文献
8.
The Korteweg–de Vries equation (KdV)[formula]is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev spaceHN(S1; ), (N?1), Hamiltonian (q):=∫S1((∂xq(x))2+q(x)3) dx, and Poisson structure ∂/∂x. The functionq≡0 is an elliptic fixed point. We prove that for anyN?1, the Korteweg–de Vries equation (and thus the entire KdV-hierarchy) admits globally defined real analytic action-angle variables. As a consequence it follows that in a neighborhood ofq≡0 inH1(S1; ), the KdV-Hamiltonian (and similarly any Hamiltonian in the KdV-hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite dimension. Moreover, using the constructed action-angle variables, we analyze the regularity properties of the Hamiltonian vectorfield of KdV. 相似文献
9.
Islam Boussaada A. Raouf Chouikha Jean-Marie Strelcyn 《Bulletin des Sciences Mathématiques》2011,135(1):89
We study the isochronicity of centers at O∈R2 for systems , , where A,B∈R[x,y], which can be reduced to the Liénard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers. 相似文献
10.
Luís Balsa Bicho António Ornelas 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(18):7061-7070
In this paper, we prove existence of radially symmetric minimizersuA(x)=UA(|x|), having UA(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |∇u(x)| is the Euclidean norm of the gradient vector and BR is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?∗∗:R×R→[0,∞] is just convex lsc and and ?∗∗(s,⋅) is even; while ρ1(⋅) and ρ2(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?∗∗(s,v)=∞ is freely allowed. 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.
Imre Patyi 《Bulletin des Sciences Mathématiques》2011,(3):43
Let X be a separable Banach space and u:X→R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h:X→Z with u(x)<‖h(x)‖ for x∈X. As a consequence we find that the sheaf cohomology group Hq(X,O) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q?1. As another consequence we prove that if f is a C1-smooth -closed (0,1)-form on the space X=L1[0,1] of summable functions, then there is a C1-smooth function u on X with on X. 相似文献
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14.
V.V. Chepyzhov M.I. Vishik S.V. Zelik 《Journal de Mathématiques Pures et Appliquées》2011,96(4):395-407
The 2D Euler equations with periodic boundary conditions and extra linear dissipative term Ru, R>0 are considered and the existence of a strong trajectory attractor in the space is established under the assumption that the external forces have bounded vorticity. This result is obtained by proving that any solution belonging the proper weak trajectory attractor has a bounded vorticity which implies its uniqueness (due to the Yudovich theorem) and allows to verify the validity of the energy equality on the weak attractor. The convergence to the attractor in the strong topology is then proved via the energy method. 相似文献
15.
In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in R3. We prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H3-framework. Moreover, if additionally the initial data belong to Lp with , the optimal convergence rates of the solutions in Lq-norm with 2≤q≤6 and its spatial derivatives in L2-norm are obtained. 相似文献
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17.
Michi-aki Inaba 《Advances in Mathematics》2011,(4):1399
For an abelian or a projective K3 surface X over an algebraically closed field k, consider the moduli space of the objects E in Db(Coh(X)) satisfying and Hom(E,E)≅k. Then we can prove that is smooth and has a symplectic structure. 相似文献
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19.
A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space Hs(R) with is developed. Provided that does not change sign, u0∈Hs () and u0∈L1(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)∈C([0,∞);Hs(R))∩C1([0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired. 相似文献
20.
Let (Rn,|⋅|,dγ) be the Gauss measure metric space, where Rn denotes the n-dimensional Euclidean space, |⋅| the Euclidean norm and for all x∈Rn the Gauss measure. In this paper, for any a∈(0,∞), the authors introduce some BLOa(γ) space, namely, the space of functions with bounded lower oscillation associated with a given class of admissible balls with parameter a. Then the authors prove that the noncentered local natural Hardy–Littlewood maximal operator is bounded from BMO(γ) of Mauceri and Meda to BLOa(γ). Moreover, a characterization of the space BLOa(γ), via the local natural maximal operator and BMO(γ), is given. The authors further prove that a class of maximal singular integrals, including the corresponding maximal operators of both imaginary powers of the Ornstein–Uhlenbeck operator and Riesz transforms of any order associated with the Ornstein–Uhlenbeck operator, are bounded from L∞(γ) to BLOa(γ). 相似文献