共查询到20条相似文献,搜索用时 31 毫秒
1.
Yoshikazu Giga Katsuya Inui Alex Mahalov Shin’ya Matsui Jürgen Saal 《Archive for Rational Mechanics and Analysis》2007,186(2):177-224
We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary
solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space for 2 < p < ∞. Here the L
p
-integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space
contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous
Besov spaces. For instance we provide and apply an operator-valued bounded H
∞-calculus for the Laplacian in for a general Banach space . 相似文献
2.
Robert Jensen Changyou Wang Yifeng Yu 《Archive for Rational Mechanics and Analysis》2008,190(2):347-370
For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation:
For H = H(p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation and
that of the Hamilton–Jacobi equation .
Supported by NSF DMS 0601162.
Supported by NSF DMS 0601403. 相似文献
3.
Crack Initiation in Brittle Materials 总被引:1,自引:0,他引:1
Antonin Chambolle Alessandro Giacomini Marcello Ponsiglione 《Archive for Rational Mechanics and Analysis》2008,188(2):309-349
In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during
a load process through a time-dependent boundary datum of the type t → t
g(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation
is brutal, that is, a big crack appears after a positive time t
i
> 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large
class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the
problem is a local minimality result for the functional where , k > 0 and f is a suitable Carathéodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (uΓ, Γ) with small enough are such that . 相似文献
4.
Let be an infinite cylinder of , n ≥ 3, with a bounded cross-section of C
1,1-class. We study resolvent estimates and maximal regularity of the Stokes operator in for 1 < q, r < ∞ and for arbitrary Muckenhoupt weights ω ∈ A
r
with respect to x′ ∈ Σ. The proofs use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions
based on the -boundedness of the family of solution operators for a system in Σ parametrized by the phase variable of the one-dimensional
partial Fourier transform.
Supported by the Gottlieb Daimler- und Karl Benz-Stiftung, grant no. S025/02-10/03. 相似文献
5.
6.
Philippe G. Ciarlet Liliana Gratie Cristinel Mardare 《Archive for Rational Mechanics and Analysis》2008,188(3):457-473
The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a
αβ
) of order two and a field of symmetric matrices (b
αβ
) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of , then there exists an immersion such that these fields are the first and second fundamental forms of the surface , and this surface is unique up to proper isometries in . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a
αβ
and b
αβ
, that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation
where A
1 and A
2 are antisymmetric matrix fields of order three that are functions of the fields (a
αβ
) and (b
αβ
), the field (a
αβ
) appearing in particular through the square root U of the matrix field The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension of the unknown immersion . In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental
theorem of surface theory set out by S. Mardare [20–22], the unknown immersion is found in the present approach to exist in function spaces “with little regularity”, such as , p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells
where rotation fields are introduced as bona fide unknowns. 相似文献
7.
In this paper, we establish the local well-posedness for the Cauchy problem of a simplified version of hydrodynamic flow of nematic liquid crystals in ${\mathbb{R}^3}$ for any initial data (u 0, d 0) having small ${L^{3}_{\rm uloc}}$ -norm of ${(u_{0}, \nabla d_{0})}$ . Here ${L^{3}_{\rm uloc}(\mathbb{R}^3)}$ is the space of uniformly locally L 3-integrable functions. For any initial data (u 0, d 0) with small ${\|(u_0, \nabla d_0)\|_{L^{3}(\mathbb{R}^3)}}$ , we show that there exists a unique, global solution to the problem under consideration which is smooth for t > 0 and has monotone deceasing L 3-energy for ${t \geqq 0}$ . 相似文献
8.
Yoichi Mito Thomas J. Hanratty Paulo Zandonade Robert D. Moser 《Flow, Turbulence and Combustion》2007,79(2):175-189
This paper uses direct numerical simulations (DNS) of turbulent flow in a channel at (Del álamo, Jiménez, Zandonade, Moser J Fluid Mech 500:135–144, 2004) to provide a picture of the turbulent structures making large contributions to the Reynolds shear stress.
Considerable work of this type has been done for the viscous wall region at smaller , for which a log-layer does not exist. Recent PIV measurements of turbulent velocity fluctuations in a plane parallel to
the direction of flow have emphasized the dominant contribution of large scale structures in the outer flow. This prompted
Hanratty and Papavassiliou (The role of wall vortices in producing turbulence. In: Panton, R.L. (ed) Self-sustaining Mechanism of Wall Turbulence. Computational Mechanics Publications, Southampton, pp. 83–108, 1997) to use DNS at to examine these structures in a plane perpendicular to the direction of flow. They identified plumes which extend from the
wall to the center of a channel. The data at are used to explore these results further, to examine the structure of the log-layer, and to test present notions about the
viscous wall layer. 相似文献
9.
Mike Cullen Wilfrid Gangbo Giovanni Pisante 《Archive for Rational Mechanics and Analysis》2007,185(2):341-363
We study the evolution of a system of n particles in . That system is a conservative system with a Hamiltonian of the form , where W
2 is the Wasserstein distance and μ is a discrete measure concentrated on the set . Typically, μ(0) is a discrete measure approximating an initial L
∞ density and can be chosen randomly. When d = 1, our results prove convergence of the discrete system to a variant of the semigeostrophic equations. We obtain that
the limiting densities are absolutely continuous with respect to the Lebesgue measure. When converges to a measure concentrated on a special d–dimensional set, we obtain the Vlasov–Monge–Ampère (VMA) system. When, d = 1 the VMA system coincides with the standard Vlasov–Poisson system. 相似文献
10.
We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation
where is a ring-shaped domain, a and μ are given positive constants, is the Heaviside maximal monotone graph: if s > 0, if s < 0. Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as
combustion. We show that under certain conditions on the initial data the level sets are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Γ
μ
is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential
equation expresses the velocity of advancement of the level surface Γ
μ
through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered
as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory
of each of the fluid particles. 相似文献
11.
Anne-Laure Dalibard 《Archive for Rational Mechanics and Analysis》2009,192(1):117-164
We study the limit as ε → 0 of the entropy solutions of the equation . We prove that the sequence u
ε
two-scale converges toward a function u(t, x, y), and u is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation
law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence
result in . 相似文献
12.
Michael Winkler 《Journal of Dynamics and Differential Equations》2008,20(1):87-113
The paper deals with positive solutions of the initial-boundary value problem for with zero Dirichlet data in a smoothly bounded domain . Here is positive on (0,∞) with f(0) = 0, and λ1 is exactly the first Dirichlet eigenvalue of −Δ in Ω. In this setting, (*) may possess oscillating solutions in presence
of a sufficiently strong degeneracy. More precisely, writing , it is shown that if then there exist global classical solutions of (*) satisfying and . Under the additional structural assumption , s > 0, this result can be sharpened: If then (*) has a global solution with its ω-limit set being the ordered arc that consists of all nonnegative multiples of the
principal Laplacian eigenfunction. On the other hand, under the above additional assumption the opposite condition ensures that all solutions of (*) will stabilize to a single equilibrium.
相似文献
13.
Under certain assumptions on f and g we prove that positive, global and bounded solutions u of the non-autonomous heat equation
in
(N ≥ 3) converge to a steady state.
Dedicated to Prof. Pavol Brunovsky on the occasion of his 70th birthday. 相似文献
14.
Georgy M. Kobelkov 《Journal of Mathematical Fluid Mechanics》2007,9(4):588-610
For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the
large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component
u
3 under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary
time interval [0, T], any viscosity coefficients and any initial conditions
a weak solution exists and is unique and and the norms are continuous in t.
The work was carried out under partial support of Russian Foundation for Basic Research (project 05-01-00864). 相似文献
15.
We consider the nonlinear elliptic system
where and is the unit ball. We show that, for every and , the above problem admits a radially symmetric solution (u
β
, v
β
) such that u
β
− v
β
changes sign precisely k times in the radial variable. Furthermore, as , after passing to a subsequence, u
β
→ w
+ and v
β
→ w
− uniformly in , where w = w
+− w
− has precisely k nodal domains and is a radially symmetric solution of the scalar equation Δw − w + w
3 = 0 in , w = 0 on . Within a Hartree–Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains
for Bose–Einstein double condensates with strong repulsion. 相似文献
16.
17.
Thierry Cazenave Flávio Dickstein Fred B. Weissler 《Journal of Dynamics and Differential Equations》2007,19(3):789-818
In this paper, we construct solutions u(t,x) of the heat equation on such that has nontrivial limit points in as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.
相似文献
18.
We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension
where , is a smooth matrix-valued map and the initial data is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate
solutions converging in to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial)
viscous parabolic approximation
as . In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in , the limit of front tracking approximations provides a weak solution of the system of conservation laws u
t
+ F(u)
x
= 0, satisfying the Liu admissibility conditions.
These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), . In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic
fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of Lax[17], or in the generalized sense of Liu[23].
Dedicated to Prof. Tai Ping Liu on the occasion of his 60
th
birthday 相似文献
19.
We study the dynamics of vortices in solutions of the Gross–Pitaevsky equation in a bounded, simply connected domain with natural boundary conditions on ∂Ω. Previous rigorous results have shown that for sequences of solutions with suitable well-prepared initial data, one can determine limiting vortex trajectories, and moreover that these trajectories
satisfy the classical ODE for point vortices in an ideal incompressible fluid. We prove that the same motion law holds for
a small, but fixed , and we give estimates of the rate of convergence and the time interval for which the result remains valid. The refined
Jacobian estimates mentioned in the title relate the Jacobian J(u) of an arbitrary function to its Ginzburg–Landau energy. In the analysis of the Gross–Pitaevsky equation, they allow us to use the Jacobian to locate
vortices with great precision, and they also provide a sort of dynamic stability of the set of multi-vortex configurations. 相似文献
20.
We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g
0(x, t) and g
1 (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g
1 (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g
0(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u
0(x, t) of the “limiting” N.–S. system with the same initial data. If the function g
1 (z, t) admits the divergence representation, the functions g
0(x, t) and g
1 (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).
相似文献