共查询到20条相似文献,搜索用时 31 毫秒
1.
Reinhard Farwig Giovanni P. Galdi Hermann Sohr 《Journal of Mathematical Fluid Mechanics》2006,8(3):423-444
We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain
. This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very
large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class
of solutions u for the more general problem with arbitrary divergence k = div u, boundary data g = u|∂Ω and an external force f, as weak as possible, but maintaining uniqueness. In principle, we will follow Amann’s approach. 相似文献
2.
Given H:ℝ3→ℝ of class C1 and bounded, we consider a sequence (un) of solutions of the H-system in the unit open disc satisfying the boundary condition un=γn on ∂. In the first part of this paper, assuming that (un) is bounded in H1(,ℝ3) we study the behavior of (un) when the boundary data γn shrink to zero. We show that either un→0 strongly in H1(,ℝ3) or un blows up at least one H-bubble ω, namely a nonconstant, conformal solution of the H-system on ℝ2. Under additional assumptions on H, we can obtain more precise information on the blow up. In the second part of this paper we investigate the multiplicity
of solutions for the Dirichlet problem on the disc with small boundary datum. We detect a family of nonconstant functions
H (even close to a nonzero constant in any reasonable topology) for which the Dirichlet problem cannot admit a ``large' solution
at a mountain pass level when the boundary datum is small. 相似文献
3.
Giovanni Alessandrini Vincenzo Nesi 《Archive for Rational Mechanics and Analysis》2001,158(2):155-171
We study mappings from ℝ2 into ℝ2 whose components are weak solutions to the elliptic equation in divergence form, div (σ∇u)= 0, which we call σ-harmonic mappings. We prove sufficient conditions for the univalence, i.e., injectivity, of such mappings.
Moreover we prove local bounds in BMO on the logarithm of the Jacobian determinant of such univalent mappings, thus obtaining
the a.e. nonvanishing of their Jacobian. In particular, our results apply to σ-harmonic mapping associated with any periodic
structure and therefore they play an important role in homogenization.
Accepted October 30, 2000?Published online April 23, 2001 相似文献
4.
Let φ:ℝ n → [0,+∞[ be a given positively one-homogeneous convex function, and let ?φ≔{φ≤ 1 }. Pursuing our interest in motion by crystalline mean curvature in three dimensions, we introduce and study the class ?φ (ℝ n ) of “smooth” boundaries in the relative geometry induced by the ambient Banach space (ℝ n , φ). It can be seen that, even when ?φ is a polytope, ?φ(ℝ n ) cannot be reduced to the class of polyhedral boundaries (locally resembling ∂?φ). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of ∂?φ) is the source of several technical difficulties related to the geometry of Lipschitz manifolds. Given a boundary δE in the class ?φ(ℝ n ), we rigorously compute the first variation of the corresponding anisotropic perimeter, which leads to a variational problem on vector fields defined on δE. It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on δE. We define such a divergence to be the φ-mean curvature κφ of δE; the function κφ is expected to be the initial velocity of δE, whenever δE is considered as the initial datum for the corresponding anisotropic mean curvature flow. We prove that κφ is bounded on δE and that its sublevel sets are characterized through a variational inequality. 相似文献
5.
Kuo-Chang Chen 《Archive for Rational Mechanics and Analysis》2006,181(2):311-331
For the planar and spatial N-body problems, it has been proved by Marchal and Chenciner that solutions for the minimizing problem with fixed ends are
free from interior collisions. This important result has been extended by Ferrario & Terracini to Newtonian-type problems
and equivariant problems. It has also been used to construct many symmetric solutions for the N-body problem.
In this paper we are interested in action minimizing solutions in function spaces with free boundaries. The function spaces
are imposed with boundary conditions, such that every mass point starts and ends on two transversal proper subspaces of ℝd, d≥2. We will prove that solutions for this minimizing problem with free boundaries are always free from collisions, including
boundary collisions. This result can be used to construct certain classes of relative periodic solutions of the N-body problem. 相似文献
6.
E. V. Rozhkova 《Mechanics of Solids》2009,44(4):526-536
The spatial problems of elasticity are mainly solved in displacements [1, 2], i.e., the Lamé equations are taken as the initial
equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses.
In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists
in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three
equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical
solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing
the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of
three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator
method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their
systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary
analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive
relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined
directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also
be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary,
special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear
combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation
method. 相似文献
7.
Summary In this paper we look for T-periodic solutions of dynamical systems. Particularly we consider the system
whereU ɛC
1(ℝ
n
x
x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ
n
, ∀t ɛ ℝ T>0. We assume that the problem is asymptotically linear with a bounded nonlinearity. Under a resonance assumption, we
find a multiplicity of T-periodic solutions for T large enough.
Presented at the VII A.I.M.E.T.A. and supported by M.P.I. (40% and 60%). 相似文献
Sommario In questo lavoro si cercano soluzioni periodiche di periodo T assegnato di sistemi dinamici. In particolare si considera un sistema di n equazioni differenziali del secondo ordine del tipo doveU ɛC 1(ℝ n x x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ n , ∀t ɛ ℝ T>0. Nel caso in cui il problema sia asintoticamente lineare, con termine nonlineare limitato e in condizioni di risonanza, troviamo che esiste tale che per il sistema ha una molteplicità di soluzioni.
Presented at the VII A.I.M.E.T.A. and supported by M.P.I. (40% and 60%). 相似文献
8.
Let Ω be a bounded Lipschitz domain in ℝ
n
with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x
n) with x′=(x1, ..., x
n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special
eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with
Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in
3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
9.
H. Beirão da Veiga 《Journal of Mathematical Fluid Mechanics》2007,9(4):506-516
In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R
3 must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference
[5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A
central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary
condition in the half-space R
+3. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition.
The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well
in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or
adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand,
by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.
相似文献
10.
We investigate the smoothness properties of local solutions of
the nonlinear Stokes problem$\begin{eqnarray*}-\diverg \{T(\eps(v))\} + \nabla \pi &=& g \msp \mbox{on $\Omega$,}\\\diverg v&\equiv & 0 \msp \mbox{on $\Omega$,}\end{eqnarray*}$where v:
n
is the velocity field, $\pi$: $ denotes the pressure
function, and g:
n
represents a system of volume forces, denoting
an open subset of
n
.
The tensor T is assumed to be the
gradient of some potential f
acting on symmetric matrices. Our main hypothesis imposed on
f is the existence of exponents
1 < p
q < \infty such that\lambda (1+|\eps|^{2})^{\frac{p-2}{2}} |\sigma|^{2} \leq D^{2}f(\eps)(\sigma ,\sigma)
\leq \Lambda (1+|\eps|^{2})^{\frac{q-2}{2}} |\sigma|^{2}holds with suitable constants , > 0, i.e.
the potential f is of anisotropic
power growth. Under natural assumptions on
p and
q we prove that velocity fields from
the space
W
1
p, loc
(;
n
)
are of class C
1, on an open subset
of with full measure. If n = 2,
then the set of interior singularities is empty.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday 相似文献
11.
We consider the three-dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition.
We discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic
behaviour at infinity. Moreover, we show that each solution with a bounded Dirichlet integral, which has a certain weak additional
decay, behaves like O(r
−2) as r=|x|→∞ and admits a wide jet region. These investigations are based on the solvability properties of the linear Stokes system
in a half space ℝ
+
3
. To investigate the Stokes problem in ℝ
+
3
, we apply the Mellin transform technique and reduce the Stokes problem to the determination of the spectrum of the corresponding
invariant Stokes-Beltrami operator on the hemisphere. 相似文献
12.
Stanislaus Maier-Paape Thomas Wanner 《Archive for Rational Mechanics and Analysis》2000,151(3):187-219
This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation
where Ω⊂ℝ
n
, n∈{1,2,3 }, is a bounded domain with sufficiently smooth boundary, and f is cubic-like, for example f(u) =u−u
3. Based on the results of [26] the nonlinear Cahn-Hilliard equation will be discussed. This equation generates a nonlinear
semiflow in certain affine subspaces of H
2(Ω). In a neighborhood U
ε with size proportional to ε
n
around the constant solution , where μ lies in the spinodal region, we observe the following behavior. Within a local inertial manifold containing there exists a finite-dimensional invariant manifold
which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1. The dimension of is proportional to ε−
n
and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to ε.
(Accepted May 25, 1999) 相似文献
13.
Sébastien Novo 《Journal of Mathematical Fluid Mechanics》2005,7(4):485-514
In the paper [7], author gives a definition of weak solution to the nonsteady Navier–Stokes system of equations which describes
compressible and isentropic flows in some bounded region Ω with influx of fluid through a part of the boundary ∂Ω. Here, we
present a way for proving existence of such solutions in the same situation as in [7] under the sole hypothesis γ > 3/2 for
the adiabatic constant. 相似文献
14.
袁镒吾 《应用数学和力学(英文版)》1987,8(11):1069-1077
In [1], under the condition that all the perfectly plastic stress components at a crack tip are functions of ϕ only, making
use of equilibrium equations, stress-strain rate relations, compatibility equations and yield condition. Lin derived the general
analytical expressions of the perfectly plastic stress field at a mixed-mode crack tip under plane and anti-plane strain.
But in [1] there were several restrictions on the proportionality factor γ in the stress-strain rate relations, such as supposing
that γ is independent of ϕ and supposing that γ=c or cr−1. In this paper, we abolish these restrictions. The cases in [1], γ=crd (n=0 or-1) are the special cases of this paper. 相似文献
15.
Juan Dávila Manuel del Pino Monica Musso Juncheng Wei 《Archive for Rational Mechanics and Analysis》2006,182(2):181-221
We consider the boundary value problem
where Ω is a smooth and bounded domain in ℝ2 and λ > 0. We prove that for any integer k ≧ 1 there exist at least two solutions u
λ
with the property that the boundary flux satisfies up to subsequences λ → 0,
where the ξ
j
are points of ∂Ω ordered clockwise in j. 相似文献
16.
We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields
are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This
toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided
the initial data belong to a “large” set in the Sobolev space H
1. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the
class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions
of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show
how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear)
inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits
a short and simple proof of the existence of strong solutions for all time. 相似文献
17.
For a nonsmooth positively one-homogeneous convex function φ:ℝ
n
→ [0,+∞[, it is possible to introduce the class ?φ (ℝ
n
) of smooth boundaries with respect to φ, to define their φ-mean curvature κφ, and to prove that, for E∈?φ (ℝ
n
), κφ∈L
∞(δE) [9]. Based on these results, we continue the analysis on the structure of δE and on the regularity properties of κφ. We prove that a facet F of δE is Lipschitz (up to negligible sets) and that κφ has bounded variation on F. Further properties of the jump set of κφ are inspected: in particular, in three space dimensions, we relate
the sublevel sets of κφ on F to the geometry of the Wulff shape ?φ≔{φ≤ 1 }.
Accepted October 11, 2000?Published online 14 February, 2001 相似文献
18.
Adriano Montanaro 《Archive for Rational Mechanics and Analysis》1998,143(4):375-400
We consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained
variational problem Here W is a double‐well potential and is a strictly convex domain. For ε small, this is closely related to the problem of partitioning Ω into two subdomains of fixed volume, where the subdomain boundaries correspond to the transitional boundary between phases.
Motivated by this geometry problem, we show that in a strictly convex domain, stable critical points of the original variational
problem have a connected, thin transition layer separating the two phases. This relates to work in [GM] where special geometries
such as cylindrical domains were treated, and is analogous to the results in [CHo] which show that
in a convex domain, stable critical points of the corresponding unconstrained problem are constant. The proof of connectivity employs tools from geometric measure theory including the co‐area formula
and the isoperimetric inequality on manifolds. The thinness of the transition layer follows from a separate calculation establishing
spatial decay of solutions to the pure phases.
(Accepted July 15, 1996) 相似文献
19.
Hiroko Morimoto 《Journal of Mathematical Fluid Mechanics》2007,9(3):411-418
Let Ω be a 2-dimensional bounded domain, symmetric with respect to the x2-axis. The boundary has several connected components, intersecting the x2-axis. The boundary value is symmetric with respect to the x2-axis satisfying the general outflow condition. The existence of the symmetric solution to the steady Navier–Stokes equations
was established by Amick [2] and Fujita [4]. Fujita [4] proved a key lemma concerning the solenoidal extension of the boundary
value by virtual drain method. In this note, we give a different proof via elementary approach by means of the stream function. 相似文献
20.
Gero Friesecke Richard D. James Stefan Müller 《Archive for Rational Mechanics and Analysis》2006,180(2):183-236
We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the
different limit models is the scaling of the elastic energy per unit volume ∼hβ, where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼hα. Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α=β=0, nonlinear bending theory to α=β=2, von Kármán theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our
rigidity result [29] which states that for maps v:(0,1)3→ℝ3, the L2 distance of ∇v from a single rotation is bounded by a multiple of the L2 distance from the set SO(3) of all rotations. 相似文献