共查询到20条相似文献,搜索用时 31 毫秒
1.
Tai‐Peng Tsai 《Archive for Rational Mechanics and Analysis》1998,143(1):29-51
This paper proves that Leray's self‐similar solutions of the three‐dimensional Navier‐Stokes equations must be trivial under
very general assumptions, for example, if they satisfy local energy estimates.
(Accepted April 7, 1997) 相似文献
2.
. We consider the problem of finding a holomorphic function in a strip with a cut ${\cal A}= \{(x,y) : \, x\in\RE,\,\,0 satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for $|x|\to\infty. We consider the problem of finding a holomorphic function in a strip with a cut satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the
class of sectionally holomorphic functions in ?, vanishing for . A special solution can be selected by fixing the value of the circulation around the cut. The problem is obtained by linearization
of the equations of the wave‐resistance problem for a “slender” cylinder submerged in a heavy fluid and moving at uniform
speed in the direction orthogonal to its generators. The results obtained, besides their own interest, are a crucial step
for the resolution of the non‐linear problem.
(Accepted October 14, 1998) 相似文献
3.
4.
We study the initial‐boundary‐value problem for the linearized equations of ideal magnetohydrodynamics defined on a bounded
domain whose boundary is a regular magnetic surface. We show that it is well‐posed in and that the loss of regularity of solutions always occurs for some initial data in
(Accepted October 23, 1997) 相似文献
5.
. We analyze a class of vector fields, called divergence‐measure fields. We establish the Gauss‐Green formula, the normal traces over subsets of
Lipschitz boundaries, and the product rule for this class of fields. Then we apply this theory to analyze entropy solutions of initial‐boundary‐value problems for hyperbolic conservation laws and to study the ways in which the
solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations,
the system of nonlinear elasticity, and a class of systems with affine characteristic hypersurfaces. The analysis in also extends to .
(Accepted July 16, 1998) 相似文献
6.
Bernadette Miara 《Archive for Rational Mechanics and Analysis》1998,142(4):331-353
We consider a family of three‐dimensional shells with the same middle surface, all composed of the same nonlinearly elastic
Saint Venant‐Kirchhoff material. Using the method of asymptotic expansions with the thickness as the “small” parameter, and
making specific assumptions on the applied forces, the geometry of the middle surface, and the kinematic boundary conditions,
we show how a “limiting”, “large‐deformation” two‐dimensional model can be identified in this fashion. By linearization, this
nonlinear membrane model reduces to the linear membrane model.
(Accepted January 13, 1997) 相似文献
7.
We prove the existence of energy‐minimizing configurations for a two‐dimensional, variational model of magnetoelastic materials
capable of large deformations. The model is based on an energy functional which is the sum of the nonlocal self‐energy (the
energy stored in the magnetic field generated by the body, and permeating the whole ambient space) and of the local anisotropy
energy, which is not weakly lower semicontinuous. A further feature of the model is the presence of a non‐convex constraint
on one of the unknowns, the magnetization, which is a unit vector field.
(Accepted November 20, 1997) 相似文献
8.
E. Carlen R. Esposito J. L. Lebowitz R. Marra A. Rokhlenko 《Archive for Rational Mechanics and Analysis》1998,142(3):193-218
We study, globally in time, the velocity distribution f(v,t) of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant
external electric field E. The density f satisfies a Boltzmann-type kinetic equation containing a fully nonlinear electron‐electron collision term as well as linear
terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant
in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the L
1 distance between f and a certain time-dependent Maxwellian stays small uniformly in t. Moreover, the mean and variance of this time‐dependent Maxwellian satisfy a coupled set of nonlinear ordinary differential
equations that constitute the “hydrodynamical” equations for this kinetic system. This remains
true even when these ordinary differential equations have non‐unique equilibria, thus proving the existence of multiple stable
stationary solutions for the full kinetic model. Our approach relies on scale‐independent estimates for the kinetic equation,
and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation
between the kinetic and hydrodynamic scales globally in time.
(Accepted September 3, 1996) 相似文献
9.
Florian Theil 《Archive for Rational Mechanics and Analysis》1998,144(1):47-78
We study the viscoelastically damped wave equation with a nonmonotone stress‐strain relation σ. This system describes the dynamics of phase transitions, which is closely related to the creation of microstructures. In
order to analyze the dynamic behavior of microstructures we consider highly oscillatory initial states. Two questions are
addressed in this work: How do oscillations propagate in space and time? What can be said about the long‐time behavior? An
appropriate tool to deal with oscillations are Young measures. They describe the local distribution or one‐point statistics
of a sequence of fast fluctuating functions. We demonstrate that highly oscillatory initial states generate in a unique fashion
an evolution in the space of Young measures and we derive the determining equations. Further on we prove a generalized dissipation
identity for Young‐measure solutions. As a consequence, it is shown that every low‐energy solution converges to a Young‐measure
equilibrium as t→∞. This is a generalization of G. Friesecke's & J. B. McLeod's [FM96] convergence result for classical solutions to the case
of Young‐measure solutions.
(Accepted November 12, 1997) 相似文献
10.
Charles M. Elliott Reiner Schätzle Barbara E. E. Stoth 《Archive for Rational Mechanics and Analysis》1998,145(2):99-127
In a Type‐II superconductor the magnetic field penetrates the superconducting body through the formation of vortices. In
an extreme Type‐II superconductor these vortices reduce to line singularities. Because the number of vortices is large it
seems feasible to model their evolution by an averaged problem, known as the mean-field model of superconductivity. We assume
that the evolution law of an individual vortex, which underlies the averaging process, involves the current of the generated
magnetic field as well as the curvature vector. In the present paper we study a two‐dimensional reduction, assuming all vortices
to be perpendicular to a given direction. Since both the magnetic field H and the averaged vorticity ω are curl‐free, we may represent them via a scalar magnetic potential q and a scalar stream function ψ, respectively. We study existence, uniqueness and asymptotic behaviour of solutions (ψ, q) of the resulting degenerate elliptic‐parabolic system (with curvature taken into account or not) by means of viscosity and
weak solutions. In addition we relate (ψ, q) to solutions (ω, H) of the mean‐field equations without curvature. Finally we construct special solutions of the corresponding stationary equations
with two or more superconducting phases.
(Accepted August 8, 1997) 相似文献
11.
We study two‐dimensional Riemann problems with piecewise constant data. We identify a class of two‐dimensional systems, including
many standard equations of compressible flow, which are simplified by a transformation to similarity variables. For equations
in this class, a two‐dimensional Riemann problem with sectorially constant data becomes a boundary‐value problem in the finite
plane. For data leading to shock interactions, this problem separates into two parts: a quasi‐one‐dimensional problem in supersonic
regions, and an equation of mixed type in subsonic regions.
We prove a theorem on local existence of solutions of quasi‐one‐dimensional Riemann problems. For 2 × 2 systems, we generalize
a theorem of Courant & Friedrichs, that any hyperbolic state adjacent to a constant state must be a simple wave.
In the subsonic regions, where the governing equation is of mixed hyperbolic‐elliptic type, we show that the elliptic part
is degenerate at the boundary, with a nonlinear variant of a degeneracy first described by Keldysh.
(Accepted December 4, 1997) 相似文献
12.
We introduce a new formulation of the Oppenheimer‐Volkoff (O‐V) equations, a system of ordinary differential equations that
models the interior of a star in general relativity, and we use this to give a completely rigorous mathematical analysis of
solutions. In particular, we prove that, under mild assumptions on the equation of state, black holes never form in solutions of the O‐V equations. As a corollary, this implies that the portion of the empty‐space Schwarzschild solution
inside the Schwarzschild radius cannot be obtained as a limit of O‐V solutions having non‐zero density. We also prove that
if the density ρ at radius r is ever larger than where M(r) is the total mass inside radius r, then M must become negative for some positive radius. We interpret M<0 as a condition for instability because we show that if the pressure
is a decreasing function of r, then M(r)<0 at some r>0 implies that the pressure tends to infinity before r=0.
(Accepted October 28, 1996) 相似文献
13.
Andrew P. Bassom Peter A. Clarkson C. K. Law J. Bryce McLeod 《Archive for Rational Mechanics and Analysis》1998,143(3):241-271
In this work we propose a new method for investigating connection problems for the class of nonlinear second‐order differential
equations known as the Painlevé equations. Such problems can be characterized by the question as to how the asymptotic behaviours
of solutions are related as the independent variable is allowed to pass towards infinity along different directions in the
complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the
ideas of isomonodromic deformation and the matching of WKB solutions. However, the implementation of these methods often tends
to be heuristic in nature and so the task of rigorising the process is complicated. The method we propose here develops uniform
approximations to solutions. This removes the need to match solutions, is rigorous, and can lead to the solution of connection
problems with minimal computational effort. Our method relies on finding uniform approximations of differ
ential equations of the generic form as the complex‐valued parameter . The details of the treatment rely heavily on the locations of the zeros of the function F in this limit. If they are isolated, then a uniform approximation to solutions can be derived in terms of Airy functions
of suitable argument. On the other hand, if two of the zeros of F coalesce as , then an approximation can be derived in terms of parabolic cylinder functions. In this paper we discuss both cases, but
illustrate our technique in action by applying the parabolic cylinder case to the “classical” connection problem associated
with the second Painlevé transcendent. Future papers will show how the technique can be applied with very little change to
the other Painlevé equations, and to the wider problem of the asymptotic behavio
ur of the general solution to any of these equations.
(Accepted May 15, 1997) 相似文献
14.
G. Rein 《Archive for Rational Mechanics and Analysis》1997,140(4):335-351
We consider the Vlasov‐Poisson system in a cosmological setting as studied in [18] and prove nonlinear stability of homogeneous
solutions against small, spatially periodic perturbations in the L
∞‐norm of the spatial mass density. This result is connected with the question of how large scale structures such as galaxies
have evolved out of the homogeneous state of the early universe.
(Accepted June 28, 1996) 相似文献
15.
. This paper is concerned with the initial‐boundary‐value problem for a nonlinear hyperbolic system of conservation laws.
We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing‐viscosity
method and finite‐difference schemes (Lax‐Friedrichs‐type schemes and the Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming
solely uniform bounds on the approximate solutions and so dealing with solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young
measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary
Young measure reduces to a Dirac mass.
From the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic
or not. Each formulation is based on a set of admissible boundary values, following the terminology of Dubois & LeFloch[15]. The local structure of these sets and the well‐posedness of the corresponding initial‐boundary‐value problem are investigated.
The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics.
(Accepted July 2, 1998) 相似文献
16.
Guy Barles Panagiotis E. Souganidis 《Archive for Rational Mechanics and Analysis》1998,141(3):237-296
In this paper we present a new definition for the global in time propagation (motion) of fronts (hypersurfaces, boundaries)
with a prescribed normal velocity, past the first time they develop singularities. We show that if this propagation satisfies
a geometric maximum principle (inclusion‐avoidance)‐type property, then the normal velocity must depend only on the position
of the front and its normal direction and principal curvatures. This new approach, which is more geometric and, as it turns
out, equivalent to the level‐set method, is then used to develop a very general and simple method to rigorously validate the
appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction‐diffusion
equations. We finally present a number of new asymptotic results. Among them are the asymptotics of (i) reaction‐diffusion
equations with rapidly oscillating coefficients, (ii) fully nonlinear nonlocal (integral differential) equations and (iii)
stochastic Ising models with long-range anisotropic interactions and general spin flip dynamics.
(Accepted July 8, 1996) 相似文献
17.
P. Poláčik 《Archive for Rational Mechanics and Analysis》2011,199(1):69-97
This paper is devoted to a class of nonautonomous parabolic equations of the form u
t
= Δu + f(t, u) on
\mathbbRN{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter
the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite
time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values
of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding
solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one
threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric
in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results
for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic
equations. 相似文献
18.
Friedemann Schuricht 《Archive for Rational Mechanics and Analysis》1997,140(2):103-159
We use variational methods to study obstacle problems for geometrically exact (Cosserat) theories for the planar deformation
of nonlinearly elastic rods. These rods can suffer flexure, extension, and shear. There is a marked difference between the
behavior of a shearable and an unshearable rod. The set of admissible deformations is not convex, because of the exact geometry
used. We first investigate the fundamental question of describing contact forces, which we necessarily treat as vector‐valued
Borel measures. Moreover, we introduce techniques for describing point obstacles. Then we prove existence for a very large
class of problems. Finally, using nonsmooth analysis for handling the obstacle, we show that the Euler‐Lagrange equations
are satisfied almost everywhere. These equations provide very detailed structural information about the contact forces.
Accepted June 3, 1996 相似文献
19.
We consider the Ginzburg‐Landau system with an applied magnetic field and analyze the behavior of solutions when the domain
is a cylinder (of radius ) and the applied field is parallel to the axis. It is shown that there is an upper critical value such that if the modulus of the applied field is greater than , the normal (nonsuperconducting) state (in which the order parameter is identically zero) is stable and if the modulus of
the applied field is slightly below , the normal state is unstable. In addition, it is shown that there is a positive lower critical value such that the normal state is unstable if the modulus of the applied field is less than and stable if the modulus is slightly above . In the case of type‐II materials for whic
h the Ginzburg‐Landau constant κ is large, it is shown that there is a discrete set of radii ℬ(κ) such that if and is sufficiently large, then for each applied field of modulus slightly less than (or slightly more than ) there is precisely one small superconducting solution (up to a gauge transformation) which is stable. Moreover for this
solution, the complex‐valued order parameter ψ is zero only on the axis of the cylinder, and its winding number is proportional
to the product of κ2 and the cross‐sectional area of the cylinder. In addition, the solution exhibits “surface superconductivity” as predicted
by the physicists de Gennes and St. James.
(Accepted July 15, 1996) 相似文献
20.
Symmetry of Ground States of Quasilinear Elliptic Equations 总被引:3,自引:0,他引:3
. We consider the problem of radial symmetry for non‐negative continuously differentiable weak solutions of elliptic equations of the form under the ground state condition Using the well‐known moving plane method of Alexandrov and Serrin, we show, under suitable conditions on A and f, that all ground states of (1) are radially symmetric about some origin O in . In particular, we obtain radial symmetry for compactly supported ground states and give sufficient conditions for the positivity
of ground states in terms of the given operator A and the nonlinearity f.
(Accepted September 21, 1998) 相似文献