共查询到20条相似文献,搜索用时 296 毫秒
1.
Nikola Popović 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,62(3):405-437
We investigate the effects of a Heaviside cutoff on the dynamics of traveling fronts in a family of scalar reaction-diffusion equations with degenerate polynomial potential that includes the classical Zeldovich equation. We prove the existence and uniqueness of front solutions in the presence of the cutoff, and we derive the leading-order asymptotics of the corresponding propagation speed in terms of the cutoff parameter. For the Zeldovich equation, an explicit solution to the equation without cutoff is known, which allows us to calculate higher-order terms in the resulting expansion for the front speed; in particular, we prove the occurrence of logarithmic (switchback) terms in that case. Our analysis relies on geometric methods from dynamical systems theory and, in particular, on the desingularization technique known as ‘blow-up.’ 相似文献
2.
Shiwang Ma 《Journal of Differential Equations》2005,212(1):129-190
We study the existence, uniqueness, global asymptotic stability and propagation failure of traveling wave fronts in a lattice delayed differential equation with global interaction for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. In the bistable case, under realistic assumptions on the birth function, we prove that the equation admits a strictly monotone increasing traveling wave front. Moreover, if the wave speed does not vanish, then the wave front is unique (up to a translation) and globally asymptotic stable with phase shift. Of particular interest is the phenomenon of “propagation failure” or “pinning” (that is, wave speed c = 0), we also give some criteria for pinning in this paper. 相似文献
3.
We consider a quasilinear equation that consists of the inviscid Burgers equation plus O(α2) nonlinear terms. As we show, these extra terms regularize the Burgers equation in the following sense: for smooth initial
data, the α > 0 equation has classical solutions globally in time. Furthermore, in the zero-α limit, solutions of the regularized
equation converge strongly to weak solutions of the Burgers equation. We present numerical evidence that the zero-α limit
satisfies the Oleinik entropy inequality. For all α ≥ 0, the regularized equation possesses a nonlocal Poisson structure.
We prove the Jacobi identity for this generalized Hamiltonian structure. 相似文献
4.
In this paper we study the homogeneous relativistic heat equation (HRHE) obtained as asymptotic limit of the so-called relativistic
heat equation (RHE) when the kinematic viscosity ν → ∞. These equations were introduced in the theory of radiation hydrodynamics
to guarantee a bounded speed of propagation of radiating energy. We shall prove that this is indeed true, and we shall construct
some explicit solutions of the HRHE exhibiting fronts propagating at light speed. 相似文献
5.
Nalini Anantharaman 《Geometric And Functional Analysis》2010,20(3):593-626
We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold.
It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the
number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally
is the “entropy” that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower
bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can
use the trace formula to prove a result going in this direction. 相似文献
6.
Yu. G. Rykov 《Mathematical Notes》1999,66(5):628-635
We study a system of equations consisting of the two-dimensional Bürgers equation and the continuity equation. In 1970 such
a system was proposed by Ya. B. Zeldovich for describing the formation of the large-scale structure of the Universe. In the
present paper, for the divergent form of this system (the zero-pressure gas dynamics system), we rigorously define the notion
of its generalized solution (in the sense of distributions) in terms of Radon measures and obtain a generalization of the
Rankine-Hugoniot relations. By using these relations, we show that, in general, the variational representation of generalized
solutions valid for the one-dimensional system of zero-pressure gas dynamics does not make sense in the two-dimensional case.
Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 760–769, November, 1999. 相似文献
7.
Interface Instability under Forced Displacements 总被引:1,自引:0,他引:1
By applying linear response theory and the Onsager principle, the power (per unit area) needed to make a planar interface
move with velocity V is found to be equal to V2/ μ, μ a mobility coefficient. To verify such a law, we study a one dimensional model where the interface is the stationary
solution of a non local evolution equation, called an instanton. We then assign a penalty functional to orbits which deviate
from solutions of the evolution equation and study the optimal way to displace the instanton. We find that the minimal penalty
has the expression V2/ μ only when V is small enough. Past a critical speed, there appear nucleations of the other phase ahead of the front, their number and
location are identified in terms of the imposed speed.
submitted 31/01/05, accepted 26/10/05 相似文献
8.
We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters
β=γ=0, δ= and 2α=(2μ-1)2 with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute
the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional
space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection
group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and
use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one
correspondence with the regular polyhedra or star-polyhedra in the three dimensional space.
Oblatum 19-III-1999 & 25-XI-1999?Published online: 21 February 2000 相似文献
9.
Vladimir Bogachev Giuseppe Da Prato Michael Röckner 《Journal of Evolution Equations》2010,10(3):487-509
We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence
and uniqueness results are known. We prove, under suitable assumptions, the existence and uniqueness of a measure valued solution,
for the corresponding Fokker–Planck equation. In particular, we verify the Chapman–Kolmogorov equations and get an evolution
system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation. 相似文献
10.
We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.This work has been supported by funds from ACI JC 1041 “Mouvements d’interfaces avec termes non-locaux”, from ACI-JC 1025 “Dynamique des dislocations” and from ONERA, Office National d’Etudes et de Recherches. The second author was also supported by the ENPC-Région Ile de France. 相似文献
11.
Xuguang Lu 《Journal of Differential Equations》2008,245(7):1705-1761
For general initial data we prove the global existence and weak stability of weak solutions of the Boltzmann equation for Fermi-Dirac particles in a periodic box for very soft potentials (−5<γ?−3) with a weak angular cutoff. In particular the Coulomb interaction (γ=−3) with the weak angular cutoff is included. The conservation of energy and moment estimates are also proven under a further angular cutoff. The proof is based on the entropy inequality, velocity averaging compactness of weak solutions, and various continuity properties of general Boltzmann collision integral operators. 相似文献
12.
We present a novel penalty approach to the Hamilton-Jacobi-Bellman (HJB) equation arising from the valuation of European options
with proportional transaction costs. We first approximate the HJB equation by a quasilinear 2nd-order partial differential
equation containing two linear penalty terms with penalty parameters λ
1 and λ
2 respectively. Then, we show that there exists a unique viscosity solution to the penalized equation. Finally, we prove that,
when both λ
1 and λ
2 approach infinity, the viscosity solution to the penalized equation converges to that of the corresponding original HJB equation. 相似文献
13.
We prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation—variational in nature—defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to “small divisors” phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical “Arnold non-degeneracy condition” of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities. 相似文献
14.
Giuseppe Da Prato Arnaud Debussche Beniamin Goldys 《Probability Theory and Related Fields》2002,123(3):355-380
We consider transition semigroups generated by stochastic partial differential equations with dissipative nonlinear terms.
We prove an integration by part formula and a Logarithmic Sobolev inequality for the invariant measure. No symmetry or reversibility
assumptions are made. Furthemore we prove some compactness results on the transition semigroup and on the embedding of the
Sobolev spaces based on the invariant measure. We use these results to derive asymptotic properties for a stochastic reaction–diffusion
equation.
Received: 29 September 2000 / Revised version: 30 May 2001 / Published online: 14 June 2002 相似文献
15.
S. M. Umarkhadzhiev 《Russian Mathematics (Iz VUZ)》2020,64(3):57-65
We give an application of so-called grand Lebesgue and grand Sobolev spaces, intensively studied during last decades, to partial differential equations. In the case of unbounded domains such spaces are defined using so-called grandizers. Under some natural assumptions on the choice of grandizers, we prove the existence, in some grand Sobolev space, of a solution to the equation Pm(D)u(x) = f(x), x ∈ ℝn, m < n, with the right-hand side in the corresponding grand Lebesgue space, where Pm(D) is an arbitrary elliptic homogeneous in the general case we improve some known facts for the fundamental solution of the operator Pm(D): we construct it in the closed form either in terms of spherical hypersingular integrals or in terms of some averages along plane sections of the unit sphere. 相似文献
16.
For multidimensional equations of flow of thin capillary films with nonlinear diffusion and convection, we prove the existence
of a strong nonnegative generalized solution of the Cauchy problem with initial function in the form of a nonnegative Radon
measure with compact support. We determine the exact upper estimate (global in time) for the rate of propagation of the support
of this solution. The cases where the degeneracy of the equation corresponds to the conditions of “strong” and “weak” slip
are analyzed separately. In particular, in the case of “ weak” slip, we establish the exact estimate of decrease in the L
2-norm of the gradient of solution. It is well known that this estimate is not true for the initial functions with noncompact
supports.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 250–271, February, 2006. 相似文献
17.
Magnus Bondesson 《Annali di Matematica Pura ed Applicata》1973,95(1):1-43
Summary We consider elliptic and parabolic difference operators and prove estimates in discrete Lp norms, 1<p<∞, which are analogues of known estimates for the corresponding differential operators. Let U be a solution in
a bounded domain Ω of an elliptic or parabolic differential equation and let Uh be a solution of the discrete equation. Using the estimates, we prove under mild regularity assumptions that if Uh converges to U in some discrete Lp normp>1, then the difference quotients of Uh converge uniformly (on compact subsets of Ω) to the corresponding derivatives of U.
Entrata in Redazione il 9 ottobre 1971. 相似文献
18.
We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is
fully non linear and naturally arises in the study of real hypersurfaces in ℂn+1, for n ≥ 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real
hypersurface ℂn+1 and it is the analogous of the equation with prescribed Gauss curvature for the complex structure. However, it is degenerate
elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow us to use elliptic techniques
such in the classical real and complex Monge-Ampère equations. By taking into account the natural geometry of the problem
we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness
of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface. 相似文献
19.
John P. D’Angelo 《Journal of Geometric Analysis》2004,14(2):215-229
We first prove a uniqueness result for certain group-invariant CR mappings to hyperquadrics. For cyclic groups these mappings
lead to a collection of polynomials ƒp,q (with integer coefficients) in two variables; here p and q are positive integers. We use the uniqueness result to prove some
surprising number-theoretic results about the ƒp,q, in particular, ƒp,q is congruent to xP + yP modulo (p) (for P ≥ 2) if and only if p is prime. We also determine recurrence relations for these polynomials for q ≤ 3
and determine a functional equation for a generating function. Finally, we discuss the invariant polynomials that arise for
certain representations of dihedral groups to illustrate the non-Abelian case. 相似文献
20.
The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions. 相似文献