共查询到20条相似文献,搜索用时 15 毫秒
1.
We study the nonlinear hyperbolic partial differential equation, ( u
t+uu x) x=1/2u
x
2
. This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are no a priori estimates on the second derivatives in any L
p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave. 相似文献
2.
We consider the initial-boundary value problem for a 2-speed system of first-order nonhomogeneous semilinear hyperbolic equations whose leading terms have a small positive parameter. Using energy estimates and a compactness lemma, we show that the diffusion limit of the sum of the solutions of the hyperbolic system, as the parameter tends to zero, verifies the nonlinear parabolic equation of the p-Laplacian type. 相似文献
4.
Archive for Rational Mechanics and Analysis - 相似文献
5.
A new method for determining the overall behavior of composite materials comprised of nonlinear inelastic constituents is presented. Upon use of an implicit time-discretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In this first part of the study the variational technique of Ponte Castañeda is used to approximate the nonuniform eigenstrains by piecewise uniform eigenstrains and to linearize the nonlinear thermoelastic problem. The resulting problem is amenable to simpler calculations and analytical results for appropriate microstructures can be obtained. The accuracy of the proposed scheme is assessed by comparison of the method with exact results. 相似文献
8.
A theory for spatial lattices is presented in a variational setting and conditions restricting stable deformations are discussed. In particular, new results on the second variation of the energy are established and used to generate pointwise necessary conditions for locally energy-minimizing configurations.
Sommario Viene presentata una teoria per i reticoli spaziali in un ambito variazionale e sono inoltre discusse condizioni che limitano deformazioni stabili. In particolare vengono stabiliti nuovi risultati sulla variazione seconda dell'energia. Tali risultati vengono usati per stabilire condizioni necessarie puntuali per configurazioni che minimizzano l'energia localmente. 相似文献
10.
Nonlinear Dynamics - The licence type in the original article was incorrect and should be CC BY and not CC BY NC. 相似文献
12.
This article deals with a boundary-layer problem arising in the kinetic theory of gases when the mean free path of molecules tends to zero. The model considered here is the stationary, nonlinear Boltzmann equation in one dimension with a slightly perturbed reflection boundary condition. We restrict our attention to the case of hard spheres collisions, with Grad's cutoff assumption. Existence, uniqueness and asymptotic behavior are derived by means of energy estimates. 相似文献
13.
Asymptotic wavefront expansions are here employed in the study of nonlinear hyperbolic waves. Numerical results based upon these methods are obtained for a particular case of interest and both these results and those obtained by the method of characteristics are presented graphically. The wavefront-based method is accurate, requires an order of magnitude less computer time, and offers a clearer understanding of the underlying wave process. 相似文献
15.
A series of problems in mechanics and physics are governed by the ordinary Poisson equation which demands linearity, isotropy,
and material homogeneity. In this paper a generalization with respect to nonliearity, anisotropy, and inhomogeneity is made.
Starting from the canonical basic equations in the primal and dual formulation respectively we derive systematically the corresponding
generalized variational principles; under certain conditions they can be extended to so calle complementary extremum principles
allowing for global bounds. For simplicity a restriction to two dimensional problems is made, including twice-connected domains. 相似文献
16.
We consider the parametrized family of equations
tt
, u-
xx
u-au+ u
2
2
u=O, x(0, L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution. 相似文献
17.
Recently, coded permutation entropy has been proposed, which enhances the noise immunity by quadratic partitioning on the basis of permutation entropy. However, coded permutation entropy and permutation entropy only consider the order of amplitude values and ignore some information related to amplitude. To overcome these defects, this paper applies the concept of quadratic partitioning to dispersion entropy (DE), takes advantage of the fact that DE can effectively measure amplitude information, and proposes coded DE (CDE), which increases the number of patterns and improves the divisibility by further coding the dispersion patterns in DE. Moreover, to reduce the computational consumption of CDE, we simplify the division criterion in quadratic partitioning while guaranteeing that no effective information is lost and propose simplified CDE (SCDE). Several simulation experiments demonstrate the advantages of SCDE and CDE over DE, permutation entropy, and coded permutation entropy in detecting the nonlinear dynamic changes within chaotic and synthetic signals. In addition, real-world experiments on electroencephalogram signals, bearing signals, and ship signals show that SCDE has better performance in medical diagnosis, fault diagnosis and signal classification, and the accuracy of SCDE-based classification methods is higher than that of other entropy-based methods. 相似文献
18.
Journal of Dynamics and Differential Equations - In a bounded smooth domain domain Ω?? n the nonlinear hyperbolic evolutionary equation depending on a small parameter... 相似文献
19.
We give sufficient conditions for the global stability of the zero solution of a functional differential equation with pulse
action and with nonlinear function satisfying the conditions of negative feedback and sublinear growth.
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Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 258–269, April–June, 2007. 相似文献
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