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1.
Marshall Slemrod Athanasios E. Tzavaras 《Archive for Rational Mechanics and Analysis》1993,122(4):353-392
This report discusses a new approach for the resolution of the fluid-dynamic limit for the Broadwell system of the kinetic theory of gases, appropriate in the case of Riemann, Maxwellian data. Since the formal limiting system is expected to have self-similar solutions, we are motivated to replace the Knudsen number in the Broadwell model so that the resulting model admits self-similar solutions =x/t and then let go to zero. The limiting procedure is justified and the resulting limit is a solution of the Riemann problem for the fluid-dynamic limit equations. A class of Riemann data for which this program can be carried out is exhibited. Furthermore, it is shown that for the Carleman model the complete program can be done successfully for arbitrary Riemann data. 相似文献
2.
We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave. 相似文献
3.
Manuel Portilheiro Athanasios E. Tzavaras 《Transactions of the American Mathematical Society》2007,359(2):529-565
We consider a class of kinetic equations, equipped with a single conservation law, which generate -contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the ``dissipative' sense, equivalent to the notion of entropy solutions for conservation laws, which permits the use of the perturbed test function method and allows for simple proofs. A general compactness framework is obtained for the diffusive scaling in . The radiative transport equations, satisfied by the Wigner function for random acoustic waves, present such a kinetic model that is endowed with conservation of energy. The general theory is used to validate the diffusive approximation of the radiative transport equation.
4.
Benoit Perthame Athanasios E. Tzavaras 《Archive for Rational Mechanics and Analysis》2000,155(1):1-48
For scalar conservation laws, the kinetic formulation makes it possible to generate all the entropies from a simple kernel.
We show how this concept replaces and simplifies greatly the concept of Young measures, avoiding the difficulties encountered
when working in L
p
. The general construction of the two kinetic functions that generate the entropies of 2 × 2 strictly hyperbolic systems
is also developed here. We show that it amounts to building a “universal” entropy, i.e., one that can be truncated by a “kinetic
value” along Riemann invariants. For elastodynamics, this construction can be completed and specialized using the additional
Galilean invariance. This allows a full characterization of convex entropies. It yields a kinetic formulation consisting of
two semi-kinetic equations which, as usual, are equivalent to the infinite family of all the entropy inequalities.
Accepted May 29, 2000?Published online November 16, 2000 相似文献
5.
We devise Lyapunov functionals and prove uniform L1 stability for one-dimensional semilinear hyperbolic systems with quadratic nonlinear source terms. These systems encompass a class of discrete velocity models for the Boltzmann equation. The Lyapunov functional is equivalent to the L1 distance between two weak solutions and non-increasing in time. They result from computations of two point interactions in the phase space. For certain models with only transversal collisional terms there exist generalizations for three and multi-point interactions. 相似文献
6.
Sophia Demoulini David M. A. Stuart Athanasios E. Tzavaras 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2000,17(6):711
It is shown that the variational approximation scheme for one-dimensional elastodynamics given by time discretisation converges, subsequentially, weakly and a.e. to a weak solution which satisfies the entropy inequalities. We also prove convergence under the restriction of positive spatial derivative (for longitudinal motions). 相似文献
7.
Athanassios E. Tzavaras 《Archive for Rational Mechanics and Analysis》1986,94(1):39-58
We consider the plastic shearing of a strain-rate dependent material exhibiting strain hardening or strain softening, subjected to steady shearing. We establish the existence of classical solutions and study the stability of uniform shearing. For materials exhibiting strain hardening or a moderate degree of strain softening we show that, as t , every solution approaches, at specific rates of convergence, uniform shearing; thus shear bands do not form. 相似文献
8.
Sophia?DemouliniEmail author David?M.?A.?Stuart Athanasios?E.?Tzavaras 《Archive for Rational Mechanics and Analysis》2012,205(3):927-961
For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of a dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data, when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging. 相似文献
9.
Sophia Demoulini David M. A. Stuart Athanasios E. Tzavaras 《Archive for Rational Mechanics and Analysis》2001,157(4):325-344
We construct a variational approximation scheme for the equations of three-dimensional elastodynamics with polyconvex stored
energy. The scheme is motivated by some recently discovered geometric identities (Qin [18]) for the null Lagrangians (the determinant and cofactor matrix), and by an associated embedding of the equations of
elastodynamics into an enlarged system which is endowed with a convex entropy. The scheme decreases the energy, and its solvability
is reduced to the solution of a constrained convex minimization problem. We prove that the approximating process admits regular
weak solutions, which in the limit produce a measure-valued solution for polyconvex elastodynamics that satisfies the classical
weak form of the geometric identities. This latter property is related to the weak continuity properties of minors of Jacobian
matrices, here exploited in a time-dependent setting.
Accepted November 18, 2000?Published online April 23, 2001 相似文献
10.
Abstract By exploring a local geometric property of the vorticity field along a vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum vortex stretching. This new understanding leads to an improved result of the global existence of the 3D Euler equation under mild assumptions. 相似文献