Hyperbolic Systems of Conservation Laws
TLDR
In this article, the authors present a survey of recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension and present some of the latest results on uniqueness and stability of entropy weak solutions.Abstract:
This is a survey paper, written in the occasion of an invited talk given by the author at the Universidad Complutense in Madrid, October 1998. Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions.read more
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Finite Volume Methods for Hyperbolic Problems
TL;DR: The CLAWPACK software as discussed by the authors is a popular tool for solving high-resolution hyperbolic problems with conservation laws and conservation laws of nonlinear scalar scalar conservation laws.
Journal ArticleDOI
Onsager and the theory of hydrodynamic turbulence
TL;DR: Onsager's contributions to the field of hydrodynamic turbulence are summarized in this paper, with a discussion of the historical context of the work and a brief speculation as to why Onsager may have chosen not to publish several significant results.
Posted Content
Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems
TL;DR: In this article, the authors considered the Cauchy problem for a strictly hyperbolic, $n\times n$ system in one space dimension, and showed that the solutions of the viscous approximations are defined globally in time and satisfy uniform BV estimates, independent of the initial data.
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Some Open Problems in Elasticity
TL;DR: In this paper, some outstanding open problems of nonlinear elasticity are described, ranging from questions of existence, uniqueness, regularity and stability of solutions in statics and dynamics to issues such as the modelling of fracture and self-contact, the status of elasticity with respect to atomistic models, the understanding of microstructure induced by phase transformations, and the passage from 3D elasticity to models of rods and shells.
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Quadratic BSDEs with convex generators and unbounded terminal conditions
Philippe Briand,Ying Hu +1 more
TL;DR: In this article, Briand and Hu proved an existence result for BSDEs with quadratic generators with respect to the variable z and with unbounded terminal conditions, but no uniqueness result was stated in that work.
References
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Measure theory and fine properties of functions
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
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Systems of Quasilinear Equations and Their Applications to Gas Dynamics
TL;DR: In this article, a new edition, revised and augmented by results of the last decade, of the work of the same title published in 1968 by 'Nauka', is devoted to mathematical questions of gas dynamics.
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Formation of singularities in one‐dimensional nonlinear wave propagation
TL;DR: In this paper, it was shown that if the system (1) is "genuinely nonlinear" in a sense defined below, and if the initial data are "sufficiently small" (but not identically 0), the first derivatives of u will become infinite for certain (x, t) with t > 0.