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- Weak and Measure-Valued Solutions of the Incompressible Euler Equations
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- Sensitivity equations for measure-valued solutions to transport equations

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Numerical analysis finite volume method mathematical analysis partial differential equations. Applied Mathematics and Mathematical Computation 13 , SIAM journal on numerical analysis 31 2 , , Journal of computational and applied mathematics 44 2 , , Commentationes Mathematicae Universitatis Carolinae 30 3 , , SFB

Citation: Azmy S. Ackleh, Nicolas Saintier, Jakub Skrzeczkowski. Sensitivity equations for measure-valued solutions to transport equations[J]. Mathematical Biosciences and Engineering, , 17 1 : Article views PDF downloads Cited by 0. Azmy S. Mathematical Biosciences and Engineering , , 17 1 :

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Journal of Applied Mathematics and Stochastic Analysis, (), WEAK AND MEASURE-VALUED SOLUTIONS TO. EVOLUTIONARY PDEs.

We propose to solve hyperbolic partial differential equations PDEs with polynomial flux using a convex optimization strategy. This approach is based on a very weak notion of solution of the nonlinear equation,namely the measure-valued mv solution,satisfying a linear equation in the space of Borel measures. The aim of this paper is,first,to provide the conditions that ensure the equivalence between the two formulations and,second,to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex,finite-dimensional,semidefinite programming problems.

Approximate solutions to hyperbolic systems of conservation laws may be generated in a variety of ways: by the method of vanishing viscosity, through difference approximations, by relaxation schemes, etc. The topic for discussion in this chapter is whether solutions may be constructed as limits of sequences of approximate solutions that are only bounded in some L p space. Since the systems are nonlinear, the difficulty lies in that the construction schemes are generally consistent only when the sequence of approximating solutions converges strongly, whereas the assumed L p bounds only guarantee weak convergence: Approximate solutions may develop high frequency oscillations of finite amplitude which play havoc with consistency. The aim is to demonstrate that entropy inequalities may save the day by quenching rapid oscillations thus enforcing strong convergence of the approximating solutions.

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We propose to solve hyperbolic partial differential equations PDEs with polynomial flux using a convex optimization strategy. This approach is based on a very weak notion of solution of the nonlinear equation,namely the measure-valued mv solution,satisfying a linear equation in the space of Borel measures. The aim of this paper is,first,to provide the conditions that ensure the equivalence between the two formulations and,second,to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex,finite-dimensional,semidefinite programming problems.

We construct suitably defined entropy solutions in the space of Radon measures. In the latter case, we describe the evolution of the singular parts. Measure-valued entropy solutions are defined similarly see Definition 3. We use an approximation procedure to construct measure-valued entropy solutions of problem P see Theorem 3. The function in 1. Hence, the following holds. Then, by Proposition 1.

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Brenier , Solution by convex minimization of the Cauchy problem for hyperbolic systems of conservation laws with convex entropy , Brezis , Functional analysis, Sobolev spaces and partial differential equations , DOI : Chernyshenko, P. Goulart, D.

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PDF | On Jan 1, , Yong Jiongmin published Weak and measure-valued solutions to evolutionary PDEs, by J. Málek, J. Nečas, M. Rokyta.