Weak and Measure-valued Solutions to Evolutionary PDEs

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Most of the results that we will need from analysis and the theory of conservation laws will be repeated, but some familiarity with measure theory and Sobolev spaces will be of help.

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Necas, J. Malek, M. Rokyta and M.

Seminars during Fall 2013

Dafermos 13 November , - , room Ulrik Skre Fjordholm Numerical approximation of measure-valued solutions to hyperbolic conservation laws. Abstract: One-dimensional hyperbolic systems of conservation laws are well-posed under the assumption that the initial data is "sufficiently small". However, there is no stability, existence or uniqueness theory for general initial data in multiple dimensions, and certain Cauchy problems might indeed be unstable with respect to initial data.

We advocate the point of view of so-called measure-valued solutions, and give numerical evidence that this might be the correct notion of solutions for hyperbolic conservation laws. We prove the existence and stability of measure-valued solutions in certain special cases, and design numerical algorithms that show strongly convergent behavior in unstable Cauchy problems. Solutions to. Evolutionary PDEs. M ALEK.

Charles University, Prague,. Czech Republic. NE CAS. Charles University, Prague.

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Necas, , available at Book Depository with free delivery worldwide. It provides a rigorous analysis of non-Newtonian. Necas, J. Ruzicka, J. Malek and M. Rokyta ,. Zimmermann, Multidimensional scalar balance laws with discontinuous flux, J. Gwiazda and E. Wiedemann, Generalized entropy method for the renewal equation with measure data, Commun.

USSR Sb. Lions, B.

Seminars during Fall 2013

Perthame and E. Tadmor, A Kinetic formulation of scalar multidimensional conservation laws, J. Michel, P. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. Paris Ser. I 2—4 , — Perthame, General relative entropy inequality: an illustration on growth models, J. Pures Appl.

A moment approach for entropy solutions to nonlinear hyperbolic PDEs

Mischler, B. Perthame and L. Ryzhik, Stability in a nonlinear population maturation model, Math. Rokyta and M. Neustupa, Measure-valued solutions of the Euler and Navier—Stokes equations for compressible barotropic fluids, Math.

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Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws, Commun. Perthame and A. Tzavaras, Kinetic formulation for systems of two conservation laws and elastodynamics, Arch.

Wiedemann, Young measures generated by ideal incompressible fluid flows, Arch. Szepessy, An existence result for scalar conservation laws using measure valued solutions, Comm. Partial Differential Equations 14 , no. Ball, ed. Reidel Publishing Col. Fefferman, J.

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Robinson and J.