Renormalized and entropy solutions of partial differential equations 1000-1M12RER
For most of equations, that are significant from the point of view of mathematical physics, it is impossible to show the existence of classical solutions, ie, solutions which are continuously differentiable as many times as the order derivatives in equations under consideration. On the other hand, the concept of distribution
solutions (or weak solutions) appears to be too poor and does not allow us to pose certain problems correctly. As an example, we can point out the lack of the uniqueness in the class distribution solutions to the simplest Burgers equation or other pathologies which appear in the case of distribution solutions to many
partial differential equations. Hence, as a certain more general idea, we can use the notion of the entropy or the renormalization, that is to postulate that (in addition
to the weak formulation of the problem) the equation satisfies certain additional weak formulation in entropic or renormalized sense (for a sufficiently rich family of entipies/renormalizations). This program (derived essentially from hyperbolic conservation laws) has found applications in many other equations: elliptic equations, parabolic equations, the Boltzmann equation, the transport equation, the Navier-Stokes equation for compressible fluid or with variable density. As another application of the entropy method, one should mention the method of relative entropy, used to analyze the long time behavior of solutions to partial differential equations. This program was successfully applied in the case of equations such as the Fokker-Planck equation, the nonlinear diffusion equation (the porous medium equation), and other equations from mathematical biology. Since this is an introductory course, we shall concentrate on the following simplest and selected issues:
- The Kruzkov theory entropy solutions for scalar hyperbolic equations [1];
- the DiPerny-Lions theory of renormalized solutions to the transport equation [2-5];
- theory of renormalized and entropy solutions to elliptic and parabolic equations [6-7];
- the method of relative entropy in dynamics of biological populations [8-9]
Type of course
Mode
Bibliography
0. P. Gwiazda Renormalized and entropy solutions of partial differential equations, http://ssdnm.mimuw.edu.pl/pliki/wyklady/PGwiazda_skrypt_UWr.pdf
1. S.N. Kru zkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243.
2. R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae Volume 98, Number 3 (1989), 511-547,
3. R. J. DiPerna and P. L. Lions, On the cauch problem for Boltzmann equations: Global existence an weak stability, Annals of Mathematics,
130 (1989), 321-366
4. L.Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004), 227-260
5. Piotr Mucha, Transport equation: Extension of classical results for div b in BMO. J. Diff. Equations 249, s. 1871-1883, 2010.
6. P. Benilan, L. Boccardo, T. Gallou¨et, R. Gariepy, M. Pierre, J.L. Vazquez, An L1 theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 241–273
7. P. Gwiazda, P. Wittbold, A. Wróblewska and Zimmermann Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces, J. Differ. Equations 253, (2) pp. 635–666 (2012)
8. Perthame, Benoît, Transport Equations in Biology, Series: Frontiers in Mathematics 2007, Birkhäuser Basel
9. G. Karch, Rownanie ciepla, Torun 2002, http://www.math.uni.wroc.pl/conferences/torun/
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