Centre de diffusion de revues académiques mathématiques


Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Ángel Castro
Mixing solutions for IPM
Journées équations aux dérivées partielles (2017), Exp. No. 3, 13 p., doi: 10.5802/jedp.653
Article PDF

Résumé - Abstract

We explain the main steps in the proof of the existence of mixing solutions of the incompressible porous media equation for all Muskat type $H^5$ initial data in the fully unstable regime which appears in [4]. Also we present some numerical simulations about these solutions.


[1] T. Beck, P. Sosoe and P. Wong, Duchon-Robert solutions for the Rayleigh-Taylor and Muskat problems. J. Differ. Equations 256, no. 1, 206-222, 2014.
[2] A. C. Bronzi, M. C. Lopes Filho and H. J. Nussenzveig Lopes, Wild solutions for 2D incompressible ideal flow with passive tracer. Commun. Math. Sci. 13, no. 5, 1333–1343 (2015).
[3] T. Buckmaster, C. De Lellis, P. Isett and L. Székelyhidi Jr., Anomalous dissipation for 1/5-Holder Euler flows. Ann. of Math. 182, 127–172 (2015).
[4] Á. Castro, D. Córodoba and D. Faraco, Mixing solutions for the Muskat problem. ArXiv:1605.04822
[5] Á. Castro, D. Córdoba, C. Fefferman, F. Gancedo and M. López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with application to water waves. Ann. of Math. 175 , 909–948 (2012).
[6] E. Chiodaroli, A counterexample to well-posedeness of entropy solutions to the compressible Euler system. J. Hyperbol. Differ. Eq. 11, 493-519 (2014).
[7] E. Chiodaroli, C. De Lellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math. 68, 1157–1190 (2015).
[8] E. Chiodaroli, E. Feireisl and O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas. Annales IHP-ANL 32, 225–243 (2015).
[9] E. Chiodaroli, O. Kreml, On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System. Arch. Ration. Mech. Anal. 214, 1019-1049 (2014).
[10] A. Choffrut, h-Principles for the Incompressible Euler Equations. Arch. Rational Mech. Anal. 210, no. 1, 133–163 (2013).
[11] P. Constantin, D. Córdoba, F. Gancedo, L. Rodriguez-Piazza and R. M. Strain, On the Muskat problem: global in time results in 2D and 3D. ArXiv:1310.0953, (2014).
[12] P. Constantin, D. Córdoba, F. Gancedo and R. M. Strain, On the global existence for the Muskat problem. J. Eur. Math. Soc. 15, 201-227, (2013).
[13] P. Constantin, F. Gancedo, V. Vicol and R. Shvydkoy, Global regularity for 2D Muskat equation with finite slop. ArXiv: 150701386, (2015).
[14] A. Córdoba, D. Córdoba and F. Gancedo, Interface evolution: the Hele-Shaw and Muskat problems. Ann. of Math. 173, no. 1, 477-542, (2011).
[15] D. Córdoba, D. Faraco, and F. Gancedo, Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch. Ration. Mech. Anal. 200, no. 3, 725–746 (2011).
[16] D. Córdoba and F. Gancedo, Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Comm. Math. Phys. 273, no. 2, 445–471 (2007).
[17] C. H. A. Cheng, R. Granero-Belinchón and S. Shkoller, Well-possednes of the Muskat problem with $H^2-$inital data. Adv. Math. 286, 32–104 (2016).
[18] S. Daneri, Cauchy Problem for Dissipative Holder Solutions to the Incompressible Euler Equations. Comm. Math. Phys. 329, no. 2, 745–786 (2014).
[19] H. Darcy, Les Fontaines Publiques de la Ville de Dijon. Dalmont, Paris, 1856.
[20] C. De Lellis and L. Székelyhidi Jr, The Euler equations as a differential inclusion. Ann. of Math. 170, 1417–1436 (2009).
[21] C. De Lellis and L. Székelyhidi Jr, On Admissibility Criteria for Weak Solutions of the Euler Equations. Arch. Rational Mech. Anal. 195, 225–260 (2010).
[22] C. De Lellis and L. Székelyhidi Jr, The h-principle and the equations of fluid dynamics. Bull. Amer. Math. Soc. (N.S.) 49, no. 3, 347–375 (2012).
[23] C. De Lellis and L. Székelyhidi Jr, Dissipative continuous Euler flows. Invent. Math. 193, no. 2, 377–407 (2013).
[24] C. De Lellis and L. Székelyhidi Jr, Dissipative Euler flows and Onsager’s conjecture. J. Eur. Math. Soc. 16, no. 7, 1467–1505 (2014).
[25] F. Clemens and L. Székelyhidi Jr, Piecewise constant subsolutions for the Muskat problem. ArXiv:1709.05155.
[26] F. Gancedo, A survey for the Muskat problem and a new estimate. SeMA J. 74, no. 1, 21–35, (2017).
[27] I. L. Hwang, The $L^2$-Boundedness of Pseudodifferential Operators. Trans. Amer. Math. Soc., 302, 1, 55–76, (1987).
[28] P. Isett and V. Vicol, Holder continuous solutions of active scalar equations. Ann. of PDE 1, no. 1, 1-77 (2015).
[29] B. Kirchheim, Rigidity and Geometry of Microstructures. Habilitation thesis, University of Leipzig, 2003.
[30] B. Kirchheim, S. Müller and V. Šverák, Studying nonlinear PDE by geometry in matrix space. Geometric analysis and nonlinear partial differential equations, 347–395.
[31] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. 157, 715–742 (2003).
[32] F. Otto, Evolution of microstructure in unstable porous media flow: a relaxational approach. Comm. Pure Appl. Math. 52, no. 7, 873–915 (1999).
[33] K. Seonghak and Y. Baisheng, Convex integration and infinitely many weak solutions to the Perona-Malik equation in all dimensions. SIAM J. Math. Anal. 47, no. 4, 2770–2794 (2015).
[34] R. Shvydkoy, Convex integration for a class of active scalar equations. J. Amer. Math. Soc. 24, 1159–1174 (2011).
[35] M. Siegel, R. Caflisch and S. Howison, Global existence, singular solutions and and ill-posedness for the Musakt Problem. Comm. Pure and Appl. Math. 57, 1374-1411, (2004).
[36] L. Székelyhidi Jr., Relaxation of the incompressible porous media equation. Ann. Sci. Éc. Norm. Supér. 45, no. 3, 491–509 (2012).
[37] L. Székelyhidi Jr., Weak solutions to the incompressible Euler equations with vortex sheet initial data. C. R. Math. Acad. Sci. Paris 349, no. 19-20, 1063–1066 (2011).
[38] L. Székelyhidi Jr and E. Wiedemann, Young measures generated by ideal incompressible fluid flows. Arch. Rational Mech. Anal. 206, no. 1, 333–366 (2012).
[39] L. Tartar, Incompressible fluid flow in a porous medium: convergence of the homogenization process. E. Sanchez-Palencia (Ed.), Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag, Berlin (1980), pp. 368-377.
Copyright Cellule MathDoc 2019 | Crédit | Plan du site