Mathematisches Forschungsinstitut Oberwolfach Report No. 39/2012 DOI: 10.4171/OWR/2012/39 Mathematical Aspects of Hydrodynamics Organised by Peter Constantin, Chicago Susan Friedlander, Los Angeles Gregory Seregin, Oxford Edriss S. Titi, Irvine 12th August – 18th August 2012 Abstract. The workshop dealt with the partial differential equations that describe fluid motion, namely the Euler equations and the Navier-Stokes equations. Many of the lectures concerned questions of existence and regularity of solutions, possibly weak solutions, of these equations and somewhat simpler variants. Mathematics Subject Classification (2000): 76, 35. Introduction by the Organisers The workshop “Mathematical Aspects of Hydrodynamics” was held in August 1218, 2012. For personal reasons the original organisers, Peter Constantin (Princeton) and Gregory Seregin (Oxford), were regrettably unable to attend the workshop. Two additional organisers, Susan Friedlander (Los Angeles) and Edriss S. Titi (Irvine and Rehovot) were added and they were present throughout the workshop. The scientific program consisted of 26 main talks (about 40 minutes each with additional 20 minutes were allocated for discussion) and 3 shorter contributions (posters plus 10 minute talks). There was plenty of time for discussions during and after the lectures and in private groups. The emphasis of the meeting was on various mathematical facets of incompressible fluid dynamics, however one session was devoted to issues connected with compressible flows. A lecture on an important, very new result concerning the existence of dissipative, Holder continuous, incompressible solutions for the Euler flows opened the workshop. The main topics discussed at the meeting included: turbulence and weak solutions of the Euler equation, regularity questions 2390 Oberwolfach Report 39/2012 for the Navier-Stokes equations, breaking solutions to the inviscid water wave equation, active scalar equations, issues of stability and instability, the limit of vanishing viscosity, aspects of magnetohydrodynamics, new numerical schemes for the Navier-Stokes equations, and stochastic systems related to fluid models. There were 42 participants from 13 different countries, namely Germany, US, UK, Russia, France, Spain, Czech Republic, Lithuania, Canada, China, Japan and Korea. One organiser and one postdoc were women. Approximately 10 participants were young researchers who came to Oberwolfach for the first time. The afternoon of the hike was very warm but a good number participated in the hike and everyone enjoyed the barbecue on the patio after the hike. A delightful musical trio provided entertainment. The organisers thank the Institute staff for their great hospitality and support before and during the conference which was very well run. Financial support for young participants from the Leibniz Association and the National Science Foundation is gratefully acknowledged. Mathematical Aspects of Hydrodynamics 2391 Workshop: Mathematical Aspects of Hydrodynamics Table of Contents László Székelyhidi Jr. (joint with Camillo De Lellis) Dissipative Hölder continuous Euler flows . . . . . . . . . . . . . . . . . . . . . . . . . . 2393 Vlad Vicol (joint with Luis Silvestre, Andrej Zlatos) Finite time blow-up in supercritical drift-diffusion equations . . . . . . . . . . . 2394 Charles R. Doering Incompressible stirring and diffusion-less mixing . . . . . . . . . . . . . . . . . . . . . 2395 Alexander Kiselev (joint with Michael Dabkowski, Luis Silvestre, Vlad Vicol) Recent results on supercritical SQG equation . . . . . . . . . . . . . . . . . . . . . . . . 2395 Josef Málek Large Data Analysis of Kolmogorov’s Two Equation Model of Turbulence 2397 Sergei Kuksin Navier-Stokes system with random force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2397 Diego Córdoba (joint with Angel Castro, Charles Fefferman, Francisco Gancedo, Javier Gomez-Serrano and Maria Lopez-Fernandez) Finite time singularities for the free boundary incompressible Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2398 Koji Ohkitani Phenomenology of intermittency and regularity problems in Navier-Stokes turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2399 C. Eugene Wayne (joint with Margaret Beck) Metastability in the two-dimensional Navier-Stokes equation . . . . . . . . . . 2401 Eduard Feireisl Weak and strong solutions to the full Navier-Stokes-Fourier system . . . . 2402 Gui-Qiang G. Chen Stability of Vortex Sheets and Related Problems . . . . . . . . . . . . . . . . . . . . . 2402 Zhouping Xin (joint with Wei Yan) Finite-Time Blow-Up of Classical Solutions to The Full Compressible Navier-Stokes System with Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2404 Claude Bardos (joint with László Székelyhidi, Edriss S. Titi and Emile Wiedemann) The vanishing viscosity limit of the Navier-Stokes equations . . . . . . . . . . . 2406 Jiahong Wu The 2D Boussinesq equations with partial dissipation . . . . . . . . . . . . . . . . . 2407 2392 Oberwolfach Report 39/2012 Yann Brenier Transport and diffusion of 3D divergence-free vector fields . . . . . . . . . . . . 2407 Vsevolod A. Solonnikov On free boundary problem of magnetohydrodynamics in multi-connected domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410 Gautam Iyer (joint with Mihaela Ignatova, James P. Kelliher, Robert L. Pego and Arghir D. Zarnescu) Existence, Stability and Coercivity of an extended Navier-Stokes system 2412 Alexis F. Vasseur (joint with Kyudong Choi, Nicholas Leger) Relative entropy applied to shocks for the compressible Euler equation . . 2413 Alexey Cheskidov (joint with Roman Shvydkoy) Euler equations and turbulence: analytical approach to intermittency . . . 2414 Thierry Gallay (joint with Yasunori Maekawa) Long-time asymptotics for the Navier-Stokes equation in a two-dimensional exterior domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2416 Francisco Gancedo (joint with A. Castro, P. Constantin, D. Córdoba, C. Fefferman, M. López-Fernández and R.M. Strain) Formation of singularities and global-in-time results for the Muskat problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2417 Hideo Kozono (joint with Takahiro Okaba, Hidenori Takahashi) Uniqueness criterion of weak solutions to the Navier-Stokes equations in general unbounded domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2419 Konstantin Pileckas On one J. Leray’s Problem in the Theory of Navier–Stokes Equations . . 2420 Jean-Pierre Eckmann Non-equilibrium steady states, a (progress) report . . . . . . . . . . . . . . . . . . . . 2422 Pierre Gilles Lemarié-Rieusset (joint with Imene Hachicha, Valeria Banica) Hyperbolic approximation for the Navier–Stokes equations : convergence under minimal regularity requirements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2422 Dongho Chae On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2423