International conference Partial Differential Equations and Applications in Honour of Mark Vishik on the occasion of his 90th birthday IITP, Moscow, Russia June 4-7, 2012 Confirmed participants 1. Mikhail Agranovich, Moscow Institute of Electronics and Mathematics (abstract) 2. Anatoli Babin, University of California -- Irvine (abstract) 3. Claude Bardos, Université Pierre et Marie Curie (abstract) 4. Sergey Bezrodnykh, Dorodnitsyn Computing Center (abstract) 5. Natalia Chalkina, Moscow State University (abstract) 6. Vladimir Chepyzhov, Institute for Information Transmission Problems (abstract) 7. Alexander Demidov, Moscow State University (abstract) 8. Sergey Dobrokhotov, Ishlinski Institute for Problems in Mechanics (abstract) 9. Stamatis Dostoglou, University of Missouri (abstract) 10. Julii Dubinskii, Moscow Power Engineering Institute (TBA) 11. Alexander Dynin, Ohio State University (abstract) 12. Grigory Eskin, University of California -- Los Angeles (abstract) 13. Mark Freidlin, University of Maryland (abstract) 14. Leonid Friedlander, University of Arizona (abstract) 15. Andrei Fursikov, Moscow State University (abstract) 16. Vakha Gishlarkaev, Chechen State University 17. Evgeny Gorin, Moscow State Pedagogical University (abstract) 18. Andrey Goritsky, Moscow State University 19. Alexander Grigoryan, University of Bielefeld (abstract) 20. Nikolay Gusev, Moscow Institute for Physics and Technology (abstract) 21. Alain Haraux, Université Pierre et Marie Curie (abstract) 22. Yulij Ilyashenko, MSU, IUM, NRU HSE, Cornell (abstract) 23. Alexei Ilyin, Keldysh Institute of Applied Mathematics (abstract) 24. Valeriy Imaykin, Research Institute of Innovative Strategies for General Education Development (abstract) 25. Valentina Ipatova, Moscow Institute for Physics and Technology (abstract) 26. Shoshana Kamin, Tel Aviv University 27. Aleksey Kapustyan, Kyiv National Taras Shevchenko University (abstract) 28. Alexander Komech, Institute for Information Transmission Problems (abstract) 29. Andrey Komech, Institute for Information Transmission Problems (abstract) 30. Elena Kopylova, Institute for Information Transmission Problems (abstract) 31. Alexander Krasnoselskii, Institute for Information Transmission Problems (abstract) 32. Victor Kozyakin, Institute for Information Transmission Problems (abstract) 33. Sergei Kuksin, Ecole Polytechnique (abstract) 34. Jean-Pierre Lohéac, Ecole Centrale de Lyon (abstract) 35. Andrey Lyapin, Russian State Technological University (abstract) 36. Vadim Malyshev, Moscow State University (abstract) 37. Victor Maslov, Moscow State University (abstract) 38. Alain Miranville, Université de Poitiers (abstract) 39. Stanislav Molchanov, UNC -- Charlotte (abstract) 40. Olga Mukina, Journal "Matematicheskii Sbornik" 41. Nikolai Nadirashvili, Institute for Information Transmission Problems (abstract) (to be confirmed) 42. Louis Nirenberg, Courant Institute (abstract) (to be confirmed) 43. Alexander Ovseevich, Institute for Problems in Mechanics (abstract) 44. Victor Palamodov, Tel Aviv University (abstract) 45. Boris Paneah, Technion (abstract) 46. Vittorino Pata, Politecnico di Milano (abstract) 47. Andrey Piatnitski, Narvik Institute of Technology and Lebedev Physical Institute (abstract) 48. Stanislav Pohozhaev, Steklov Mathematical Institute (abstract) 49. Olga Pyrkova, Moscow Institute for Physics and Technology (abstract) 50. Evgeny Radkevich, Moscow State University (abstract) 51. Nikita Ratanov, Universidad del Rosario (abstract) 52. Igor Rudakov, Bryansk State University (abstract) 53. Vsevolod Sakbaev, Moscow Institute for Physics and Technology (abstract) 54. Andrei Shafarevich, Moscow State University (abstract) 55. George Sell, University of Minnesota (abstract) 56. Armen Shirikyan, University of Cergy-Pontoise (abstract) 57. Alexander Shnirelman, Concordia University (abstract) 58. Mikhail Shubin, Northeastern University (abstract) 59. Yakov Sinai, Princeton University (to be confirmed) (abstract) 60. Elena Sitnikova, Moscow State University of Civil Engineering (abstract) 61. Sergey Skorokhodov, Dorodnitsyn Computing Centre 62. Alexander Skubachevskii, Peoples' Friendship University of Russia (abstract) 63. Vsevolod Solonnikov, Steklov Mathematical Institute, St.-Petersburg (abstract) 64. Tatiana Suslina, St.-Petersburg State University (abstracts) 65. Nikolai Tarkhanov, Universität Potsdam (abstract) 66. Roger Temam, Indiana University (abstract) 67. Vladimir Tikhomirov, Moscow State University 68. Edriss Titi, University of California -- Irvine (abstract) 69. Dmitry Treschev, Steklov Mathematical Institute (abstract) 70. Gérard Tronel, UPMC Paris VI 71. Boris Vainberg, University of North Carolina -- Charlotte (abstract) 72. Vladimir Vlasov, Dorodnitsyn Computing Center (abstract) 73. Nikita Vvedenskaya, Institute for Information Transmission Problems (abstract) 74. Wolfgang Wendland, University of Stuttgart (abstract) 75. Ingo Witt, Universität Göttingen (abstract) 76. Hisao Fujita Yashima, Università di Torino (abstract) 77. Vladimir Zakharov, University of Arizona, Lebedev Institute of Physics, Novosibirsk State University (abstract) 78. Sergey Zelik, University of Surrey (abstract) Remarks on strongly elliptic systems in Lipschitz domains M.S. Agranovich Moscow Institute of Electronics and Mathematics, Moscow, Russia [email protected] We discuss some fundamental facts of the theory of strongly elliptic second- order systems in bounded Lipschitz domains. We propose a simplified choice of the right-hand side of the system and the conormal derivative in the Green formula. Using “Weyl’s decomposition” of the space of solutions, we obtain two- sided estimates for solutions of the Dirichlet and Neumann problems. We remove the algebraic restriction in the generalized Savar´etheorem on the regularity of solutions of these problems for systems with Hermitian principal part. The corollaries for potential type operators and Poincar´e–Steklov operators on the boundary are strengthened. We consider the transmission problems for two systems in domains with common Lipschitz boundary without assumption of the absence of jumps in the coefficients on this boundary. We construct examples of strongly elliptic second-order systems, for which the Neumann problem does not have the Fredholm property. Infinite energy solutions for damped Navier–Stokes equations in R2 Sergey Zelik University of Surrey, Guildford, United Kingdom [email protected] The so-called damped Navier–Stokes equations in the whole 2D space: ∂tu +(u, ∇x)u = ∆xu − αu + ∇xp + g, u , u u , (div = 0 t=0 = 0 where α is a positive parameter, will be considered and the results on the global well-posedness, dissipativity and further regularity of weak solutions of this 2 2 problem in the uniformly-local spaces Lb (R ) will be presented. These results are obtained based on the further development of the weighted energy theory for the Navier–Stokes type problems. Note that any divergent free vector field ∞ 2 u0 ∈ L (R ) is allowed and no assumptions on the spatial decay of solutions as |x| → ∞ are posed. In addition, the applications to the classical Navier–Stokes problem in R2 (which corresponds to α = 0) will be also considered. In particular, the improved estimate on the possible growth rate of spatially non-decaying solutions as time goes to infinity: 5 u t 2 2 C t , ( )Lb (R ) ≤ ( + 1) where C depends on u0 and g, but is independent of t, will be presented. Note that the previous best known estimate was super-exponential in time: 2 C2t u(t)L∞(R2) ≤ C1e , see [1]. References [1] O. Sawada and Y. Taniuchi, A remark on L∞-solutions to the 2D Navier– Stokes equations, J. Math. Fluid Mech., 9 (2007), 533–542. Is free surface deep water hydrodynamics an integrable system? Vladimir Zakharov University of Arizona, Lebedev Institute of Physics, Novosibirsk State University [email protected] Equation of coagulation process of falling drops Hisao Fujita Yashima Universit´e8 Mai 1945, Guelma, Alg´erie and Universit`adi Torino, Italy [email protected] We consider coagulation process of water drops which fall in the air. This process is described by an integro-differential equation for the density σ(m,t,x) of the water liquid contained in drops of mass m. For the motion of drops we consider the their velocity u(m) determined by the mass m and the velocity of the air; on the other hand for the coagulation process we consider a probability β(m1,m2) of meeting between a drop of mass m1 and one of mass m2 (see for example [3]). First we prove the existence of a stationay solution with a constant horizontal wind [2]. Secondly we prove the existence and uniqueness of the global solution in the absence of the wind [1]; the convergence of the global solution to the stationary solution is a corollary of this result. Уравнение процесса когуляции падающих капель Мы рассматриваем процесс когуляции падающих в воздухе водяных ка- пель, который описывается интегро-дифференциальным уравнением для плотности σ(m,t,x) воды, которая содержится в каплях массы m. Для дви- жения капель мы рассматриваем их скорость u(m), которое определяется массой m и скоростью воздуха, а для процесса когуляции мы рассматри- ваем вероятность β(m1,m2) встречи капли массы m1 и капли массы m2 (см. например [3]). Во-первых будем доказывать существование стационарного решения с постоянным ветром [2]. Во-вторых будем доказывать существование и един- ственность глобального решения в отсутствие ветра [1]; отсюда следует так- же сходимость глобального решения к стационарному решению. References