{"id":2098,"date":"2021-01-12T09:52:01","date_gmt":"2021-01-12T17:52:01","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=2098"},"modified":"2021-01-26T10:06:07","modified_gmt":"2021-01-26T18:06:07","slug":"applied-math-talk-searching-for-singularities-in-navier-stokes-flows-using-variational-optimization-methods-by-di-kang-mcmaster-university-canada","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/applied-math-talk-searching-for-singularities-in-navier-stokes-flows-using-variational-optimization-methods-by-di-kang-mcmaster-university-canada\/","title":{"rendered":"Applied math. talk: Searching for singularities in Navier-Stokes flows using variational optimization methods by Di Kang, McMaster University, Canada"},"content":{"rendered":"
Abstract: In the presentation we will discuss our research program\r\nconcerning the search for the most singular behaviors possible in viscous\r\nincompressible flows. These events are characterized by extremal growth of \r\nvarious quantities, such as the enstrophy, which control the regularity of the solution. \r\nThey are therefore intimately related to the question of possible singularity formation \r\nin the 3D Navier-Stokes system, known as the\r\nhydrodynamic blow-up problem. We demonstrate how new insights\r\nconcerning such questions can be obtained by formulating them as\r\nvariational PDE optimization problems which can be solved\r\ncomputationally using suitable discrete gradient flows. More\r\nspecifically, such an optimization formulation allows one to identify\r\n\"extreme\" initial data which, subject to certain constraints, leads to\r\nthe most singular flow evolution.  In offering a systematic approach\r\nto finding flow solutions which may saturate known estimates, the\r\nproposed paradigm provides a bridge between mathematical analysis and\r\nscientific computation. In particular, it makes it possible to\r\ndetermine whether or not certain mathematical estimates are \"sharp\",\r\nin the sense that they can be realized by actual vector fields, or if\r\nthese estimates may still be improved. In the presentation we will\r\nreview a number of results concerning 1D and 2D flows characterized by\r\nthe maximum possible growth of different Sobolev norms of the\r\nsolutions.  As regards 3D flows, we focus on the enstrophy which is a\r\nwell-known indicator of the regularity of the solution. We find a family of initial \r\ndata with fixed enstrophy which leads to the largest possible growth of this quantity \r\nat some prescribed final time. Since even with such worst-case initial data the\r\nenstrophy remains finite, this indicates that the 3D Navier-Stokes\r\nsystem reveals no tendency for singularity formation in finite time.\r\n\r\n[joint work with Dongfang Yun and Bartosz Protas]<\/pre>\n","protected":false},"excerpt":{"rendered":"
Abstract: In the presentation we will discuss our research program concerning the search for the most singular behaviors possible in viscous incompressible flows. 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