Youtor
Singularities in Partial Differential Equations Marco Fontelos LMS AGM 2024
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=u72yfK-b_JI
Abstract: Starting from smooth initial data, solutions to evolutionary nonlinear Partial Differential Equations may develop singularities in finite time. These singularities can be such that solutions diverge (blow-up), or loose continuity (shocks), etc. A fundamental concept in the context of singularities is that of selfsimilarity. We use a similarity transformation of the original equation with respect to the singular point, such that self-similar behaviour is mapped to the fixed point of a dynamical system. We point out that analysing the dynamics close to the fixed point is a useful way of characterizing the singularity, in that the dynamics frequently reduces to very few dimensions. • Once the problem reduces to a finite number of degrees of freedom, one can use ideas from finite dimensional dynamical systems: attractors, center manifolds, bifurcations, etc. Finally, once a singularity develops, the issue fn how one can continue the solution to a PDE after the singularity arises. We will present these problems with examples of well-known Partial Differential Equations and systems and point to the many open mathematical problems, as well as to the current state of the art in the attempts to solve them. We put special emphasis on problems arising in connection to the Euler and Navier-Stokes equations of Fluid Mechanics. • Part of the LMS Annual General Meeting 2024, which took place at BMA House on Friday 22 November 2024. • ========== • The London Mathematical Society has, since 1865, been the UK's learned society for the advancement, dissemination and promotion of mathematical knowledge. Our mission is to advance mathematics through our members and the broader scientific community worldwide. • For further information: • ► Website: https://www.lms.ac.uk • ► Events: https://www.lms.ac.uk/events • ► Grants and Prizes: https://www.lms.ac.uk/grants-prizes • ► Publications: https://www.lms.ac.uk/publications • ► Membership: https://www.lms.ac.uk/membership • Follow us: • ► Twitter: / londmathsoc • ► Facebook: / londonmathematicalsociety • ► LinkedIn: / the-london-mathematical-society • ► Youtube: @LondonMathematicalSociety
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