Forthcoming Seminars › Seminar Series

OxPDE Lunchtime Seminar

Thu, 23/05
12:00 		Francesco Solombrino (Technical University of Munich) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Quasistatic evolution problems in perfect plasticity for generalized multiphase materials
Inspired by some recents developments in the theory of small-strain elastoplasticity, we both revisit and generalize the formulation of the quasistatic evolutionary problem in perfect plasticity for heterogeneous materials recently given by Francfort and Giacomini. We show that their definition of the plastic dissipation measure is equivalent to an abstract one, where it is defined as the supremum of the dualities between the deviatoric parts of admissible stress fields and the plastic strains. By means of this abstract definition, a viscoplastic approximation and variational techniques from the theory of rate-independent processes give the existence of an evolution statisfying an energy- dissipation balance and consequently Hill's maximum plastic work principle for an abstract and very large class of yield conditions.
Thu, 23/05
12:00 		Daniela Tonon (Imperial College & Institut de Mathématiques de Jussieu) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
SEMINAR MOVED (details will follow)
Thu, 30/05
12:00 		James Robinson (University of Warwick) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
A coupled parabolic-elliptic system arising in the theory of magnetic relaxation
    In 1985 Moffatt suggested that stationary flows of the 3D Euler equations with non-trivial topology could be obtained as the time-asymptotic limits of certain solutions of the equations of magnetohydrodynamics. Heuristic arguments also suggest that the same is true of the system
    \[ -\Delta u+\nabla p=(B\cdot\nabla)B\qquad\nabla\cdot u=0\qquad \]
    \[ B_t-\eta\Delta B+(u\cdot\nabla)B=(B\cdot\nabla)u \]
    when $ \eta=0 $. In this talk I will discuss well posedness of this coupled elliptic-parabolic equation in the two-dimensional case when $ B(0)\in L^2 $ and $ \eta>0 $.
    Crucial to the analysis is a strengthened version of the 2D Ladyzhenskaya inequality: $ \|f\|_{L^4}\le c\|f\|_{L^{2,\infty}}^{1/2}\|\nabla f\|_{L^2}^{1/2} $, where $ L^{2,\infty} $ is the weak $ L^2 $ space. I will also discuss the problems that arise in the case $ \eta=0 $.
    This is joint work with David McCormick and Jose Rodrigo.
Thu, 06/06
12:00 		Mayte Pérez-Llanos (Universidad Autonoma de Madrid) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Numerical approximations for a nonloncal model for sandpiles
    In this talk we study numerical approximations of continuous solutions to a nonlocal $ p $-Laplacian type diffusion equation,
    \[ u_t (t, x) = \int_\Omega J(x − y)|u(t, y) − u(t, x)|^{p-2} (u(t, y) − u(t, x)) dy. \]
    First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one, as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties with the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition, as $ t $ goes to infinity.
    Next, we discretize also the time variable and present a totally discrete method which also enjoys the above mentioned properties.
    In addition, we investigate the limit as $ p $ goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile.
    Finally, we present some numerical experiments that illustrate our results.
    This is a joint work with J. D. Rossi.

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