Past OxPDE Special Seminar

17 May 2013
10:00
Michel Chipot
Abstract
<p>A mini-lecture series consisting of four 1 hour lectures.</p> We would like to consider asymptotic behaviour of various problems set in cylinders. Let $\Omega_\ell = (-\ell,\ell)\times (-1,1)$ be the simplest cylinder possible. A good model problem is the following. Consider $u_\ell$ the weak solution to $$ \cases{ -\partial_{x_1}^2 u_\ell - \partial_{x_2}^2 u_\ell = f(x_2) \quad \hbox{in } \Omega_\ell, \quad \cr \cr u_\ell = 0 \quad \hbox{ on } \quad \partial \Omega_\ell. \cr} $$ When $\ell \to \infty$ is it trues that the solution converges toward $u_\infty$ the solution of the lower dimensional problem below ? $$ \cases{ - \partial_{x_2}^2 u_\infty = f(x_2) \quad \hbox{in }(-1,1), \quad \cr \cr u_\infty = 0 \quad \hbox{ on } \quad \partial (-1,1). \cr} $$ If so in what sense ? With what speed of convergence with respect to $\ell$ ? What happens when $f$ is also allowed to depend on $x_1$ ? What happens if $f$ is periodic in $x_1$, is the solution forced to be periodic at the limit ? What happens for general elliptic operators ? For more general cylinders ? For nonlinear problems ? For variational inequalities ? For systems like the Stokes problem or the system of elasticity ? For general problems ? ... We will try to give an update on all these issues and bridge these questions with anisotropic singular perturbations problems. \smallskip \noindent {\bf Prerequisites} : Elementary knowledge on Sobolev Spaces and weak formulation of elliptic problems.
  • OxPDE Special Seminar
10 May 2013
10:00
Michel Chipot
Abstract
<p>A mini-lecture series consisting of four 1 hour lectures.</p> We would like to consider asymptotic behaviour of various problems set in cylinders. Let $\Omega_\ell = (-\ell,\ell)\times (-1,1)$ be the simplest cylinder possible. A good model problem is the following. Consider $u_\ell$ the weak solution to $$ \cases{ -\partial_{x_1}^2 u_\ell - \partial_{x_2}^2 u_\ell = f(x_2) \quad \hbox{in } \Omega_\ell, \quad \cr \cr u_\ell = 0 \quad \hbox{ on } \quad \partial \Omega_\ell. \cr} $$ When $\ell \to \infty$ is it trues that the solution converges toward $u_\infty$ the solution of the lower dimensional problem below ? $$ \cases{ - \partial_{x_2}^2 u_\infty = f(x_2) \quad \hbox{in }(-1,1), \quad \cr \cr u_\infty = 0 \quad \hbox{ on } \quad \partial (-1,1). \cr} $$ If so in what sense ? With what speed of convergence with respect to $\ell$ ? What happens when $f$ is also allowed to depend on $x_1$ ? What happens if $f$ is periodic in $x_1$, is the solution forced to be periodic at the limit ? What happens for general elliptic operators ? For more general cylinders ? For nonlinear problems ? For variational inequalities ? For systems like the Stokes problem or the system of elasticity ? For general problems ? ... We will try to give an update on all these issues and bridge these questions with anisotropic singular perturbations problems. \smallskip \noindent {\bf Prerequisites} : Elementary knowledge on Sobolev Spaces and weak formulation of elliptic problems.
  • OxPDE Special Seminar
29 April 2013
14:00
Abstract
<p>In this talk, we will introduce how to apply Green's function method to get &nbsp;pointwise estimates for solutions of the Cauchy problem of nonlinear evolution equations with dissipative &nbsp;structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and exhibit the generalized Huygen's principle. Then, for other nonlinear dissipative evolution equations, we will &nbsp;introduce some recent results and give brief explanations. Our approach is based on the detailed analysis of the Green's function of the linearized system and micro-local analysis, such as frequency decomposition and so on.</p>
  • OxPDE Special Seminar
27 November 2012
17:00
Abstract
Several shape optimization problems, e.g. in image processing, biology, or discrete geometry, involve the Willmore functional, which is for a surface the integrated squared mean curvature. Due to its singularity, minimizing this functional under constraints is a delicate issue. More precisely, it is difficult to characterize precisely the structure of the minimizers and to provide an explicit formulation of their energy. In a joint work with Giacomo Nardi (Paris-Dauphine), we have studied an "integrated" version of the Willmore functional, i.e. a version defined for functions and not only for sets. In this talk, I will describe the tools, based on Young measures and varifolds, that we have introduced to address the relaxation issue. I will also discuss some connections with the phase-field numerical approximation of the Willmore flow, that we have investigated with Elie Bretin (Lyon) and Edouard Oudet (Grenoble).
  • OxPDE Special Seminar
16 November 2012
12:30
Dorian Goldman
Abstract
Notice that the time is 12:30, not 12:00! \newline \vskip\baselineskip The following is joint work with Sylvia Serfaty and Cyrill Muratov. We study the asymptotic behavior of the screened sharp interface Ohta-Kawasaki energy in dimension 2 using the framework of Γ-convergence. In that model, two phases appear, and they interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating ``droplets" of that phase in a sea of the other phase. We consider perturbations to the critical volume fraction where droplets first appear, show the number of droplets increases monotonically with respect to the perturbation factor, and describe their arrangement in all regimes, whether their number is bounded or unbounded. When their number is unbounded, the most interesting case we compute the Γ limit of the `zeroth' order energy and yield averaged information for almost minimizers, namely that the density of droplets should be uniform. We then go to the next order, and derive a next order Γ-limit energy, which is exactly the ``Coulombian renormalized energy W" introduced in the work of Sandier/Serfaty, and obtained there as a limiting interaction energy for vortices in Ginzburg-Landau. The derivation is based on their abstract scheme, that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing at all to the Euler-Lagrange equation, we establish here for all configurations which have ``almost minimal energy," the asymptotic roundness and radius of the droplets as done by Muratov, and the fact that they asymptotically shrink to points whose arrangement should minimize the renormalized energy W, in some averaged sense. This leads to expecting to see hexagonal lattices of droplets.
  • OxPDE Special Seminar
13 November 2012
17:00
Miao Shuang (with D. Christodoulou)
Abstract
In this talk I shall discuss about the classical compressible Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. We considered initial data which outside a sphere coincide with the data corresponding to a constant state, we established theorems which gave a complete description of the maximal development. In particular, we showed that the boundary of the domain of the maximal development has a singular part where the inverse density of the wave fronts vanishes, signaling shock formation.
  • OxPDE Special Seminar
19 September 2012
12:00
Laura Spinolo
Abstract
I will be concerned with initial-boundary value problems for systems of conservation laws in one space variable. First, I will go over some of the most relevant features of these problems. In particular, I will stress that different viscous approximation lead, in general, to different limits. Next, I will discuss possible ways of characterizing the limit of a given viscous approximation. Also, I will establish a uniqueness criterion that allows to conclude that the limit of a self-similar approximation introduced by Dafermos et al. actually coincide with the limit of the physical viscous approximation. Finally, if time allows I will mention consequences on the design of numerical schemes. The talk will be based on joint works with S. Bianchini, C. Christoforou and S. Mishra.
  • OxPDE Special Seminar
6 August 2012
17:00
Phoebus Rosakis
Abstract
The energy of a deformed crystal is calculated in the context of a central force lattice model in two dimensions. When the crystal shape is a lattice polygon, it is shown that the energy equals the bulk elastic energy, plus the boundary integral of a surface energy density, plus the sum over the vertices of a corner energy function. This is an exact result when the interatomic potential has finite range; for an infinite-range potential it is asymptotically valid as the lattice parameter tends to zero. The surface energy density is obtained explicitly as a function of the deformation gradient and boundary normal. The corner energy is found as an explicit function of the deformation gradient and the normals of the two facets meeting at the corner. A new bond counting approach is used, which reduces the problem to certain lattice point problems of number theory. The approach is then extended to more general convex regions with possibly curved boundary. The resulting surface energy density depends on the unit normal in a striking way. It is continuous at irrational directions, discontinuous at rational ones and nowhere differentiable. The method also yields an explicit interfacial energy for twin and phase boundaries.
  • OxPDE Special Seminar

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