Past Seminars › Seminar Series

OxPDE Lunchtime Seminar (past)

Thu, 16/05
12:00 		Paolo Secchi (University of Brescia) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
The plasma-vacuum interface problem with external excitation
    We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma density does not go to zero continuously at the interface, but has a jump, meaning that it is bounded away from zero in the plasma region and it is identically zero in the vacuum region. The plasma-vacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations.
    Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev spaces.
    The proof follows from the well-posedness of the homogeneous linearized problem and a basic a priori energy estimate, the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash-Moser iteration.
    This is a joint work with Y. Trakhinin (Novosibirsk).
Wed, 15/05
12:00 		Cleopatra Christoforou (University of Cyprus) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Decay of positive waves to hyperbolic systems of balance laws
Historically, decay rates have been used to provide quantitative and qualitative information on the solutions to hyperbolic conservation laws. Quantitative results include the establishment of convergence rates for approximating procedures and numerical schemes. Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time. In this talk, I will present two decay estimates on the positive waves for systems of hyperbolic and genuinely nonlinear balance laws satisfying a dissipative mechanism. The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics. Using this result on the spreading of rarefaction waves, the rate of convergence for vanishing viscosity approximations to hyperbolic balance laws will also be established. The proof relies on error estimates that measure the interaction of waves using suitable Lyapunov functionals. If time allows, a further application of the recent developments in the theory of balance laws to differential geometry will be addressed.
Thu, 09/05
12:01 		Šárka Nečasová (Academy of Sciences of the Czech Republic) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains and incompressible limits
We consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behaviour, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained. Secondly, we suppose that the characteristic speed of the fluid is domi- nated by the speed of sound and perform the low Mach number limit in the framework of weak solutions. The standard incompressible Navier-Stokes system is identified as the target problem. References:
    [1] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains. J. Differential Equations, 254:125–140, 2013.
    [2] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Incompressible limits of fluids excited by moving boundaries. Submitted
Thu, 02/05
12:00 		Christopher Hopper (OxPDE, University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Partial Regularity for constrained minimisers of quasi convex functionals with $ p $-growth

We consider minimisers of integral functionals $F$ over a ‘constrained’ class of $W^{1,p}$-mappings from a bounded domain into a compact Riemannian manifold $M$, i.e. minimisers of $F$ subject to holonomic constraints. Integrands of the form $f(Du)$ and the general $f(x,u,Du)$ are considered under natural strict $p$-quasiconvexity and $p$-growth assumptions for any exponent $1 < p <+\infty$. Unlike the harmonic and $p$-harmonic map case, the quasiconvexity condition requires one to linearise the map at the level of the gradient. In a bid to give a direct proof of partial $C^{1,α}-regularity for such minimisers, we developing an appropriate notion of a tangential harmonic approximation together with a discussion on the difficulties in establishing Caccioppoli-type inequalities. The need in the latter problem to construct suitable competitors to the minimiser via the so-called Luckhaus Lemma presents difficulties quite separate to that of the unconstrained case. We will give a proof of this lemma together with a discussion on the implications for higher integrability.
Thu, 25/04
12:00 		Konstantinos Koumatos (OxPDE, University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
From nonlinear to linearized elasticity via $ \Gamma $-convergence: the case of multi-well energies satisfying weak coercivity conditions
We derive geometrically linear elasticity theories as $ \Gamma $-limits of rescaled nonlinear multi-well energies satisfying a weak coercivity condition, in the sense that the standard quadratic growth from below of the energy density $ W $ is replaced by the weaker p-growth far from the energy wells, where $1
Thu, 07/03
12:00 		Marc Briane (Université de Rennes) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Characterisation of electric fields in periodic composites
This is work done in collaboration with G.W. Milton and A. Treibergs (University of Utah). Our purpose is to characterise, among all the regular periodic gradient fields, the ones which are isotropically realisable electric fields, namely solutions of a conduction equation with a suitable isotropic conductivity. In any dimension a sufficient condition of realisability is that the gradient field does not vanish. This condition is also necessary in dimension two but not in dimension three. However, when the conductivity also needs to be periodic, the previous condition is shown to be not sufficient. Then, using the associated gradient flow a necessary and sufficient condition for the isotropic realisability in the torus is established and illustrated by several examples. The realisability of the matrix gradient fields and the less regular laminate fields is also investigated.
Thu, 28/02
12:00 		Stefano Bianchini (SISSA-ISAS) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Quadratic interaction functional and structure of solutions to hyperbolic conservation laws
The proof of several properties of solutions of hyperbolic systems of conservation laws in one space dimension (existence, stability, regularity) depends on the existence of a decreasing functional, controlling the nonlinear interactions of waves. In a special case (genuinely nonlinear systems) the interaction functional is quadratic, while in the general case it is cubic. Several attempts to prove the existence of a a quadratic functional also in the most general case have been done. I will present the approach we follow in order to prove this result, an some of its implication we hope to exploit.

Work in collaboration with Stefano Modena.
Thu, 21/02
12:00 		Alexandre Boritchev (Ecole Polytechnique) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
1D Burgers Turbulence as a model case for the Kolmogorov Theory
The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all models of turbulence. In particular, K41 and corrections to it provide estimates of small-scale quantities such as increments and energy spectrum for a 3D turbulent flow. However, because of the well-known difficulties involved in studying 3D turbulent flow, there are no rigorous results confirming or infirming those predictions. Here, we consider a well-known simplified model for 3D turbulence: Burgulence, or turbulence for the 1D Burgers equation. In the space-periodic case with a stochastic white in time and smooth in space forcing term, we give sharp estimates for small-scale quantities such as increments and energy spectrum.
Thu, 14/02
12:15 		Paul Tod (OxPDE) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
CANCELLED!
The new schedule will follow shortly
Wed, 06/02
14:00 		Scott Armstrong (Université Paris Dauphine) OxPDE Lunchtime Seminar Add to calendar Gibson Grd floor SR
Regularity theory of degenerate elliptic equations in nondivergence form with applications to homogenization
We will present a regularity result for degenerate elliptic equations in nondivergence form. In joint work with Charlie Smart, we extend the regularity theory of Caffarelli to equations with possibly unbounded ellipticity– provided that the ellipticity satisfies an averaging condition. As an application we obtain a stochastic homogenization result for such equations (and a new estimate for the effective coefficients) as well as an invariance principle for random diffusions in random environments. The degenerate equations homogenize to uniformly elliptic equations, and we give an estimate of the ellipticity.
Thu, 31/01
12:00 		Timothy Blass (Carnegie Mellon University & OxPDE) OxPDE Lunchtime Seminar Add to calendar Gibson Grd floor SR
Dynamics for Screw Dislocations with Antiplane Shear
I will discuss the motion of screw dislocations in an elastic body under antiplane shear. In this setting, dislocations are viewed as points in a two-dimensional domain where the strain field fails to be a gradient. The motion is determined by the Peach-Koehler force and the slip-planes in the material. This leads to a system of discontinuous ODE, where the vector field depends on the solution to an elliptic PDE with Neumann data. We show short-time existence of solutions; we also have uniqueness for a restricted class of domains. In general, global solutions do not exist because of collisions.
Thu, 24/01
12:00 		Bernard Dacorogna (Ecole Polytechnique Federale de Lausanne) OxPDE Lunchtime Seminar Add to calendar Gibson Grd floor SR
The pullback equation for differential forms
This seminar is at ground floor!
An important question in geometry and analysis is to know when two $ k- $forms $ f $ and $ g $ are equivalent. The problem is therefore to find a map $ \varphi $ such that
\[ \varphi^{\ast}\left( g\right) =f. \]
We will mostly discuss the symplectic case $ k=2 $ and the case of volume forms $ k=n. $ We will give some results when $ 3\leq k\leq n-2, $ the case $ k=n-1 $ will also be considered.
The results have been obtained in collaboration with S. Bandyopadhyay, G. Csato and O. Kneuss and can be found, in part, in the book below.

Csato G., Dacorogna B. et Kneuss O., The pullback equation for differential forms, Birkhaüser, PNLDE Series, New York, 83 (2012).
Thu, 17/01
12:00 		Parth Soneji (OxPDE) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Relaxation in BV via polyhedral approximation
We first provide a brief overview of some of the key properties of the space $ \textrm{BV}(\Omega;\mathbb{R}^{N}) $ of functions of Bounded Variation, and the motivation for its use in the Calculus of Variations. Now consider the variational integral
\[ F(u;\Omega):=\int_{\Omega}f(Du(x))\,\textrm{d} x\,\textrm{,} \]
where $ \Omega\subset\mathbb{R}^{n} $ is open and bounded, and $ f\colon\mathbb{R}^{N\times n}\rightarrow\mathbb{R} $ is a continuous function satisfying the growth condition $ 0\leq f(\xi)\leq L(1+|\xi|^{r}) $ for some exponent $ r $. When $ u\in\textrm{BV}(\Omega;\mathbb{R}^{N}) $, we extend the definition of $ F(u;\Omega) $ by introducing the functional
\[ \mathscr{F}(u,\Omega):= \inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(Du_{j})\,\textrm{d} x\, \left| \!\!\begin{array}{r} (u_{j})\subset W_{\textrm{loc}}^{1,r}(\Omega, \mathbb{R}^{N}) \\ u_{j} \stackrel{\ast}{\rightharpoonup} u\,\,\textrm{in }\textrm{BV}(\Omega, \mathbb{R}^{N}) \end{array} \right. \bigg\} \,\textrm{.} \]
For $ r\in [1,\frac{n}{n-1}) $, we prove that $ \mathscr{F} $ satisfies the lower bound
\[ \mathscr{F}(u,\Omega) \geq \int_{\Omega} f(\nabla u (x))\,\textrm{d} x + \int_{\Omega}f_{\infty} \bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|\,\textrm{,} \]
provided $ f $ is quasiconvex, and the recession function $ f_{\infty} $ ($ := \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t $) is assumed to be finite in certain rank-one directions. This result is a natural extension of work by Ambrosio and Dal Maso, which deals with the case $ r=1 $; it involves combining work of Kristensen, Braides and Coscia with some new techniques, including a polyhedral approximation result and a blow-up argument that exploits fine properties of BV functions.
Thu, 13/12/2012
12:00 		Po Lam Yung (Rutgers University) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Two nonlinear wave equations with conformal invariance

In this talk, we will look at two non-linear wave equations in 2+1 dimensions, whose elliptic parts exhibit conformal invariance.

These equations have their origins in prescribing the Gaussian and mean curvatures respectively, and the goal is to understand well-posedness, blow-up and bubbling for these equations.

This is a joint work with Sagun Chanillo.
Thu, 29/11/2012
12:00 		Heikki Pekka Hakkarainen (University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Aspects of variational problems with linear growth condition on metric measure spaces
Thu, 22/11/2012
12:00 		Charles Stuart (l'École Polytechnique Fédérale de Lausanne) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Bifurcation for some non-Fréchet differentiable problems
Thu, 08/11/2012
12:00 		Carsten Carstensen (Humboldt-Universität zu Berlin) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Numerical analysis of some two well problem
Thu, 01/11/2012
12:30 		Arghir D. Zarnescu (University of Sussex) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Analytical and numerical aspects of an extended Navier-Stokes system
H. Johnston and J.G. Liu proposed in 2004 a numerical scheme for approximating numerically solutions of the incompressible Navier-Stokes system. The scheme worked very well in practice but its analytic properties remained elusive.
In order to understand these analytical aspects they considered together with R. Pego a continuous version of it that appears as an extension of the incompressible Navier-Stokes to vector-fields that are not necessarily divergence-free. For divergence-free initial data one has precisely the incompressible Navier-Stokes, while for non-divergence free initial data, the divergence is damped exponentially.
We present analytical results concerning this extended system and discuss numerical implications. This is joint work with R. Pego, G. Iyer (Carnegie Mellon) and J. Kelliher, M. Ignatova (UC Riverside).
Thu, 25/10/2012
12:00 		Eylem Öztürk (Hacettepe Üniversitesi) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Investigation of a class of reaction-diffusion equations
We investigate a mixed problem with Robin boundary conditions for a diffusion-reaction equation. We investigate the problem in the sublinear, linear and super linear cases, depending on the nonlinear part. We obtain relations between the parameters of the problem which are sufficient conditions for the existence of generalized solutions to the problem and, in a special case, for their uniqueness. The proof relies on a general existence theorem by Soltanov. Finally we investıgate the time-behaviour of solutions. We show that boundedness of solutions holds under some additional conditions as t is convergent to infinity. This study is joint work with Kamal Soltanov (Hacettepe University).
Thu, 18/10/2012
12:00 		Qilong Gu (University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Exact boundary controllability on a tree-like network
We establish the exact boundary controllability of nodal profile for general first order quasi linear hyperbolic systems in 1-D. And we apply the result in a tree-like network with general nonlinear boundary conditions and interface conditions. The basic principles of choosing the controls and getting the controllability are given.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

Syndicate content