# Past OxPDE Lunchtime Seminar

15 May 2013
12:00
Cleopatra Christoforou
Abstract
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.
• OxPDE Lunchtime Seminar
9 May 2013
12:01
Abstract
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: <ul>[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.</ul> <ul>[2] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Incompressible limits of ﬂuids excited by moving boundaries. Submitted</ul>
• OxPDE Lunchtime Seminar
2 May 2013
12:00
Christopher Hopper
Abstract

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 $C^{1,α}-regularity for such minimisers, we developing an appropriate notion of a tangential harmonic approximation together with a discussion on the difficulties in establishing Caccioppoli-type inequalities. The need in the latter problem to construct suitable competitors to the minimiser via the so-called Luckhaus Lemma presents difficulties quite separate to that of the unconstrained case. We will give a proof of this lemma together with a discussion on the implications for higher integrability. • OxPDE Lunchtime Seminar 25 April 2013 12:00 Konstantinos Koumatos Abstract We derive geometrically linear elasticity theories as$\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<p\leq 2\$. This study is motivated by energies arising in nematic elastomers and is joint work with V. Agostiniani and T. Blass.
• OxPDE Lunchtime Seminar
7 March 2013
12:00
Marc Briane
Abstract
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.
• OxPDE Lunchtime Seminar
28 February 2013
12:00
Stefano Bianchini
Abstract
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.
• OxPDE Lunchtime Seminar
21 February 2013
12:00
Alexandre Boritchev
Abstract
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.
• OxPDE Lunchtime Seminar
14 February 2013
12:15
Paul Tod
Abstract
The new schedule will follow shortly
• OxPDE Lunchtime Seminar
6 February 2013
14:00
Abstract
<p>We will present a regularity result for degenerate elliptic equations in nondivergence form. </p> <p>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.</p>
• OxPDE Lunchtime Seminar
31 January 2013
12:00
Abstract
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.
• OxPDE Lunchtime Seminar