Past PDE CDT Lunchtime Seminar

16 June 2016
12:00
Ben Sharp
Abstract
An embedded hypersurface in a Riemannian manifold is said to be minimal if it is a critical point with respect to the induced area. The index of a minimal hypersurface (roughly speaking) tells us how many ways one can locally deform the surface to decrease area (so that strict local area-minimisers have index zero). We will give an overview of recent works linking the index, topology and geometry of closed and embedded minimal hypersurfaces. The talk will involve separate joint works with Reto Buzano, Lucas Ambrozio and Alessandro Carlotto. 
  • PDE CDT Lunchtime Seminar
9 June 2016
12:00
Panagiota Daskalopoulos
Abstract
Some of the most important problems in geometric flows are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the partial differential equation involved. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time $-\infty < t \leq T$ for some $T \leq +\infty$. The classification of such solutions often sheds new insight to the singularity analysis. 
In this talk we will discuss Uniqueness Theorems for ancient solutions to geometric partial differential equations such as the Mean curvature flow, the Ricci flow and the Yamabe flow. We will also discuss the construction of new ancient solutions from the parabolic gluing of one or more solitons.
  • PDE CDT Lunchtime Seminar
2 June 2016
12:00
Franz Gmeineder
Abstract
In this talk I will report on regularity results for convex autonomous functionals of linear growth which depend on the symmetric gradients. Here, generalised minimisers will be attained in the space BD of functions of bounded of deformation which consists of those summable functions for which the distributional symmetric gradient is a Radon measure of finite total variation. Due to Ornstein's Non--Inequality, BD contains BV as a proper subspace and thus the full weak gradients of BD--functions might not exist even as Radon measures. In this talk, I will discuss conditions on the variational integrand under which partial regularity or higher Sobolev regularity for minima and hence the existence and higher integrability of the full gradients of minima can be established. This is joint work with Jan Kristensen.
  • PDE CDT Lunchtime Seminar
25 May 2016
16:00
Abstract
In 1973 D. G. Schaeffer established an interesting regularity result that applies to scalar conservation laws with uniformly convex fluxes. Loosely speaking, it can be formulated as follows: for a generic smooth initial datum, the admissible solution is smooth outside a locally finite number of curves in the time-space plane. Here the term ``generic`` should be interpreted in a suitable technical sense, related to the Baire Category Theorem. Several author improved later his result, also for numerical purposes, while only C. M. Dafermos and X. Cheng extended it in 1991 to a special 2x2 system with coinciding shock and rarefaction curves and which satisfies an assumption that reframes what in the scalar case is the assumption of uniformly convex flux, called `genuine nonlinearity'. My talk will aim at discussing a recent explicit counterexample that shows that for systems of at least three equations, even when the flux satisfies the assumption of genuinely nonlinearity, Schaeffer`s Theorem does not extend because countably many shocks might develop from a ``big`` family of smooth initial data. I will then mention related works in progress.
  • PDE CDT Lunchtime Seminar
19 May 2016
12:00
Kenneth Karlsen
Abstract
Stochastic partial differential equations arise in many fields, such as biology, physics, engineering, and economics, in which random phenomena play a crucial role. Recently many researchers have been interested in studying the effect of stochastic perturbations on hyperbolic conservation laws and other related nonlinear PDEs possessing shock wave solutions, with particular emphasis on existence and uniqueness questions (well-posedness). In this talk I will attempt to review parts of this activity.
  • PDE CDT Lunchtime Seminar
12 May 2016
12:00
Roland Grinis
Abstract
In this talk, we shall discuss how building upon the threshold theorem for wave maps, techniques inspired by the blow-up analysis of supercritical harmonic maps, can lead to a decomposition of the map into a decoupled sum of rescaled solitons, along a suitably chosen sequence of time slices converging to the maximal time of existence, with a term having asymptotically vanishing energy in the interior of the light cone, and when the target manifold is an Euclidean sphere. This work is motivated by the soliton resolution conjecture, on which spectacular progress has been achieved recently for equivariant wave maps, radial Yang-Mills fields and semi-linear critical wave equations.
  • PDE CDT Lunchtime Seminar
29 April 2016
12:00
Abstract
The classical result of Oleinik and her collaborators in 1960s on the Prandtl equations shows that in two space dimensions, the monotonicity condition on the tangential component of the velocity field in the normal direction yields local in time well-posedness of the system. Recently, the well-posedness of Prandtl equations in Sobolev spaces has also been obtained under the same monotonicity condition. Without this monotonicity condition, it is well expected that boundary separation will be developed. And the work of Gerard-Varet and Dormy gives the ill-posedness, in particular in Sobolev spaces, of the linearized systemaround a shear flow with a non-degenerate critical point under when the boundary layer tends to the Euler flow exponentially in the normal direction. In this talk, we will first show that this exponential decay condition is not necessary and then in some sense it shows that the monotonicity condition is sufficient and necessary for the well-posedness of the Prandtl equations in two space dimensions in Sobolev spaces. Finally, we will discuss the problem in three space dimensions.
  • PDE CDT Lunchtime Seminar
10 March 2016
12:00
Dejan Gajic
Abstract
Price’s law postulates inverse-power polynomial decay rates for solutions to the wave equation on Schwarzschild backgrounds with respect to appropriately normalized null coordinates. Polynomial decay rates as a lower bound are known in the physics literature as “late-time power law tails”. I will discuss new physical space methods for proving sharp decay rates for solutions to the wave equation on a class of asymptotically flat, stationary, spherically symmetric spacetimes, establishing in particular the upper bounds and lower bounds in Price’s law on Schwarzschild. This work has been done jointly with Yannis Angelopoulos and Stefanos Aretakis.
  • PDE CDT Lunchtime Seminar
3 March 2016
12:00
Abstract
We address regularity properties of (vector-valued) weak solutions to quasilinear elliptic systems, for the special situation that the inhomogeneity grows naturally in the gradient variable of the unknown (which is a setting appearing for various applications). It is well-known that such systems may admit discontinuous and even unbounded solutions, when no additional structural assumption on the inhomogeneity or on the leading elliptic operator or on the solution is imposed. In this talk we discuss two conceptionally different types of such structure conditions. First, we consider weak solutions in the space $W^{1,p}$ in the limiting case $p=n$ (with $n$ the space dimension), where the embedding into the space of continuous functions just fails, and we assume on the inhomogeneity a one-sided condition. Via a double approximation procedure based on variational inequalities, we establish the existence of a weak solution and prove simultaneously its continuity (which, however, does not exclude in general the existence of irregular solutions). Secondly, we consider diagonal systems (with $p=2$) and assume on the inhomogeneity sum coerciveness. Via blow-up techniques we here establish the existence of a regular weak solution and Liouville-type properties. All results presented in this talk are based on joint projects with Jens Frehse (Bonn) and Miroslav Bulíček (Prague).
  • PDE CDT Lunchtime Seminar

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