Past 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
25 February 2016
12:00
Jonas Lührmann
Abstract
The Maxwell-Klein-Gordon equation models the interaction of an electromagnetic field with a charged particle field. We discuss a proof of global regularity, scattering and a priori bounds for solutions to the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for essentially arbitrary smooth data of finite energy. The proof is based upon a novel "twisted" Bahouri-Gérard type profile decomposition and a concentration compactness/rigidity argument by Kenig-Merle, following the method developed by Krieger-Schlag in the context of critical wave maps. This is joint work with Joachim Krieger.
  • PDE CDT Lunchtime Seminar
18 February 2016
12:00
Yakov Shlapentokh-Rothman
Abstract

For a positive measure set of Klein-Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein-Klein-Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein-Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein-Klein-Gordon equations. This is joint work with Otis Chodosh.

 
  • PDE CDT Lunchtime Seminar
4 February 2016
12:00
Herbert Koch
Abstract
Level sets of solutions to elliptic and parabolic problems are often much more regular than the equation suggests. I will discuss partial analyticity and consequences for level sets, the regularity of solutions to elliptic PDEs in some limit cases, and the regularity of flow lines for bounded stationary solutions to the Euler equation. This is joint work with Nikolai Nadirashvili.
  • PDE CDT Lunchtime Seminar

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