Thu, 10 Mar 2016
12:00
L6

Sharp decay estimates for waves on black holes and Price's law

Dejan Gajic
(Cambridge)
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.
Fri, 29 Apr 2016
12:00
L6

Prandtl equations in Sobolev Spaces

Tong Yang
(City University of Hong Kong)
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.
Thu, 25 Feb 2016
12:00
L6

Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation

Jonas Lührmann
(ETH Zurich)
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.
Thu, 04 Feb 2016
12:00
L6

Regularity of level sets and flow lines

Herbert Koch
(Universitat Bonn)
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.
Thu, 03 Mar 2016
12:00
L6

Some regularity results for classes of elliptic systems with "structure"

Lisa Beck
(Universitat Ausburg)
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).
Thu, 18 Feb 2016
12:00
L6

Time-Periodic Einstein-Klein-Gordon Bifurcations Of Kerr

Yakov Shlapentokh-Rothman
(Princeton University)
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.

 
Thu, 11 Feb 2016
12:00
L6

Blow up by bubbling in critical parabolic problems

Manuel del Pino
(Universidad de Chile)
Abstract
We report some new results on construction of blowing up solutions by scalings of a finite energy entire steady states in two parabolic equations: the semilinear heat equation with critical nonlinearity and the 2d harmonic map flow into S2.
Thu, 28 Jan 2016
12:00
L6

Meaning of infinities in singular SPDEs

Wei-Jun Xu
(Warwick University)
Abstract
Many interesting stochastic PDEs arising from statistical physics are ill-posed in the sense that they involve products between distributions. Hence, the solutions to these equations are obtained after suitable renormalisations, which typically changes the original equation by a quantity that is infinity. In this talk, I will use KPZ and Phi^4_3 equations as two examples to explain the physical meanings of these infinities. As a consequence, we will see how these two equations, interpreted after suitable renormalisations, arise naturally as universal limits for two distinct classes of statistical physics systems. Part of the talk based on joint work with Martin Hairer.
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