Past PDE CDT Lunchtime Seminar | Mathematical Institute

Thu, 27 Oct 2016
12:00

The inverse Calderón problem with Lipschitz conductivities

Pedro Caro

(Basque Center for Applied Mathematics)

Abstract

In this talk I will present a recent uniqueness result for an inverse boundary value problem consisting of recovering the conductivity of a medium from boundary measurements. This inverse problem was proposed by Calderón in 1980 and is the mathematical model for a medical imaging technique called Electrical Impedance Tomography which has promising applications in monitoring lung functions and as an alternative/complementary technique to mammography and Magnetic Resonance Imaging for breast cancer detection. Since in real applications, the medium to be imaged may present quite rough electrical properties, it seems of capital relevance to know what are the minimal regularity assumptions on the conductivity to ensure the unique determination of the conductivity from the boundary measurements. This question is challenging and has been brought to the attention of many analysts. The result I will present provides uniqueness for Lipschitz conductivities and was proved in collaboration with Keith Rogers.

Thu, 20 Oct 2016
12:00

Long-time existence for Yang-Mills flow

Alex Waldron

(Stony Brook University)

Abstract

I'll discuss the problem of controlling energy concentration in YM flow over a four-manifold. Based on a study of the rotationally symmetric case, it was conjectured in 1997 that bubbling can only occur at infinite time. My thesis contained some strong elementary results on this problem, which I've now solved in full generality by a more involved method.

Thu, 13 Oct 2016
12:00

Boundary regularity for strong local minimizers and Weierstrass problem

Judith Campos Cordero

(Ausburg University)

Abstract

We prove partial regularity up to the boundary for strong local minimizers in the case of non-homogeneous integrands and a full regularity result for Lipschitz extremals with gradients of vanishing mean oscillation. As a consequence, we also establish a sufficiency result for this class of extremals, in connection with Grabovsky-Mengesha theorem (2009), which states that $C^1$ extremals at which the second variation is positive, are strong local minimizers.

Thu, 16 Jun 2016
12:00

Minimal hypersurfaces with bounded index

Ben Sharp

(University of Pisa)

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.

Thu, 09 Jun 2016
12:00

Ancient solutions of Geometric Flows

Panagiota Daskalopoulos

(Columbia University)

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.

Thu, 02 Jun 2016
12:00

Regularity Theory for Symmetric-Convex Functionals of Linear Growth

Franz Gmeineder

(Oxford)

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.

Wed, 25 May 2016
16:00

A counterexample concerning regularity properties for systems of conservation laws

Laura Caravenna

(Università degli Studi di Padova)

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.

Thu, 19 May 2016
12:00

Stochastic Conservation Laws

Kenneth Karlsen

(University of Oslo)

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.

Thu, 12 May 2016
12:00

Quantization of time-like energy for wave maps into spheres

Roland Grinis

(Oxford)

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.

Thu, 05 May 2016
12:00

Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions

Marshall Slemrod

(University of Wisconsin)

Abstract

We will discuss some underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the interconnected nonlinear partial differential equations.

Fri, 29 Apr 2016
12:00

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, 10 Mar 2016
12:00

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.

Thu, 03 Mar 2016
12:00

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, 25 Feb 2016
12:00

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, 18 Feb 2016
12:00

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

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, 04 Feb 2016
12:00

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, 28 Jan 2016
12:00

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.

Thu, 21 Jan 2016
12:00

Obstacle problems of Signorini type, and for non-local operators

Nicola Garofalo

(Universita' degli studi di Padova)

Abstract

In this talk I will overview what is presently known about various types of obstacle problems. The focus will be on elliptic and parabolic problems of Signorini type, and on problems for non-local operators. I will discuss the role of monotonicity formulas in such problems, as well as (in the time-independent case) of some new epiperimetric inequalities.

Thu, 03 Dec 2015

12:00 - 13:00

Weak solutions to the Navier-Stokes initial boundary value problem in exterior domains with initial data in L(3,∞)

Paolo Maremonti

(Seconda Università degli Studi di Napoli)

Abstract

We consider the Navier-Stokes initial boundary value problem (NS-IBVP) in a smooth exterior domain. We are interested in establishing existence of weak solutions (we mean weak solutions as synonym of solutions global in time) with an initial data in L(3,∞)

(Lorentz space). Apart from its own analytical interest, the research is connected with questions related to the space-time asymptotic properties of solutions to the NS-IBVP. However these questions are not discussed. The assumption on the initial data in L(3,∞) cuts the L2-theory out, which is the unique known for weak solutions. We ﬁnd a simple strategy to bypass the diﬃculties of an initial data /∈ L2, and we take care to perform the same “regularity properties” of Leary’s weak solutions, hence to furnish a structure theorem of a weak solution.

Thu, 26 Nov 2015

12:00 - 13:00

Non-orientable line defects in the Landau-de Gennes theory of nematic liquid crystals

Giacomo Canevari

(University of Oxford)

Abstract

Nematic liquid crystals are composed by rod-shaped molecules with long-range orientation order. These materials admit topological defect lines, some of which are associated with non-orientable configurations. In this talk, we consider the Landau-de Gennes variational theory of nematics. We study the asymptotic behaviour of minimizers as the elastic constant tends to zero. We assume that the energy of minimizers is of the same order as the logarithm of the elastic constant. This happens, for instance, if the boundary datum has finitely many singular points. We prove convergence to a locally harmonic map with singularities of dimension one (non-orientable line defects) and, possibly, zero (point defects).

Thu, 19 Nov 2015

12:00 - 13:00

Why gradient flows of some energies good for defect equilibria are not good for dynamics, and an improvement

Amit Acharya

(Carnegie Mellon Univeristy)

Abstract

Straight screw dislocations are line defects in crystalline materials and wedge disclinations are line defects in nematic liquid crystals. In this talk, I will discuss the development and implications of a single pde model intended to describe equilibrium states and dynamics of these defects. These topological defects are classically treated as singularities that result in infinite total energy in bodies of finite extent that behave linearly in their elastic response. I will explain how such singularities can be alleviated by the introduction of an additional 'eigendeformation' field, beyond the fundamental fields of the classical theories involved. The eigendeformation field bears much similarity to gauge fields in high- energy physics, but arises from an entirely different standpoint not involving the notion of gauge invariance in our considerations. It will then be shown that an (L2) gradient flow of a 'canonical', phase- field type (up to details) energy function coupling the deformation to the eigendeformation field that succeeds in predicting the defect equilibrium states of interest necessarily has to fail in predicting particular types of physically important defect dynamics. Instead, a dynamical model based on the same
energy but involving a conservation statement for topological charge of the line defect field for its evolution will be shown to succeed. This is joint work with Chiqun Zhang, graduate student at CMU.

Thu, 12 Nov 2015

12:00 - 13:00

Energy decay in a 1D coupled heat-wave system

David Seifert

(University of Oxford)

Abstract

We study a simple one-dimensional coupled heat wave system, obtaining a sharp estimate for the rate of energy decay of classical solutions. Our approach is based on the asymptotic theory of $C_0$-semigroups and in particular on a result due to Borichev and Tomilov (2010), which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the semigroup generator. This technique not only leads to an optimal result, it is also simpler than the methods used by other authors in similar situations and moreover extends to problems on higher-dimensional domains. Joint work with C.J.K. Batty (Oxford) and L. Paunonen (Tampere).

Thu, 05 Nov 2015

12:00 - 13:00

Ancient Solutions to Navier-Stokes Equations in Half Space

Tobias Barker

(University of Oxford)

Abstract

The relationship between the so-called ancient (backwards) solutions to the Navier-Stokes equations in the space or in a half space and the global well-posedness of initial boundary value problems for these equations will be explained. If time permits I will sketch details of an equivalence theorem and a proof of smoothness properties of mild bounded ancient solutions in the half space, which is a joint work with Gregory Seregin

Thu, 29 Oct 2015

12:00 - 13:00

Quantitative flatness results for nonlocal minimal surfaces in low dimensions

Eleonora Cinti

(WIAS Berlin)

Abstract

We consider minimizers of nonlocal functionals, like the fractional perimeter, or the fractional anisotropic perimeter, in low dimensions. It is known that a minimizer for the nonlocal perimeter in $\mathbb{R}^2 $ is necessarily an halfplane. We give a quantitative version of this result, in the following sense: we prove that minimizers in a ball of radius $R$ are nearly flat in $B_1$, when $R$ is large enough. More precisely, we establish a quantitative estimate on how "close" these sets are (in the $L^{1}$-sense and in the $L^{\infty}$ -sense) to be a halfplane, depending on $R$. This is a joint work with Joaquim Serra and Enrico Valdinoci.