Past PDE CDT Lunchtime Seminar

It is nowadays well understood that the multidimensional isentropic Euler system is desperately ill–posed. Even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both global existence and uniqueness. We propose a different approach to well–posedness of this system based on ideas from the theory of Markov semigroups: we show the existence of a Borel measurable solution semiflow. To this end, we introduce a notion of dissipative solution which is understood as time dependent trajectories of the basic state variables - the mass density, the linear momentum, and the energy - in a suitable phase space. The underlying system of PDEs is satisfied in a generalized sense. The solution semiflow enjoys the standard semigroup property and the solutions coincide with the strong solutions as long as the latter exist. Moreover, they minimize the energy (maximize the energy dissipation) among all dissipative solutions.

The classical wave equation is derived from the system of three equations: The equation of motion of a (one-dimensional) deformable body, the Hook law as a constitutive equation, and the  strain measure, and describes wave propagation in elastic media. 
Fractional wave equations describe wave phenomena when viscoelasticity of a material or non-local effects of a material comes into an account. For waves in viscoelastic media, instead of Hook's law, a constitutive equation for viscoelastic body,  for example, Fractional Zener model or distributed order model of viscoelastic body, is used. To consider non-local effects of a media, one may replace classical strain measure by non-local strain measure. There are other constitutive equations and other ways to describe non-local effects which will be discussed within the talk.  
The system of three equations subject to initial conditions, initial displacement and initial velocity, is equivalent to one single equation, called fractional wave equation. Using different models for constitutive equations, and non-local measures, different fractional wave equations are obtained. After derivation of such equations, existence and uniqueness of their solution in the spaces of distributions is proved by the use of Laplace and Fourier transforms as main tool. Plots of solutions are presented. For some of derived equations microlocal analysis of the solution is conducted. 

The Einstein equations in wave coordinates are an example of a system 
which does not obey the "null condition". This leads to many 
difficulties, most famously when attempting to prove global existence, 
otherwise known as the "nonlinear stability of Minkowski space". 
Previous approaches to overcoming these problems suffer from a lack of 
generalisability - among other things, they make the a priori assumption 
that the space is approximately scale-invariant. Given the current 
interest in studying the stability of black holes and other related 
problems, removing this assumption is of great importance.

The p-weighted energy method of Dafermos and Rodnianski promises to 
overcome this difficulty by providing a flexible and robust tool to 
prove decay. However, so far it has mainly been used to treat linear 
equations. In this talk I will explain how to modify this method so that 
it can be applied to nonlinear systems which only obey the "weak null 
condition" - a large class of systems that includes, as a special case, 
the Einstein equations. This involves combining the p-weighted energy 
method with many of the geometric methods originally used by 
Christodoulou and Klainerman. Among other things, this allows us to 
enlarge the class of wave equations which are known to admit small-data 
global solutions, it gives a new proof of the stability of Minkowski 
space, and it also yields detailed asymptotics. In particular, in some 
situations we can understand the geometric origin of the slow decay 
towards null infinity exhibited by some of these systems: it is due to 
the formation of "shocks at infinity".

The finite element exterior calculus is a powerful approach to study many problems under the same lens. The canonical finite element spaces (see Arnold, Falk and Winther) are tied together with an exact sequence and have the required smoothness to define the exterior derivatives weakly. However, some applications require spaces that are more smooth (e.g. plate bending problems, incompressible flows). In this talk we will discuss some recent results in developing finite element spaceson simplicial triangulations with more smoothness, that also fit in an exact sequence. This is joint work with Guosheng Fu, Anna Lischke and Michael Neilan.

We present some regularity results for the gradient of solutions to very degenerate equations, which exhibit a great lack of ellipticity.
In particular we show that local weak solutions of the orthotropic p−harmonic equation are locally Lipschitz, for every $p\geq 2$ and in every dimension.
The results presented in this talk have been obtained in collaboration with Pierre Bousquet (Toulouse), Lorenzo Brasco (Ferrara) and Anna Verde (Napoli).
 

Motivated by recent problems on mixing flows, it is useful to characterize Besov spaces via oscillation of functions (averages) and minimization problems for bounded variation functions (Bianchini-type norms). In this talk, we discuss various descriptions of Besov spaces in terms of different kinds of averages, as well as Bianchini-type norms. Our method relies on the K-functional of the theory of real interpolation. This is a joint work with S. Tikhonov (Barcelona).

We consider a non-linear PDE on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.

In this talk, we demonstrate that formation of Type I singularities of suitable weak solutions of the Navier-Stokes equations occur if there exists non-zero mild bounded ancient solutions satisfying a 'Type I' decay condition. We will also discuss some new Liouville type Theorems. Joint work with Dallas Albritton (University of Minnesota).

In this talk, we study the long time behaviour of a cloud of weakly interacting Brownian particles, conditionally on the observation of their initial and final configuration. In particular, we connect this problem, which may be regarded as a nonlinear version of the Schrödinger problem, to the study of the long time behaviour of Mean Field Games. Combining tools from optimal transport and stochastic control we prove convergence towards the equilibrium configuration and establish convergence rates. A key ingredient to derive these results is a new functional inequality, which generalises Talagrand’s inequality to the entropic transportation cost.

Stein ($1981$) proved the borderline Sobolev embedding result which states that for $n \geq 2,$ $u \in L^{1}(\mathbb{R}^{n})$ and $\nabla u \in L^{(n,1)}(\mathbb{R}^{n}; \mathbb{R}^{n})$ implies $u$ is continuous. Coupled with standard Calderon-Zygmund estimates for Lorentz spaces, this implies $u \in C^{1}(\mathbb{R}^{n})$ if $\Delta u \in L^{(n,1)}(\mathbb{R}^{n}).$ The search for a nonlinear generalization of this result culminated in the work of Kuusi-Mingione ($2014$), which proves the same result for $p$-Laplacian type systems. \paragraph{} In this talk, we shall discuss how these results can be extended to differential forms. In particular, we can prove that if $u$ is an $\mathbb{R}^{N}$-valued $W^{1,p}_{loc}$ $k$-differential form with $\delta \left( a(x) \lvert du \rvert^{p-2} du \right) \in L^{(n,1)}_{loc}$ in a domain of $\mathbb{R}^{n}$ for $N \geq 1,$ $n \geq 2,$ $0 \leq k \leq n-1, $ $1 < p < \infty, $ with uniformly positive, bounded, Dini continuous scalar function $a$, then $du$ is continuous.

In this talk, which is based on joint work with Benjamin Gess, I will describe a pathwise well-posedness theory for stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, as an approximation to the Dean–Kawasaki equation in fluctuating hydrodynamics, to describe the fluctuating hydrodynamics of a zero range process, and as a model for the evolution of a thin film in the regime of negligible surface tension.  Our methods are motivated by the theory of stochastic viscosity solutions, which are applied after passing to the equation’s kinetic formulation, for which the noise enters linearly and can be inverted using the theory of rough paths.  I will also mention the application of these methods to nonlinear diffusion equations with linear, multiplicative noise.

In this talk I will discuss recent work with P. Daskalopoulos on sufficient conditions to prove uniqueness of complete graphs evolving by mean curvature flow. It is interesting to remark that the behaviour of solutions to mean curvature flow differs from the heat equation, where non-uniqueness may occur even for smooth initial conditions if the behaviour at infinity is not prescribed for all times. 

In this talk I will illustrate how solutions of nonlinear nonlocal Fokker-Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined outside such domain. Two different approaches are analysed, making crucial use of uniform estimates for energy and entropy functionals respectively. In both cases we prove that the problem in a bounded domain can be seen as a limit problem in the whole space involving a suitably chosen sequence of large confining potentials.
This is joint work with Maria Bruna and José Antonio Carrillo.
 

This talk will be an introduction to the use of conformal methods in asymptotic analysis in general relativity. We shall consider the explicit example of flat spacetime (Minkowski spacetime). The full conformal compactification will be constructed. For a simple example of a conformally invariant equation (we'll take the wave equation), we shall see how the compactification allows to infer precise informations on the asymptotic behaviour of the solution in all directions, for a certain class of data at any rate. Then, depending on time and questions, I will either describe how a scattering theory can be constructed using the same method or, explain how conformal methods can be used on other asymptotically flat geometries.

Harmonic map equations are an elliptic PDE system arising from the  
minimisation problem of Dirichlet energies between two manifolds. In  
this talk we present some some recent works concerning the symmetry  
and stability of harmonic maps. We construct a new family of  
''twisting'' examples of harmonic maps and discuss the existence,  
uniqueness and regularity issues. In particular, we characterise of  
singularities of minimising general axially symmetric harmonic maps,  
and construct non-minimising general axially symmetric harmonic maps  
with arbitrary 0- or 1-dimensional singular sets on the symmetry axis.  
Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$  
to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data, for  
$p \geq 2$. The stability fails for $p <2$ due to Almgren--Lieb and  
Mazowiecka--Strzelecki.

(Joint work with Prof. Robert M. Hardt.)

I will prove the 2d Biot-Savart law for the vorticity being an unbounded measure $\mu$, i.e. such that $\mu(\mathbb{R}^2)=\infty$, and show how can one infer some useful information concerning Kaden's spirals using it. Vorticities being unbounded measures appear naturally in the engineering literature as self-similar approximations of 2d Euler flows, see for instance Kaden's or Prandtl's spirals. Mathematicians are interested in such objects since they seem to be related to the questions of well-posedness of Delort's solutions of the 2d vortex sheet problem for the Euler equation. My talk is based on a common paper with K.Oleszkiewicz, M. Preisner and M. Szumanska.

In this talk we consider several scenarios involving the interaction among incompressible fluids of different nature. The main concern is the dynamics of the free boundary separating the fluids, which evolves with the velocity flow. The important question to address is whether the regularity is preserved in time or, on the other hand, the system develops singularities. We focus on Navier-Stokes models, where the viscosity of the fluids play a crucial role. At first showing results of finite time blow-up for the case of vacuum-fluid interaction. Later discussing new recent results on global existence for 1996 P.L. Lions' conjecture for density patches evolving by inhomogeneous Navier-Stokes equations.

For the North Analysis British Seminar, see: https://www.maths.ox.ac.uk/node/29687

We will talk about the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations. When viscosity coefficients are given as a constant multiple of density's power, based on some analysis  of  the nonlinear structure of this system, by introducing some new variables and the initial layer compatibility conditions, we identify the class of initial data admitting a local regular solution with far field vacuum and  finite energy  in some inhomogeneous Sobolev spaces, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier (2006, Anal. Simi. Fluid Dynam.),  Jiu-Wang-Xin (2014, JMFM) and so on. Moreover, in contrast to the classical well-posedness theory in the case of  the constant viscosity,   we show   that one can not obtain any global classical solution whose $L^\infty$  norm of $u$ decays to zero as time $t$ goes to infinity under the assumptions on the conservation laws of total mass and momentum.

In this talk I will present the recent developments in the topic of existence of solutions to the two-fluid systems. I will discuss the application of approach developed by P.-L. Lions and E. Feireisl and explain the limitations of this technique in the context of multi-component flow models. A particular example of such a model is two-fluids Stokes system with single velocity field and two densities, and with an algebraic pressure law closure. The existence result that uses the compactness criterion introduced for the Navier-Stokes system by D. Bresch and P.-E. Jabin will be presented. I will also mention an innovative construction of solutions relying on the G. Crippa and C. DeLellis stability estimates for the transport equation.

Novel machine learning techniques based on deep learning, i.e., the data-driven manipulation of neural networks, have reported remarkable results in many areas such as image classification, game intelligence, or speech recognition. Driven by these successes, many scholars have started using them in areas which do not focus on traditional machine learning tasks. For instance, more and more researchers are employing neural networks to develop tools for the discretisation and solution of partial differential equations. Two reasons can be identified to be the driving forces behind the increased interest in neural networks in the area of the numerical analysis of PDEs. On the one hand, powerful approximation theoretical results have been established which demonstrate that neural networks can represent functions from the most relevant function classes with a minimal number of parameters. On the other hand, highly efficient machine learning techniques for the training of these networks are now available and can be used as a black box. In this talk, we will give an overview of some approaches towards the numerical treatment of PDEs with neural networks and study the two aspects above. We will recall some classical and some novel approximation theoretical results and tie these results to PDE discretisation. Afterwards, providing a counterpoint, we analyse the structure of network spaces and deduce considerable problems for the black box solver. In particular, we will identify a number of structural properties of the set of neural networks that render optimisation over this set especially challenging and sometimes impossible. The talk is based on joint work with Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok, Felix Voigtlaender, and Mones Raslan

The focus of this talk is the regularity theory for time-harmonic Maxwell's equations with complex anisotropic parameters. By using the Helmholtz decomposition of the fields, we show how the problem can be completely reduced to a regularity question for elliptic equations, for which classical results may be applied. In particular, we prove the Hölder regularity of solutions under minimal assumptions on the coefficients.
 

I am going to report on some developments in regularity theory of nonlinear, degenerate equations, with special emphasis on estimates involving linear and nonlinear potentials. I will cover three main cases: degenerate nonlinear equations, systems, non-uniformly elliptic operators. 

The Vlasov-Poisson system is a kinetic equation that models collisionless plasma. A plasma has a characteristic scale called the Debye length, which is typically much shorter than the scale of observation. In this case the plasma is called ‘quasineutral’. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling, the formal limit of the Vlasov-Poisson system is the Kinetic Isothermal Euler system. The Vlasov-Poisson system itself can formally be derived as the limit of a system of ODEs describing the dynamics of a system of N interacting particles, as the number of particles approaches infinity. The rigorous justification of this mean field limit remains a fundamental open problem. In this talk we present the rigorous justification of the quasineutral limit for very small but rough perturbations of analytic initial data for the Vlasov-Poisson equation in dimensions 1, 2, and 3. Also, we discuss a recent result in which we derive the Kinetic Isothermal Euler system from a regularised particle model. Our approach uses a combined mean field and quasineutral limit.