We will discuss the similarity and difference between deterministic and stochastic NLS. Different notions (or possible formulations) of local solutions will also be discussed. We will also present a global well posedness result for stochastic mass critical NLS. Joint work with Weijun Xu (Oxford)

# Past PDE CDT Lunchtime Seminar

We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position and its orientation vector, which lies on the unit sphere. We prove that, in the low temperature regime, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into the sphere. This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals and the existence of its global weak solution was first obtained by Y.M Chen, using Ginzburg-Landau approximation. The key ingredient of our result is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the Q-tensor), a spectral decomposition of the linearized operator near the limiting local equilibrium distribution, as well as the energy dissipation estimates. This is a joint work with Wei Wang in Zhejiang university.

Function solutions to linear PDEs often carry rigidity properties directly associated to the equation they satsify. However, the realm of solutions covers a much larger sets of solutions. For instance, we can speak of measure solutions, as opposed to classical $C^\infty$ functions or even $L^p$ functions. It is only logical to expect that the “better” space the solution lives in, the more rigid its properties will be.

Measure solutions lie just at a comfortable half of this threshold: it is a sufficently large space which allows for a rich range of new structures; but is sufficiently rigid to preserve a meaningful geometrical pattern. For example, have you ever wondered how gradients look like in the space of measures? What about other PDE structures? In this talk I will discuss these general questions, a few examples of them, and a new theoretical approach to its understanding via PDE theory, harmonic analysis, and geometric measure theory methods.

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.