# Partial Differential Equations and their Applications Seminar

### Meetings are held on Wednesdays at 15.00-16.00 in B3.03.

#### Organisers: Charlie Elliott, Jose Rodrigo, Ben Pooley & Wojciech Ożański

These meetings provide an opportunity for individuals to discuss in an informal manner progress on their current work, and describe interesting new problems related to PDEs and their applications and computation.

We have chosen this time to allow us to make a pub or restaurant visit afterwards if we feel inclined.

Group emails can be sent via applied_maths_pde_workgroup at listserv dot warwick dot ac dot uk.

## Term 2

16th January Manuel Torrilhon (RWTH Aachen)

Title: Non-Equilibrium Gases and Shallow Flows
Abstract: While numerical methods typically come with a refinement strategy that allows to control accuracy, mathematical models for physical processes mostly come as monolithic theories and the actual model errors are difficult to assess. Using hierarchical models the physical accuracy can be increased successively by traversing the model cascade. Hence, model error estimates can easily be obtained allowing true predictivity including model adaptivity. We will construct and investigate hierarchical models for two different applications: Slow non-equilibrium gases, where the model hierarchy bridges from classical fluid dynamics to Boltzmann equation, and shallow flows where the hierarchy bridges from depth-averaged St. Venant equations to free-surface Navier-Stokes. In both cases we construct a cascade of models based on function expansions and projections of the respective complex reference model mimicking the approach of numerical methods.

23rd January Stefano Modena (University of Leipzig)

Title: Non-uniqueness for the transport equation with Sobolev vector fields
Abstract: One of the main questions in the theory of the linear transport equation is whether uniqueness of weak solutions holds in the case the given vector field is not smooth. We show that even for incompressible, Sobolev (thus quite “well-behaved”) vector fields, uniqueness of solutions can drastically fail. Our result can be seen as a counterpart to DiPerna and Lions’ well-posedness theorem (joint with G. Sattig and L. Székelyhidi).

30th January Andre Guerra (University of Oxford)

Title: Morrey’s Problem and a Conjecture of Šverák
Abstract: Quasiconvexity and rank-one convexity play a fundamental role in the vectorial Calculus of Variations; Morrey’s problem is to decide whether these notions coincide. In this talk we will see that to solve this problem it is sufficient to check whether all extremal rank-one convex functions are quasiconvex. We will also identify a class of such extremal functions, thus proving a conjecture made by Šverák in 1992.

6th February Michele Coti Zelati (Imperial College London)

Title: TBC
Abstract: TBC

13th February Jean Van Schaftingen (Université catholique de Louvain)

Title: TBC
Abstract: TBC

20th February Augustin Moinat (University of Warwick)

Title: Space-time localisation for dynamical $\Phi^4_3$
Abstract: TBC

27th February TBC

Title: TBC
Abstract: TBC

Title: TBC
Abstract: TBC

13th March Tomasz Cieślak (Institute of Mathematics, Polish Academy of Sciences)

Title: TBC
Abstract: TBC

## Term 1

10th October Aneta Wróblewska-Kamińska (Imperial)

Title: Flow of heat conducting fluid in a time dependent domain

Abstact: We consider a flow of heat conducting fluid inside a moving domain whose shape in time is prescribed by a given velocity field. The flow in this case is governed by the compressible Navier-Stokes-Fourier system consisting of equation of continuity, momentum balance, entropy balance and energy equality. The velocity is supposed to fulfil the full-slip boundary condition and we assume that the fluid is thermally isolated. In the presented article we show the existence of a variational solution. To this end we construct proper penalising approximation. This result is a joint work with O. Kreml, V. Macha, and S. Necasova.

17th October Manuel del Pino (Bath)

Title: Gluing methods for Vortex dynamics in Euler flows
Abstact: We consider the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. We construct smooth solutions with concentrated vorticities around 𝑘 points which evolve according to the Hamiltonian system for the Kirkhoff-Routh energy, using an outer-inner solution gluing approach. The asymptotically singular profile around each point resembles a scaled finite mass solution of Liouville's equation.

24th October Daoguo Zhou (Oxford)

Title: Some Regularity Results for the Navier-Stokes Equations
Abstact: First, we prove some one scale regularity criteria for the three dimensional Navier-Stokes equations, which improve previous results due to Caffarelli, Kohn, Nirenberg, et al. As an application, we refine the estimate of the Minkowski dimension of the potential singular set of NSE. Then we discuss the regularity of NSE in in the largest critical space.

31th October Nick Sharples (Middlesex)

Title: On Solutions of the Transport Equation in the Presence of Singularities
Abstact: In this talk we will consider the transport equation when the vector field has a set of non-BV singularities: we will demonstrate the existence and uniqueness of solutions provided that the set of singularities has a sufficiently small anisotropic fractal dimension and the normal component of the vector field is sufficiently integrable near the singularities. This result extends the DiPerna-Lions-Ambrosio uniqueness theory, which requires the vector field to be BV. We will then establish some qualitative properties of solutions, before examining an application to vector fields generated by point-vortex-like singularities.

7th November Mahir Hadzic (King's College London)

Title: Gravitational collapse for the Euler-Poisson system
Abstact: The compressible Euler-Poisson system is the fundamental Newtonian model of a self gravitating star. Apart from some very special cases, almost nothing is rigorously known about the existence of compactly supported collapsing stars surrounded by vacuum.

For any value of the adiabatic index in the (supercritical) range (1,4/3), we construct an infinite-dimensional family of initial data that lead to finite-time gravitational collapse. By choosing a suitable set of geometrically motivated coordinates, we show that all of the star content is gradually absorbed into the singularity. This is joint work with Yan Guo (Brown) and Juhi Jang (USC).

14th November ﻿No talk scheduled

Abstact: For linear PDEs, think of Laplace or heat equations, regularity of their right-hand-sides i.e. given forces, say ${\rm div} F$, translates into an appropriate regularity of their solutions over a wide range of function spaces for F, e.g. $L_q$ spaces. When we turn our attention to basic nonlinear equations, take p-laplacian (stationary or evolutionary), such optimal regularity property is known only within a certain range of q's ('within duality'). To make matters worse, main problems appear already on the level of attempts to provide an existence theory beyond that range. Nevertheless, taking a half-step back and considering certain nonlinear systems of quadratic growth, motivated by Carreau fluids, one can provide well-posedness, i.e. existence, uniqueness and optimal regularity theory in the entire range of q's for right-hand sides, also 'beyond duality'. In my talk I will present our recent relevant results.