# 2018-19

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

#### Organisers: Charlie Elliott & Jose Rodrigo

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 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 Naiver-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 singularity 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*

21th November **Matias Delgadino** (Imperial)

**Title:** *A quantitative relationship between mixing and enhanced dissipation*

**Abstact:** We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a time-scale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation time-scales. We will give a general overview on the subject, and present a few specific applications that include passive scalar evolution in both planar and radial settings, fractional diffusion, linearized two-dimensional Navier-Stokes equations, and even simple examples in kinetic theory.

28th November **Susana Gutierrez** (Birmingham)

**Title:** *Self-similar solutions of the Landau-Lifshitz-Gilbert equation and related problems*

**Abstact:** The Landau-Lifshitz-Gilbert equation (LLG) is a continuum model describing the dynamics for the spin in ferromagnetic materials. In the first part of this talk we describe our work concerning the properties and dynamical behaviour of the family of self-similar solutions under the one-dimensional LLG-equation. Motivated by the properties of this family of self-similar solutions, in the second part of this talk we consider the Cauchy problem for the LLG-equation and provide a global well-posedness result provided that the BMO norm of the initial data is small.

5th December **Jan Burczak** (Oxford)

**Title:** *TBC*

**Abstact:** TBC