# 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) (slides)

**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:** *Metastability results for the Navier-Stokes equations and related models*

**Abstract:** We study diffusion and mixing in the incompressible Navier-Stokes equations and related scalar models. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequency. In turn, mixing acts to enhance the dissipative forces, giving rise to what we refer to as enhanced dissipation: this can be understood by the identification of a time-scale faster than the purely diffusive one. This talk is based on two recently obtained results: (1) a general quantitative criterion that links mixing rates (in terms of decay of negative Sobolev norms) to enhanced dissipation time-scales, and (2) a precise identification of the enhanced dissipation time-scale for the Navier-Stokes equations linearized around the Poiseuille flow, along with metastability results and nonlinear transition stability thresholds.

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

**Title:** *Endpoint estimates in Lorentz spaces for vector fields*

**Abstract:** Endpoint Gagliardo-Nirenberg-Sobolev embeddings have been known to hold for vector fields with an estimate that depends on a smaller set of combinations of the full derivative of the vector field, such as the differential and codifferential (J. Bourgain and H. Brezis, 2004) or the symmetric part of the derivative (M.J. Strauss, 1973). On the other hand, A. Alvino has obtained in 1977 the corresponding scalar endpoint estimate in the scale of Lorentz spaces. I will present a new family of endpoint Lorentz estimates for vector fields.

20th February **Augustin Moinat** (University of Warwick) (slides)

**Title:** *Space-time localisation for the dynamic $\Phi^4_3$ model.*

**Abstract: **We give an a priori bound for solutions of the dynamic $\Phi^4_3$ equation. This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions. We will show the different techniques used for the small and large scale bounds. For small scales we use techniques derived from Hairer’s theory of regularity structures. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure.

27th February **Juan Luis Vázquez** (Universidad Autónoma de Madrid)

**Title:** *Results and Open Problems in Nonlocal Porous Medium and Nonlocal Thin Film Flows.*

**Abstract:** The talk presents work on the existence, regularity and typical behaviour of solutions of nonlinear parabolic equations driven by fractional operators, which introduce nonlocal effects into classical settings. The models we discuss are related to porous medium and thin film equations. The problems in bounded domains offer new challenges.

6th March *Cancelled*

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

**Title:** *Fully parabolic chemorepulsion system.*

**Abstract: **Chemorepulsion system is similar in its form to the Keller-Segel system. The main difference being a sign in the aggregation equation. However, due to the difference in a sign, a different type of properties of solutions is expected, i.e. no finite-time aggregation. In 2008, together with Ph. Laurencot and C. Morales-Rodrigo, we have shown that indeed in 2d solutions are regular, global-in-time and bounded. The higher dimensional case is still open. In my talk I'll be speaking about recent results with K.Fujie, in particular about a new energy-like identity (obtained via hydrodymics analogy) and its connections to the Li-Yau-Hamilton type inequalities.

## 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*

21st 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:** Well-posedness of nonlinear systems beyond duality

**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.