# Applied PDEs Working Seminar

**Meetings are held on Tuesdays at 16.00-17.00 in D1.07.**

**Organisers: ****Andreas Dedner,**** Charlie Elliott, Björn Stinner**

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 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 2013**

**3rd December - David O'Connor (Warwick) - On the Surface Cahn-Hilliard Equation**

I will discuss the Cahn-Hilliard equation on moving n-dimensional hypersurfaces embedded in n+1 dimensional space. We will show the derivation through use of conservation laws and discuss existence and uniqueness of solutions for a class of potential functions, covering the basics of geometric analysis required as well as the appropriate function spaces for such solutions. I will then discuss an extension of a previously known result for the Cahn-Hilliard equation in flat space, showing that a similar interface evolution law holds on fixed surfaces.

**26th November - Huan Wu (Warwick) - Atomistic-to-Continuum coupling: A quasi-non-local model in 1D**

The study of defects is an essential component of materials science research. Molecular simulation provides a way to study material behaviour at a microscopic scale. However crystal defects affect elastic fields far beyond their close neighbourhood, which presents a challenge in atomistic simulation, that is, the strongly coupled multiscale problems.

Atomistic-to-continuum (a/c) coupling methods are a class of computational multiscale schemes that combine the accuracy of atomistic models and the efficiency of continuum models. In short, these schemes admit the atomistic model in a neighbourhood of defect cores and approximate using a continuum elasticity model at far fields. These methods contain various approximation errors, including pure modelling errors and computational errors. A rigorous numerical analysis approach is therefore desired to optimize the level of accuracy of the schemes and to reduce the computational cost.

I will be talking about the quasi-non-local coupling method in 1D, where a coarse-grained mesh is used to optimize the efficiency of the numerical simulations. It involves the estimation of the pure modelling errors and the finite element errors using P1 and Pk schemes.

**19th November (double seminar)**

**4pm - Michael White (Minnesota) - Implicit solvent fluid dynamics with fluctuations**

Biomolecules such as proteins and nucleic acids are composed of long chains of charged particles and are usually found in a solvent such as salted water. Some of the popular models for their behavior will be introduced, including the more recent Variational Implicit Solvent Model (VISM). Then new work on a dynamic version of the implicit-solvent model will be presented. The method introduces fluctuations in the implicit solvent through the Landau-Lifshitz Navier-Stokes equations, a stochastic version of the Navier-Stokes equations. This allows us to model the dielectric boundary between the biomolecule and the solvent as a distribution of surfaces. The model is tested on some example problems and compared against molecular dynamics simulations.

**5pm - ****Tom Ranner (Leeds) - Solving partial differential equations on surfaces generated by biological imaging**

In this talk, I will explain how one can solve a partial differential equation on an evolving surface which is defined by biological images. I will explain how one can transform the image into a parametric form which can then be approximated by a triangulated surface: This allows the use of an arbitrary Lagrangian-Eulerian formulation of the surface finite element method to solve different partial differential equations. The methodology will be explained from a practical point of view with some numerical examples to show the efficacy of this approach.

**12th November - Faizan Nazar (Warwick) - Local Defects in the Thomas-Fermi-von Weiszäcker Theory of Crystals.**

In this talk I will discuss the existence and uniqueness of a coupled system of partial differential equations that arises from minimising the Thomas-Fermi-von Weiszäcker energy functional for a periodic crystal lattice. I will then discuss the effect of introducing a local defect which distorts the lattice, and obtain global estimates for the difference of solutions between the defective and periodic electron ground-state densities.

**5th November - Mike Scott (Warwick) - A Parabolic PDE on an evolving surface with finite time singularity.**

We consider the heat equation on an evolving hyperboloid, which undergoes a cone singularity at finite time. We prove that the projection of the density of some physical quantity, which is conserved, vanishes at the singular point and obeys a power law in one regime. If we have time, I will describe how one can continue the solution past the singularity for small times, at a microscopic level; this uses the theory of Stochastic Differential Equations. Since the proofs of the statements before the singular time uses probabilistic methods, we will not go into the details here, but will first give an overview of how probabilistic methods can be used to derive point-wise estimates on solutions to parabolic PDE. This is joint work with Martin Hairer.

**29th October - Graham Hobbs (Warwick) - Regularity theory a fourth order PDE with delta right hand side**

In this talk I will formulate a fourth order PDE with delta right hand side under homogeneous Dirichlet or Navier boundary conditions and establish the solution's regularity. I will then discuss an application in modelling the deformation of Biomembranes; that this PDE arises from an energy minimisation problem and the regularity is required to study the related gradient flow.

**22nd October**** - Stefano Bosia (Politecnico di Milano) - On some models for binary fluids with non-local interactions**

We present some results from an on-going project dedicated to diffuse interface models for the flow of binary fluids with non-local interactions. More precisely, we discuss variations on the well-known model H, which can be seen as resulting from the coupling between a Navier-Stokes equation for the velocity field of the mixture and a Cahn-Hilliard equation describing the evolution in the composition.

We first consider the effects of nonlocal interactions between the components of the fluid. In the case of 2D chemically reacting fluids, the existence and uniqueness of solutions is obtained. Moreover, we show that a family of exponential attractors exists, which is continuous with respect to the reaction rate. These equations model, e.g., the evolution of chemically reacting fluids as polymers.

In the last part of this talk we introduce some recent results and open problems concerning a nonlocal version of the Cahn-Hilliard equation with singular interaction kernels. This is a preliminary step in the study of this kind of nonlocal models for binary fluid flows.

These results are the outcome of joint work with H.Abels, M.Grasselli and A.Miranville.

**15th October - Yulong Yu (Warwick) - Direct and inverse scattering problems by unbounded surfaces**

Scattering theory has been an active research area in the past 50 years, to which many applied mathematicians pay their great attention. In this talk, we present some new theoretical and numerical results on direct and inverse scattering problems by unbounded surfaces.

This talk mainly consists of two parts. In the first part, we consider an acoustic scattering problem by an unbounded interface with buried obstacles. Under certain conditions on wave numbers and types of obstacles, we show that the direct scattering problem is well-posed. Particularly, we continue to analyze scattering problems due to point source waves (PSWs) and hyper singular point source waves (HSPSWs). We give some nice properties shared by the scattering solutions in the two cases above, which will contribute us to prove uniqueness result for the inverse scattering problem.

In the last part, we focus on inverse scattering of elastic waves from rigid periodic structures. We establish the factorization method to identify an unknown grating surface from knowledge of the scattered compressional or shear waves measured on a line above the scattering surface. A number of computational examples are provided to show the accuracy of the inversion algorithms.

These results are joint work with Prof. B. Zhang and Dr. G. Hu.

**8th October - James Sprittles (Warwick) - Modelling of Singular Capillary Flows**

Understanding the dynamics of liquids that are influenced by strong interfacial forces is key for the development of many emerging technologies in the microfluidic domain, such as '3D-printers'. Many of the flow processes occurring in such technologies can be categorised as 'singular', in the sense that the conventional fluid mechanical equations either give no solution, or predict unphysical blow up in physical quantities. In such cases, it is clear that the mathematical modelling must be re-considered to account for some additional physics.

In this talk, I will focus on two examples of singular capillary flows, namely the spreading of liquids over solids (dynamic wetting) and the merging of liquid drops (coalescence), explaining in each case how these problems are formulated in a continuum mechanics framework and then demonstrating how the resulting solutions are fundamentally flawed. Solutions from the literature will be discussed alongside a relatively new model, whose derivation I will briefly describe, which overcomes the aforementioned issues by extending the surface equations to account for the physics of interface formation.

Finally, a comparison of the predictions of the proposed models against recent experimental measurements will be presented that allow the limits of applicability of the various models to be ascertained.

**1st October - Josef Weber (Regensburg) - Existence of weak solutions for a diffuse interface model with soluble surfactants in two-phase flow**

We derive an appropriate time-discretization for the diffuse interface model and show existence for the time-discrete problem with the help of the Leray-Schauder principle. Using compactness results and an energy-estimate we show convergence of the interpolant functions to a weak solution of the problem.