Analysis of Partial Differential Equations (APDE)http://hdl.handle.net/20.500.11824/12022-11-27T19:46:43Z2022-11-27T19:46:43ZVarious results concerning homogenisation of nematic liquid crystalsCeuca, R.D.http://hdl.handle.net/20.500.11824/15272022-10-24T22:20:55Z2022-10-17T00:00:00ZVarious results concerning homogenisation of nematic liquid crystals
Ceuca, R.D.
In this thesis we present various results concerning homogenisation of nematic liquid crystals: two of them are in perforated domains, while the other one concerns rates of convergence for boundary homogenisation. The first work described in this thesis is a $\Gamma$-convergence result for the Landau-de Gennes model in $3D$ domains with connected perforations. The goal of the analysis is to find new terms in the energy functional that are independent of the gradient. The second result is an error estimate for a $2$D toy model used to describe rugosity effects in nematic liquid crystals via homogenisation problems, using once again the Landau-de Gennes model. The last problem is a local $L^2$-convergence result for a homogenisation problem in $\mathbb{R}^2$ with isolated perforations. Here we use the Oseen-Frank model, with the goal of finding new gradient-dependent terms in the energy functional.
We start, in Chapter 1, with a brief introduction to nematic liquid crystals. We introduce two major variational models used to describe nematic liquid crystals: Landau-de Gennes (LdG) and Oseen-Frank (OF). For LdG theory, which uses $Q$-tensors as the order parameter, we present typical choices for each type of energy contribution (bulk, elastic and surface). For OF theory based on the order parameter $\textbf{n}\in\mathbb{S}^2$ we discuss the elastic energy, that depends on the director and its gradient. We then present a short summary of the main mathematical results obtained for LdG and OF.
In Chapter 2, we analyse a homogenisation problem in $\mathbb{R}^3$ using the Landau-de Gennes model in which the perforations form a cubic microlattice. We assume a dillute regime, that is the volume of the cubic microlattice tends to $0$ as its characteristic length scale tends to $0$. The goal of this problem is to show that, given this geometrical setting, by choosing various types of surface energies one can obtain a new material in the limit of vanishing characteristic size of the microlattice. This material also behaves like a nematic liquid crystal, but now with different bulk coefficients. At the end of this chapter, we discuss a rate of convergence of the approximating surface energies to a homogenised term.
In Chapter 3, we concentrate on achieving and improving error estimates in homogenisation problems, since they can give us crucial information for manufacturing processes. Here, we consider a simplified $2$D model in which we highlight how one could replace a rugose boundary with the imposed homeotropic alignment by a flat boundary with an effective alignment depending on the initial geometry of the rugosity. We are able to improve an $L^2$ error estimate for a class of linear nonhomogeneous Robin problems.
In Chapter 4, we consider a general elastic energy in a $2$D case for the Oseen-Frank model in domains with isolated perforations. The goal of this study is to analyse how one could obtain a nematic liquid crystal with novel elastic properties via homogenisation procedures. Under suitable conditions we can analyse the $\mathbb{S}^1$-valued homogenisation problem via a scalar problem obtained through the lifting procedure. We also prove a local $L^2$ convergence result.
2022-10-17T00:00:00ZMotion of a rigid body in a compressible fluid with Navier-slip boundary conditionNecasova, S.Ramaswamy, M.Roy, A.Schlömerkemper, A.http://hdl.handle.net/20.500.11824/15182022-09-08T22:21:11Z2022-11-25T00:00:00ZMotion of a rigid body in a compressible fluid with Navier-slip boundary condition
Necasova, S.; Ramaswamy, M.; Roy, A.; Schlömerkemper, A.
In this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions are considered. We prove existence of a weak solution of the fluid-structure system up to collision.
2022-11-25T00:00:00ZSharp local smoothing estimates for Fourier integral operatorsBeltran D.Hickman J.Sogge C.D.http://hdl.handle.net/20.500.11824/15132022-08-28T22:20:56Z2019-01-01T00:00:00ZSharp local smoothing estimates for Fourier integral operators
Beltran D.; Hickman J.; Sogge C.D.
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory
integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions
arising from the structure of the Fourier integrals.
2019-01-01T00:00:00ZA∞ condition for general bases revisited: complete classification of definitionsKosz, D.http://hdl.handle.net/20.500.11824/15042022-08-18T22:20:44Z2022-05-27T00:00:00ZA∞ condition for general bases revisited: complete classification of definitions
Kosz, D.
We refer to the discussion on different characterizations of the
A∞ class of weights, initiated by Duoandikoetxea, Martín-Reyes, and Ombrosi
[Math. Z. 282 (2016), pp. 955–972]. Twelve definitions of the A∞ condition are
considered. For cubes in Rd every two conditions are known to be equivalent,
while for general bases we have a trichotomy: equivalence, one-way implication,
or no dependency may occur. In most cases the relations between different
conditions have already been established. Here all the unsolved cases are
treated and, as a result, a full diagram of the said relations is presented.
2022-05-27T00:00:00Z