No / This title provides detailed description of how Partial Differential Equations are used in the field of geometric design, and supplies clear and concise explanations of how to implement the techniques described. It also offers extensive discussions (with examples) or practical applications of Partial Differential Equations in geometric design.

yes / The spine of an object is an entity that can characterise the object¿s topology and describes the object by a lower dimension. It has an intuitive appeal for supporting geometric modelling operations. The aim of this paper is to show how a spine for a PDE surface can be generated. For the purpose of the work presented here an analytic solution form for the chosen PDE is utilised. It is shown that the spine of the PDE surface is then computed as a by-product of this analytic solution. This paper also discusses how the of a PDE surface can be used to manipulate the shape. The solution technique adopted here caters for periodic surfaces with general boundary conditions allowing the possibility of the spine based shape manipulation for a wide variety of free-form PDE surface shapes.

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Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD. / Master of Science / The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field. Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.

In this thesis, we consider two problems in the Q-tensor theory of nematic liquid crystals. The first concerns eigenvalue constraints on the Q-tensor order parameter. In particular, by employing a singular potential constructed by Ball and Majumdar, we consider the existence, regularity and "strict physicality" of weak solutions to the Beris-Edwards equations of nemato-hydrodynamics. In the second part of the thesis, we consider a gradient flow of the well-studied Landau-de Gennes energy. We prove some rigorous results on the average long-time statistical behaviour of its solutions, which are in agreement with experimental observations in the condensed matter physics literature.

Chain dynamics have gained renewed interest recently, following the release of a viral YouTube video showcasing a phenomenon called the chain fountain. Recent work in the field shows that there exists unexplained behavior in newly proposed chain systems. We consider a general system of a chain traveling at constant velocity in an external force field and derive steady state solutions for the time invariant shape of the chain. Perturbing the solution introduces moving waves along the steady state shape with components that propagate along and against the direction of travel of the chain. Furthermore, we develop a numerical model using a discrete approximation of the chain in order to empirically test our results. The behavior of the chain fountain and related chain systems is discussed in the context of these findings.

I consider a model for the control of criminality in cities. The model was developed during my REU at UCLA. The model is a system of partial differential equations that simulates the behavior of criminals and where they may accumulate, hot spots. I have proved a prior bounds for the partial differential equations in both one-dimensional and higher dimensional case, which proves the attractiveness and density of criminals in the given area will not be unlimitedly high. In addition, I have found some local bifurcation points in the model.

In this work, we study the application both of optimal control techniques and a numerical method to a system of partial differential equations arising from a problem in wound healing. Optimal control theory is a generalization of calculus of variations, as well as the method of Lagrange Multipliers. Both of these techniques have seen prevalent use in the modern theories of Physics, Economics, as well as in the study of Partial Differential Equations. The numerical method we consider is the method of lines, a prominent method for solving partial differential equations. This method uses finite difference schemes to discretize the spatial variable over an N-point mesh, thereby converting each partial differential equation into N ordinary differential equations. These equations can then be solved using numerical routines defined for ordinary differential equations.

In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H1-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, we connect these large-scale estimates with local regulartity results at the microscopic-level to achieve interior Lipschitz regularity at every scale.

This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem? The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions. The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported on H^s for s<1/2, and local solutions to the quartic KdV cannot be obtained below H^{1/2} by using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}. Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}. The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.