Three Problems in Nonlinear Analysis

Biswas, Mrinmay (2020) Three Problems in Nonlinear Analysis. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Mrinmay Biswas (13RS044)) 13RS044.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In this thesis, we consider three problems in nonlinear analysis. The first problem is about the best constant of Generalized Hardy Inequalities. We be- gin our study in one dimension where we find its best constant and show that it is never achieved except in one case. We refer Chapter two for details, and see Biswas [14] as well. Note that, in the one dimensional case, we refer the inequality as the Bliss Inequality. In higher dimensions, while we have not been able to find the best constant in general, we show that the best constant remains same for convex domains. We also calculate the best constant in the homogenous case. See Chapter three and Bandyopadhyay-Biswas [7] for more details in this context. The second problem of this thesis, discussed in the fourth chapter, is an approximate con- trollability problem of a biological model, namely of FitzHugh-Nagumo Equation in one dimension, which is a simplified model of a nerve axon. We investigate the approximate controllability of this model using a localized interior control. The null controllability of the linearized system is not possible by using a localized interior control since the spectrum of the linearized system has an accumulation point, though it is approximate controllable. We show that while the solution of the FHN Equation fails to be globally approximate controllable in a given time, it is possible to move from one steady state to arbitrarily close to another steady state, after some appropriate time by a localized interior control, provided that both the aforementioned steady states stay in the same connected component of the set of steady states. See Chowdhury-Biswas-Dutta [25] for more details in this regard. Finally, in the fifth chapter, we consider the backward uniqueness problem of the one- dimensional compressible Navier-Stokes Equation linearized around a constant steady state (Q₀; V₀);Q₀ > 0; V₀ > 0, with periodic boundary conditions in (0; 2π). We prove that the linearized system with homogeneous periodic boundary conditions has backward uniqueness property when initial conditions lie in H¹per(0; 2π)x L²(0, 2π) satisfying an additional technical condition, where H¹per(0; 2π) denotes the Sobolev space of periodic functions with mean value zero. In a nutshell, we discuss the following three problems: 1. Invariance property of best constants of Generalized Hardy Inequalities, 2. Approximate controllability of FitzHugh-Nagumo System, and 3. Backward uniqueness problem of the linearized compressible Navier-Stokes Equation.

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