Renormalization Group Methods in Applied Mathematical Problems

by Merdan, Huseyin

egin{abstract} This work presents the application of the methods known as renormalization group (RG) and scaling in the physics literature to applied mathematics problems after a brief review of the methodology. The first part of the thesis involves an application to a class of nonlinear parabolic differential equations. We consider equations of the form $u_{t}=frac{1}{2}u_{xx}+varepsilon N(x,u,u_{x},u_{xx})$ where $varepsilon$ is a small positive number and $N$ is dimensionally consistent without additional dimensional constants. First, RG methods are described for determining the key exponents related to the decay of solutions to these equations. The determination of decay exponents is viewed as an asymptotically self-similar process that facilitates an RG approach. These methods are extended to higher order in the small coefficient of the nonlinearity. The RG calculations lead to the result that for large space and time, the solution is characterized by $u(x,t)sim t^{-frac{1}{2}-alpha }u^{ast }(xt^{-1/2},1)$ where the exponent $alpha$ is a simple function of the exponents of the terms in $N$. Finally, the RG results are verified in some cases by rigorous proofs and other calculations. In the second part, the application of renormalization technique to systems of equations describing interface problems are presented. The temporal evaluation of an interface separating two phases is analyzed for large time. We study the standard sharp interface problem in the quasi-static regime. The characteristic length, $R(t)$, of a self-similar system that is the time dependent length scale characterizing the pattern growth is calculated by implementing a renormalization procedure. It behaves as $t^{eta}$ where $eta $ has values in the continuous spectrum $[1/3,1/2]$ when the dynamical undercooling is non-zero, and $eta$ in $[1/3,infty )$ when the undercooling is set at zero. The single value of $eta =1$ is extracted from this continuous spectrum as a consequence of boundary conditions that impose a plane wave. It is also shown that in almost all of these cases, the capillarity length (arising from surface tension) is irrelevant for the large time behavior even though it has a crucial role at the early stage evolution of an interface. end{abstract}