arXiv subjects
Publications and source records attributed to Shibi Vasudevan.
We study the point spectrum of a second order difference operator with complex potential on the half-line via Fredholm determinants of the corresponding Birman-Schwinger operator pencils, the Evans and the Jost functions. An application is given to instability of a generalization of the Kolmogorov flow for the Euler equation of ideal fluid on the two dimensional torus.
For a second order difference equation that arises in the study of stability of unidirectional (generalized Kolmogorov) flows for the Euler equations of ideal fluids on the two dimensional torus, we relate the following five functions of the spectral parameter: the Fredholm determinants of the Birman-Schwinger operator pencils associated with the second order equation and the equivalent system of the first order equations; the Jost function constructed by means of the Jost solutions of the second order equation; the Evans function constructed by means of the matrix valued Jost solutions of the first order system, and, finally, to backward and forward continued fractions associated with the second order difference equation. We prove that all five functions are equal, and that their zeros are the discrete eigenvalues of the second order difference equation. We use this to improve an instability result for a generalization of the Kolmogorov (unidirectional) flow for the Euler equation on the 2D torus.
We study instability of unidirectional flows for the linearized 2D Navier-Stokes equations on the torus. Unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a single vector $\mathbf p \in \mathbb Z^{2}$. Using Fourier series and a geometric decomposition allows us to decompose the linearized operator $L_{B}$ acting on the space $\ell^{2}(\mathbb Z^{2})$ about this steady state as a direct sum of linear operators $L_{B,\mathbf q}$ acting on $\ell^{2}(\mathbb Z)$ parametrized by some vectors $\mathbf q\in\mathbb Z^2$. Using the method of continued fractions we prove that the linearized operator $L_{B,\mathbf q}$ about this steady state has an eigenvalue with positive real part thereby implying exponential instability of the linearized equations about this steady state. We further obtain a characterization of unstable eigenvalues of $L_{B,\mathbf q}$ in terms of the zeros of a perturbation determinant (Fredholm determinant) associated with a trace class operator $K_λ$. We also extend our main instability result to cover regularized variants (involving a parameter $α>0$) of the Navier-Stokes equations, namely the second grade fluid model, the Navier-Stokes-$α$ and the Navier-Stokes-Voigt models.
We study stability of unidirectional flows for the linearized 2D $α$-Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector $\mathbf p \in \mathbb Z^{2}$. We linearize the $α$-Euler equation and write the linearized operator $L_{B} $ in $\ell^{2}(\mathbb Z^{2})$ as a direct sum of one-dimensional difference operators $L_{B,\mathbf q}$ in $\ell^{2}(\mathbb Z)$ parametrized by some vectors $\mathbf q\in\mathbb Z^2$ such that the set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ covers the entire grid $\mathbb Z^{2}$. The set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ can have zero, one, or two points inside the disk of radius $\|\mathbf p\|$. We consider the case where the set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ has exactly one point in the open disc of radius $\mathbf p$. We show that unidirectional flows that satisfy this condition are linearly unstable. Our main result is an instability theorem that provides a necessary and sufficient condition for the existence of a positive eigenvalue to the operator $L_{B,\mathbf q}$ in terms of equations involving certain continued fractions. Moreover, we are also able to provide a complete characterization of the corresponding eigenvector. The proof is based on the use of continued fractions techniques expanding upon the ideas of Friedlander and Howard.
We study spectral instability of steady states to the linearized 2D Euler equations on the torus written in vorticity form via certain Birman-Schwinger type operators $K_λ(μ)$ and their associated 2-modified perturbation determinants $\mathcal D(λ,μ)$. Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator $L_{\rm vor}$ in terms of zeros of the 2-modified Fredholm determinant $\mathcal D(λ,0)=\det_{2}(I-K_λ(0))$ associated with the Hilbert Schmidt operator $K_λ(μ)$ for $μ=0$. As a consequence, we are also able to provide an alternative proof to an instability theorem first proved by Zhiwu Lin which relates existence of an unstable eigenvalue for $L_{\rm vor}$ to the number of negative eigenvalues of a limiting elliptic dispersion operator $A_{0}$.
We derive analogues of the classical Rayleigh, Fjortoft and Arnold stability and instability theorems in the context of the 2D $α$-Euler equations.