Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn computing the instability index of a non-selfadjoint differential operator associated with coating and rimming flows
323
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the problem of finding the instability index of certain non-selfadjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. The main result reduces the computation of the instability index to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunovs method to associate the differential operator with a quadratic form, whose maximal positive subspace has dimension equal to the instability index. The quadratic form is given by a solution of Lyapunovs equation, which here takes the form of a fourth order linear PDE in two variables. Elliptic estimates for the solution of this PDE play a key role. We include some numerical examples.
rate research
Read More
We investigate minimal operator corresponding to operator differential expression with exit from space, study its selfadjoint extensions, also for one particular selfadjoint extension corresponding to boundary value problem with some rational function of eigenparameter in boundary condition establish asymptotics of spectrum and derive trace formula
This paper is an expanded version of the lectures I delivered at the Indian Statistical Institute, Bangalore, during the OTOA 2014 conference.
We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $partial_{ u} u - gamma(x) u = 0$ on the boundary $Gamma$ and $gamma(x) > 0.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, : t geq 0.$ The eigenvalues $lambda_k$ of $G$ with ${rm Re}: lambda_k < 0$ yield asymptotically disappearing solutions $u(t, x) = e^{lambda_k t} f(x)$ having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case $min_{xin Gamma} gamma(x) > 1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $G.$
We prove two improv
We consider a Schrodinger hamiltonian $H(A,a)$ with scaling critical and time independent external electromagnetic potential, and assume that the angular operator $L$ associated to $H$ is positive definite. We prove the following: if $|e^{-itH(A,a)}|_{L^1to L^infty}lesssim t^{-n/2}$, then $ ||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}|_{L^1to L^infty}lesssim t^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on the ground level of $L$ and the space dimension $n$. We prove similar results also for the heat semi-group generated by $H(A,a)$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
