Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the problem of finding the instability index of certain non-self-adjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. Our main result shows that the instability index of such operators is determined by its restriction to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunov's 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 Lyapunov's 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.  include a numerical example.
['Almut Burchard', 'Marina Chugunova']
I am writing a research paper on a idea On Computing the Instability Index of a Non-Self-Adjoint Differential Operator Associated with Coating and Rimming Flows
[{"from": "human", "value": "I am writing a research paper on a idea On Computing the Instability Index of a Non-Self-Adjoint Differential Operator Associated with Coating and Rimming Flows"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the problem of finding the instability index of certain non-self-adjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. Our main result shows that the instability index of such operators is determined by its restriction to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunov's 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 Lyapunov's 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.  include a numerical example."}]