In this presentation we will discuss examples of how carefully designed numerical computations involving solution of variational optimization problems can provide insights about some open questions in the analysis of Euler flows. The first problem we consider concerns the possibility of singularity formation in incompressible Euler flows. Based on the local well-posedness result of Kato (1972) asserting existence of smooth solutions in the Sobolev space , , we search for potentially singular flows systematically by solving a PDE optimization problem where the norm of the solution is maximized at a certain time . Local maximizers of this problem found using an adjoint-based Riemannian gradient descent method indicate the possibility of singularity formation if the time is sufficiently long. We will also review the results obtained by solving analogous problems for the viscous Navier-Stokes equation where no evidence for singularity formation was found. The second question we consider concerns the stability of the Taylor-Green vortex in 2D Euler flows. Numerical evidence is provided for the fact that such flows possess unstable eigenvalues embedded in the band of the essential spectrum of the linearized operator. However, the unstable eigenfunction is discontinuous at the hyperbolic stagnation points of the base flow and its regularity is consistent with the prediction of Lin (2004). This eigenfunction gives rise to an exponential transient growth with the rate given by the real part of the eigenvalue followed by passage to a nonlinear instability. We also illustrate a fundamentally different, nonmodal, growth mechanism involving a continuous family of uncorrelated functions, rather than an eigenfunction of the linearized operator, such that the resulting flows saturate the known estimates on the growth of the semigroup related to the essential spectrum of the linearized Euler operator.
[Joint work with Xinyu Zhao and Roman Shvydkoy]
