2023/10/04 by Albritton, Dallas, Ogden, W. Jacob · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2310.03120
We consider a complexification of the Euler equations introduced by Šverák which conserves energy. We prove that these complex Euler equations are nonlinearly ill-posed below analytic regularity and, moreover, we exhibit solutions which lose analyticity in finite time. Our examples are complex shear flows and, hence, one-dimensional. This motivates us to consider fully nonlinear systems in one spatial dimension which are non-hyperbolic near a constant equilibrium. We prove nonlinear ill-posedness and finite-time singularity for these models. Our approach is to construct an infinite-dimensional unstable manifold to capture the high frequency instability at the nonlinear level.