The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains

This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used

Magnetic Relaxation of a Voigt-MHD System

We construct solutions of the magnetohydrostatic (MHS) equations in bounded domains and on the torus in three spatial dimensions, as infinite time limits of Voigt approximations of viscous, non-resistive incompressible magnetohydrodynamics equations. The Voigt approximations modify the time evolution without introducing artificial viscosity. We show that the obtained MHS solutions are regular, nontrivial, and are not Beltrami fields.Comment: 16 page

Navier-Stokes Flow for a Fluid Jet with a Free Surface

The three-dimensional Navier-Stokes flow of a viscous fluid jet bounded by a moving free surface under isothermal conditions and without surface tension is considered. The fluid domain is assumed to be periodic in the axial direction and initially axisymmetric. A local-in-time existence and regularity result is proven for the full governing equations using a contraction argument in an appropriate function space. Here a Lagrangian specification of the flow field is employedin order to mitigate the difficulties involved in dealing with an evolving fluid domain. It is also shown that the associated linear problem gives rise to an analytic semigroup of contractions on the space of divergence-free Lebesgue-square-integrable vector fields

The spectrum of the Poincar{\'e} operator in an ellipsoid

We reprove the fact, due to Backus, that the Poincar{\'e} operator in ellipsoids admits a pure point spectrum with polynomial eigenfunctions.We then show that the eigenvalues of the Poincar{\'e} operator restricted to polynomial vector fields of fixed degree admitsa limit repartition given by a probability measure that we construct explicitely. For that, we use Fourier integral operators and ideas comingfrom Alan Weinstein and the first author in the seventies. The starting observation is that the orthogonal polynomials in ellipsoids satisfy a PDE

On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow

In this article, we study the axisymmetric surface diffusion flow (ASD), a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older regular surfaces of revolution embedded in R^3 and satisfying periodic boundary conditions. We also give conditions for global existence of solutions and prove that solutions are real analytic in time and space. Further, we investigate the geometric properties of solutions to ASD. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. Then, focusing on the family of cylinders, we establish results regarding stability, instability and bifurcation behavior, with the radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana

Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

We generalize two results in the Navier-Stokes regularity theory whose proofs rely on `zooming in' on a presumed singularity to the local setting near a curved portion $\Gamma \subset \partial\Omega$ of the boundary. Suppose that $u$ is a boundary suitable weak solution with singularity $z^* = (x^*,T^*)$, where $x^* \in \Omega \cup \Gamma$. Then, under weak background assumptions, the $L_3$ norm of $u$ tends to infinity in every ball centered at $x^*$: \begin{equation*} \lim_{t \to T^*_-} \lVert u(\cdot, t)\rVert_{L_{3}\left(\Omega \cap B(x^*,r)\right)} = \infty \quad \forall r > 0. \end{equation*} Additionally, $u$ generates a non-trivial `mild bounded ancient solution' in $\mathbb{R}^3$ or $\mathbb{R}^3_+$ through a rescaling procedure that `zooms in' on the singularity. Our proofs rely on a truncation procedure for boundary suitable weak solutions. The former result is based on energy estimates for $L_3$ initial data and a Liouville theorem. For the latter result, we apply perturbation theory for $L_\infty$ initial data based on linear estimates due to K. Abe and Y. Giga