Semi-discrete optimal transport methods for the semi-geostrophic equations

We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

On classical solutions of Rayleigh–Taylor instability in inhomogeneous viscoelastic fluids

We study the nonlinear Rayleigh–Taylor (RT) instability of an inhomogeneous incompressible viscoelastic fluid in a bounded domain. It is well known that there exist strong solutions of RT instability in $H^{2}$ H 2 -norm in inhomogeneous incompressible viscoelastic fluids, when the elasticity coefficient κ is less than some threshold $\kappa _{\mathrm{C}}$ κ C . In this paper, we prove the existence of classical solutions of RT instability in $L^{1}$ L 1 -norm in Lagrangian coordinates based on a bootstrap instability method with finer analysis, if $\kappa <\kappa _{\mathrm{C}}$ κ < κ C . Moreover, we also get classical solutions of RT instability in $L^{1}$ L 1 -norm in Eulerian coordinates by further applying an inverse transformation of Lagrangian coordinates.

Theoretical Study on Fluid Velocity for Viscous Fluid in a Circular Cylindrical Shell

A theoretical algorithm by united Lagrangian-Eulerian method for the problem in dealing with viscous fluid and a circular cylindrical shell is presented. In this approach, each material is described in its preferred reference frame. Fluid flows are given in Eulerian coordinates whereas the elastic circular cylindrical shell is treated in a Lagrangian framework. The fluid velocity in a two-dimensional uniform elastic circular cylindrical shell filled with viscous fluid is studied under the assumption of low Reynolds number. The coupling between the viscous fluid and the elastic circular cylindrical shell shows kinematic conditions at the shell surface. Also, the radial velocity and axial velocity of the fluid are discussed with the help of graphs.