In this brief note we show that the author’s previous result in [1] on the nonexistence of self-similar singularities for the 3D incompressible Euler equations implies actually the nonexistence of ‘locally self-similar’ singular solution, which has been sought by many physicists. Nonexistence of locally self-similar solution We are concerned here on the following Euler equations for the homogen...

We consider finite element methods based on simplices to solve the problem of nearly incompressible elasticity. Two different approaches based respectively on dual meshes and dual bases are presented, where in both approaches pressure is discontinuous and can be statically condensed out from the system. These novel approaches lead to displacement-based low order finite element methods for nearl...

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When an elastic filament spins in a viscous incompressible fluid it may undergo a whirling instability, as studied asymptotically by Wolgemuth, Powers, and Goldstein [Phys. Rev. Lett., 84 (2000), pp. 16–23]. We use the immersed boundary (IB) method to study the interaction between the elastic filament and the surrounding viscous fluid as governed by the incompressible Navier–Stokes equations. T...

We develop the density-matrix renormalization group (DMRG) technique for numerically studying incompressible fractional quantum Hall (FQH) states on the sphere. We calculate accurate estimates for ground-state energies and excitation gaps at FQH filling fractions nu=1/3 and nu=5/2 for systems that are considerably larger than the largest ever studied by exact diagonalization. We establish, by c...

The membrane of biological cells consists of tightly packed lipid molecules with hydrophilic heads facing the exterior and interior fluids and hydrophobic tails hiding in the middle, forming a bilayer structure. Most cell membranes (and other biological membranes) can deform but resist area dilation and are often modelled as incompressible fluid interfaces [12]. The aqueous solution outside the...

The semi-geostrophic equations are used in meteorology. They appear as a variant of the two-dimensional Euler incompressible equations in vorticity form, where the Poisson equation that relates the stream function and the vorticity field is just replaced by the fully non linear elliptic Monge-Ampère equation. This work gathers new results concerning the semi-geostrophic equations: Existence and...

In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in R3. This result is analogous to the celebrated Beale-KatoMajda’s breakdown criterion for the inviscid Eluer equations of incompressible fluids. In R2 we establish global weak solutions to the magnetohydrodynamic equation...

Biological uid dynamics typically involves geometrically complicated structures which are often deforming in time. We give a brief overview of some approaches based on using xed Cartesian grids instead of attempting to use a grid which conforms to the boundary. Both nite-diierence and nite-volume methods are discussed, as well as a combined approach which has recently been used for computing in...