This paper studies the three-dimensional density-dependent incompressible magnetohydrodynamic equations. First, a regularity criterion is proved which allows the initial density to contain vacuum. Then we establish another blow-up criterion in the Besov space Ḃ0 ∞,2 when the positive initial density is bounded away from zero. Third, we prove a global nonexistence result for initial density with...

We present two finite element methods for simplicial meshes to approximate the solution of the problem of nearly incompressible elasticity. Although both approaches are based on mixed formulations of linear elastic equations and biorthogonal systems, one of them is nonsymmetric, and the other symmetric. An interesting feature of both approaches is that displacement-based formulations can be obt...

To date it has not been possible to prove whether or not the three-dimensional incompressible Euler equations develop singular behaviour in finite time. Some possible singular scenarios, as for instance shock-waves, are very important from a physical point of view, since they imply the connection among the macroscopic and the microscopic scale. Therefore, the appearence of this type of singular...

This article addresses a fundamental concern regarding the incompressible approximation of fluid motions, one of the most widely used approximations in fluid mechanics. Common belief is that its accuracy is O , where denotes the Mach number. In this article, however, we prove an O 2 accuracy for the incompressible approximation of the isentropic, compressible Euler equations thanks to several d...

We propose a numerical method for modeling highly deformable nonlinear incompressible solids that conserves the volume locally near each node in a finite element simulation. Our method works with arbitrary constitutive models, is applicable to both passive materials and active materials (e.g. muscles), and works with simple tetrahedra without the need for multiple quadrature points or stabiliza...

The membrane’s dynamics is very important for cells. A membrane in 2dimensional space can be seen as an incompressible closed curve in a plane or a cylindrical surface in 3-dimensional space. In this paper, we design a second-order accurate numerical algorithm to simulate the shape transformation of the membrane. In the algorithm, we use the tangent angles to present the curve and avoid the dif...

This paper presents the development of a 2D high-order solver with unstructured spectral difference method for unsteady incompressible Navier-Stokes equations. Timemarching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditiona...

We present a second-order accurate finite difference method for numerical solution of the incompressible Navier-Stokes equations in deforming domains. Our approach is a generalization of the Bell-Colella-Glaz predictor-corrector method for incompressible flow. In order to treat the time-dependence and inhomogeneities in the incompressibility constraint introduced by presence of deforming bounda...