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The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid....
The time-ordered exponential of a time-dependent matrix is defined as the function of that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in . The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by . Yet, the existence of such inverses, crucial to...
Fundamental solutions to linear partial differential equations with constant coefficients are represented in the form of Laplace type integrals.
We prove the large time existence of solutions to the magnetohydrodynamics equations with slip boundary conditions in a cylindrical domain. Assuming smallness of the L₂-norms of the derivatives of the initial velocity and of the magnetic field with respect to the variable along the axis of the cylinder, we are able to obtain an estimate for the velocity and the magnetic field in without restriction on their magnitude. Then the existence follows from the Leray-Schauder fixed point theorem.
Let , let be a hypersurface of , be a submanifold of . Denote by the Levi form of at . In a previous paper [3] two numbers , are defined; for they are the numbers of positive and negative eigenvalues for . For , , we show here that are still the numbers of positive and negative eigenvalues for when restricted to . Applications to the concentration in degree for microfunctions at the boundary are given.
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