We first study the so-called Heat equation with two families of elliptic operators which are fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equation with operators including the “large” eigenvalues has strong similarities with a Heat equation in lower dimension whereas, surprisingly, for operators including “small” eigenvalues it shares some properties with some transport equations.
We propose a fast method for high order approximations of the solution of the Cauchy problem for the linear non-stationary Stokes system in R^3 in the unknown velocity u and kinematic pressure P. The density f(x,t) and the divergence vector-free initial value g(x) are smooth and rapidly decreasing as |x| tends to infinity. We construct the vector u in the form u=u1+u2 where u1 solves a system of homogeneous heat equations and u2 solves a system of non-homogeneous heat equations with right-hand side f-graf P. Moreover, P=-L( div f)) where L denotes the harmonic potential.
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