A CLASS OF ANISOTROPIC DIFFUSION-CONVECTION EQUATIONS in NON-DIVERGENCE FORM

We generalize Einstein's probabilistic method for the Brownian motion to study the fluids in porous media. By considering the general probability density distribution in the multi-dimensional case and relating the average jump with the velocity of the fluid, we derive an anisotropic diffusion equation in non-divergence form for the fluid's density that contains a convection term. This is then combined with the Darcy and the constitutive laws for compressible fluid flows to yield a nonlinear partial differential equations for the density function. We established the maximum and strong maximum principles for its solution. For the latter, transformations of the Bernstein–Cole–Hopf type are constructed and utilized. Then the initial boundary value problem is studied.We establish the lemma of growth in time for the truncated linear operator and investigate the long-time behavior of the classical solutions