Degenerate Nonlinear Einstein Modeling of Brownean Motion in Fluid Flows and Chemotaxis

We will discuss two projects:
1.The movement of the living organism in a band form towards the presence of chemical substrates based on a system of the PDE. We incorporate Einstein’s method of Brownian motion to deduce the chemotactic model exhibiting a traveling band. To our knowledge this was the first time that Einstein’s method has been used to motivate equations describing the mutual interaction of the chemotactic system. We have shown that in the presence of limited and unlimited substrate, traveling bands are achievable and it has been explained accordingly. We also study the stability of the constant steady states for the system. The linearized system about a steady state is obtained. We are able to find explicit conditions for linear instability and stability under homogeneous boundary conditions. 2. A modification is proposed to Einstein’s model by introducing the dependence of the diffusion matrix on the concentration of particles. By assuming a qualified decrease of diffusion for small concentrations, the modified model successfully resolves the paradox and establishes the existence ofa finite propagation speed. However, recent advancements in stochastic processes have brought to light a paradox associated with this model. Specifically, it predicts an infinite propagation speed, which contradicts the second law of thermodynamics. Therefore, our approach extends Einstein’s model by integrating the essential modification, which resolves the paradox. This modification enables a more precise depiction of the behavior of nonlinear flows in porous media while upholding adherence to the fundamental principles of thermodynamics The method employed in this analysis involves solving a nonlinear degenerate parabolic partial differential equation in divergent form