Asymptotic Analysis

A rigorous analytical justification of turbulence observed in active fluids and caused by self-propulsion is presented. We prove existence of unstable wave modes for the generalized Stokes and Navier–Stokes systems by developing an approach in spaces of Fourier transformed Radon measures.

The objective of this paper is two fold. Firstly, a strictly hyperbolic system of conservation laws known as Euler equation of compressible fluid flow is considered and limiting behavior of its solutions as the pressure like term vanishes is studied. Secondly, the limiting behavior of the solutions for another strictly hyperbolic system which is a perturbed model of the system introduced by Korchinski [Solution of a Riemann problem for a system of conservation laws possessing no classical weak solution, 1977, Adelphi University] is also studied as the perturbation parameter vanishes.

We establish two types of characterizations for high order anisotropic Sobolev spaces. In particular, we prove high order anisotropic versions of Bourgain–Brezis–Mironescu’s formula and Nguyen’s formula.