Analysis of a Living Fluid Continuum Model

Abstract

Abstract. Generalized Navier-Stokes equations which were proposed recently to describe active turbulence in living fluids are analyzed rigorously. Results on wellposedness and stability in the L 2 (R n )-setting are derived. Due to the presence of a Swift-Hohenberg term global wellposedness in a strong setting for arbitrary initial data in L 2 σ (R n ) is available. Based on the existence of global strong solutions, results on linear and nonlinear (in-) stability for the disordered steady state and the manifold of ordered polar steady states are derived, depending on the involved parameters.

Turbulence in active fluids caused by self-propulsion

“…It is well-known that then maximal regularity remains true for A LF +B(·). In fact, based on a Neumann series argument very similar as, e.g., in [19,Theorem 2.5] or [24,Lemma 3] it can be proved that…”

“…with a pressure constant p 0 and, if α < 0, additionally In order to include the steady states, as in [24] we consider the following generalized system:…”

“…Utilizing the local inverse theorem, the construction of a unique local strong solution follows now verbatim the lines of the proof of [24,Theorem 1].…”

“…Last not least, active turbulence was also suggested as a power source for various microfluidic applications [14,15,16,20]. We refer to those papers and to [24] for a more detailed description of the physics behind the additional occuring terms.…”

“…In [24] an analytical approach to (1.1) in L 2 (R n ) is presented. The aim of this note is prove well-posedness and significant results on stability and instability (turbulence) in the FM(R n )-setting, i.e., in spaces of Fourier transformed Radon measures.…”