2018/10/09 by Julien Lequeurre, Lequeurre, Julien, Alexandre Munnier +1 · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1810.04222
openalex publication_date 2018/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main purpose of this work is to provide a Hilbertian functional framework\nfor the analysis of the planar Navier-Stokes (NS) equations either in vorticity\nor in stream function formulation. The fluid is assumed to occupy a bounded\npossibly multiply connected domain. The velocity field satisfies either\nhomogeneous (no-slip boundary conditions) or prescribed Dirichlet boundary\nconditions. We prove that the analysis of the 2D Navier-Stokes equations can be\ncarried out in terms of the so-called nonprimitive variables only (vorticity\nfield and stream function) without resorting to the classical NS theory (stated\nin primitive variables, i.e. velocity and pressure fields). Both approaches (in\nprimitive and nonprimitive variables) are shown to be equivalent for weak\n(Leray) and strong (Kato) solutions. Explicit Bernoulli-like formulas are\nderived and allow recovering the pressure field from the vorticity fields or\nthe stream function. In the last section, the functional framework described\nearlier leads to a simplified rephrasing of the vorticity dynamics, as\nintroduced by Maekawa in [52]. At this level of regularity, the vorticity\nequation splits into a coupling between a parabolic and an elliptic equation\ncorresponding respectively to the non-harmonic and harmonic parts of the\nvorticity equation. By exploiting this structure it is possible to prove new\nexistence and uniqueness results, as well as the exponential decay of the\npalinstrophy (that is, loosely speaking, the H1 norm of the vorticity) for\nlarge time, an estimate which was not known so far.\n