An explicit Eulerian vortex solution of the compressible Euler equations at large Mach numbers resolves the classical problem of turbulence

Source abstract

A statistically closed analytical theory of compressible vortex turbulence is developed based on an explicit Eulerian classical vortex solution of the three-dimensional (3D) inviscid Burgers equations or, equivalently in the large-Mach-number limit, of the compressible 3D Euler equations. The finite-time blow-up of the solution is avoided, and the solution becomes globally smooth without introducing any additional hyperviscous dissipation at small scales, either through supercritical homogeneous linear friction or after stochastic averaging over a homogeneous Wiener process representing an arbitrarily small effective viscosity. Thus, in contrast to the known counterexample to the Clay Millennium Prize Problem in the incompressible case, the fundamental question of whether ordinary viscous dissipation can suppress the singularity arising from non-linear vortex stretching for any smooth initial data is positively resolved within the present compressible model. A rigorous assessment of the range of validity of the Millionshchikov hypothesis and of Onsager's dissipative anomaly, or Kolmogorov's zeroth hypothesis, is also obtained in arbitrary spatial dimensions. In particular, the strong dissipative anomaly is found to be valid only in six dimensions, whereas in the three-dimensional case only the weak dissipative anomaly is possible. MSC 2020 76F50, 76F02, 35Q35, 60H30 PACS 47.27; 47.10.-g; 05.20.-y; 05.40.-a