Searching for Singularities in Navier-Stokes Flows Based on the Ladyzhenskaya-Prodi-Serrin Conditions

Abstract

In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity ${\int }_0^T\Vert \mathbf{u}(t){\Vert }_{L^q(\Omega )}^{p}dt$, where $2/p+3/q\le 1$, $q>3$, is bounded, then the solution $\mathbf{u}(t)$ of the Navier-Stokes system is smooth on the interval $[0,T]$. In other words, if a singularity should occur at some time $t\in [0,T]$, then this quantity must be unbounded. We have probed this condition by studying a family of variational PDE optimization problems where initial conditions ${\mathbf{u}}_0$ are sought to maximize ${\int }_0^T\Vert \mathbf{u}(t){\Vert }_{L^4(\Omega )}^{8}dt$ for different $T$ subject to suitable constraints. These problems are solved numerically using a large-scale adjoint-based gradient approach. Even in the flows corresponding to the optimal initial conditions determined in this way no evidence has been found for singularity formation, which would be manifested by unbounded growth of $\Vert \mathbf{u}(t){\Vert }_{L^4(\Omega )}$. However, the maximum enstrophy attained in these extreme flows scales in proportion to ${\mathcal{E}}_0^{3/2}$, the same as found by Kang et al. (2020) when maximizing the finite-time growth of enstrophy. In addition, we also consider sharpness of an a priori estimate on the time evolution of $\Vert \mathbf{u}(t){\Vert }_{L^4(\Omega )}$ by solving another PDE optimization problem and demonstrate that the upper bound in this estimate could be improved.

arXiv comment31 pages, 15 figures. arXiv admin note: text overlap with arXiv:1909.00041

References in corpus (23)

• Computing Nearly Singular Solutions Using Pseudo-Spectral Methods
• Potentially Singular Solutions of the 3D Incompressible Euler Equations
• 3D Euler about a 2D Symmetry Plane
• Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods
• Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces
• Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem
• Towards a finite-time singularity of the Navier-Stokes equations. Part 2. Vortex reconnection and singularity evasion
• Chaotic blowup in the 3D incompressible Euler equations on a logarithmic lattice
• Vorticity moments in four numerical simulations of the 3D Navier-Stokes equations
• Maximum Palinstrophy Growth in 2D Incompressible Flows
• Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations
• Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows
• Quantitative bounds for critically bounded solutions to the Navier-Stokes equations
• Vortices, Maximum Growth and the Problem of Finite-Time Singularity Formation
• Complex-space singularities of 2D Euler flow in Lagrangian coordinates
• Weak and strong solutions of the $3D$ Navier-Stokes equations and their relation to a chessboard of convergent inverse length scales
• Sharp bounds on enstrophy growth in the viscous Burgers equation
• Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows
• Finite-time Singularity formation for Strong Solutions to the axi-symmetric $3D$ Euler Equations
• Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics
• Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation
• Maximum palinstrophy amplification in the two-dimensional Navier-Stokes equations
• Potential singularity formation of incompressible axisymmetric Euler equations with degenerate viscosity coefficients

Cited by in corpus (3)

• Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics
• Systematic search for singularities in 3D Euler flows
• Unraveling Self-Similar Energy Transfer Dynamics: a Case Study for 1D Burgers System