On Spherically Symmetric Solutions¶of the Compressible Isentropic Navier–Stokes Equations

We prove the global existence of weak solutions to the Cauchy problem for the compressible isentropic Navier–Stokes equations in ℝ ⁿ (n= 2, 3) when the Cauchy data are spherically symmetric. The proof is based on the exploitation of the one-dimensional feature of symmetric solutions and use of a new (multidimensional) property induced by the viscous flux. The present paper extends Lions' existence theorem [15] to the case 1< γ <γ _n for spherically symmetric initial data, where γ is the specific heat ratio in the pressure, γ _n = 3/2 for n= 2 and γ _n = 9/5 for n= 3. Dedicated to Professor Rolf Leis on the occasion of his 70th birthday Received: 17 January 2000 / Accepted: 3 July 2000     

Forward Discretely Self-Similar Solutions of the Navier–Stokes Equations

Extending the work of Jia and ?verák on self-similar solutions of the Navier–Stokes equations, we show the existence of large, forward, discretely self-similar solutions.     

Cosmological and Quantum Solutions of the Navier–Stokes Equations

Russian Physics Journal - It is shown that the vector Navier–Stokes equation has a variety of quantum solutions, so the scope of this equation is not limited to the field of classical...     

On physics-based preconditioning of the Navier–Stokes equations

We develop a fully implicit scheme for the Navier–Stokes equations, in conservative form, for low to intermediate Mach number flows. Simulations in this range of flow regime produce stiff wave systems in which slow dynamical (advective) modes coexist with fast acoustic modes. Viscous and thermal diffusion effects in refined boundary layers can also produce stiffness. Implicit schemes allow one to step over the fast wave phenomena (or unresolved viscous time scales), while resolving advective time scales. In this study we employ the Jacobian-free Newton–Krylov (JFNK) method and develop a new physics-based preconditioner. To aid in overcoming numerical stiffness caused by the disparity between acoustic and advective modes, the governing equations are transformed into the primitive-variable form in a preconditioning step. The physics-based preconditioning incorporates traditional semi-implicit and physics-based splitting approaches without a loss of consistency between the original and preconditioned systems. The resulting algorithm is capable of solving low-speed natural circulation problems (M∼10^-4)$(M\sim {10}^{-4})$ with significant heat flux as well as intermediate speed (M∼1)$(M\sim 1)$ flows efficiently by following dynamical (advective) time scales of the problem.     

On the Weak Solutions to the Maxwell–Landau–Lifshitz Equations and to the Hall–Magneto–Hydrodynamic Equations

In this paper we deal with weak solutions to the Maxwell–Landau–Lifshitz equations and to the Hall–Magneto–Hydrodynamic equations. First we prove that these solutions satisfy some weak-strong uniqueness property. Then we investigate the validity of energy identities. In particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. In the case of the Hall–Magneto–Hydrodynamic equations we also give a sufficient condition to guarantee the magneto-helicity identity. Our conditions correspond to the same heuristic scaling as the one introduced by Onsager in hydrodynamic theory. Finally we examine the sign, locally, of the anomalous dissipations of weak solutions obtained by some natural approximation processes.     

Liouville-Type Theorems for the Forced Euler Equations and the Navier–Stokes Equations

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in ${\mathbb {R}^N}$ . If we assume “single signedness condition” on the force, then we can show that a ${C^1 (\mathbb {R}^N)}$ solution (v, p) with ${|v|^2+ |p| \in L^{\frac{q}{2}}(\mathbb {R}^N),\,q \in (\frac{3N}{N-1}, \infty)}$ is trivial, v = 0. For the solution of the steady Navier–Stokes equations, satisfying ${v(x) \to 0}$ as ${|x| \to \infty}$ , the condition ${\int_{\mathbb {R}^3} |\Delta v|^{\frac{6}{5}} dx < \infty}$ , which is stronger than the important D-condition, ${\int_{\mathbb {R}^3} |\nabla v|^2 dx < \infty}$ , but both having the same scaling property, implies that v = 0. In the appendix we reprove Theorem 1.1 (Chae, Commun Math Phys 273:203–215, 2007), using the self-similar Euler equations directly.     

Slip Boundary Conditions for the Compressible Navier–Stokes Equations

The slip boundary conditions for the compressible Navier–Stokes equations are derived systematically from the Boltzmann equation on the basis of the Chapman–Enskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar–Gross–Krook model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.     

Probabilistic Estimates for the Two-Dimensional Stochastic Navier–Stokes Equations

We consider the Navier–Stokes equation on a two-dimensional torus with a random force, white noise in time, and analytic in space, for arbitrary Reynolds number R. We prove probabilistic estimates for the long-time behavior of the solutions that imply bounds for the dissipation scale and energy spectrum as R.     

On the Helicity in 3D-Periodic Navier–Stokes Equations II: The Statistical Case

We study the asymptotic behavior of the statistical solutions to the Navier–Stokes equations using the normalization map [9]. It is then applied to the study of mean energy, mean dissipation rate of energy, and mean helicity of the spatial periodic flows driven by potential body forces. The statistical distribution of the asymptotic Beltrami flows are also investigated. We connect our mathematical analysis with the empirical theory of decaying turbulence. With appropriate mathematically defined ensemble averages, the Kolmogorov universal features are shown to be transient in time. We provide an estimate for the time interval in which those features may still be present.

Our collaborator and friend Basil Nicolaenko passed away in September of 2007, after this work was completed. Honoring his contribution and friendship, we dedicate this article to him.

     

Non-Formation of Vacuum States for Compressible Navier–Stokes Equations

We prove that weak solutions of the Navier–Stokes equations for compressible fluid flow in one space dimension do not exhibit vacuum states, provided that no vacuum states are present initially. The solutions and external forces that we consider are quite general: the essential requirements are that the mass and energy densities of the fluid be locally integrable at each time, and that the L ² _loc-norm of the velocity gradient be locally integrable in time. Our analysis shows that, if a vacuum state were to occur, the viscous force would impose an impulse of infinite magnitude on the adjacent fluid, thus violating the hypothesis that the momentum remains locally finite. Received: 20 March 2000 / Accepted: 16 July 2000     

The Small Scales of the Stochastic Navier–Stokes Equations Under Rough Forcing

We prove that the small scale structures of the stochastically forced Navier–Stokes equations approach those of the naturally associated Ornstein–Uhlenbeck process as the scales get smaller. Precisely, we prove that the rescaled kth spatial Fourier mode converges weakly on path space to an associated Ornstein–Uhlenbeck process as |k| . In addition, we prove that the Navier–Stokes equations and the naturally associated Ornstein–Uhlenbeck process induce equivalent transition densities if the viscosity is replaced with hyper-viscosity. This gives a simple proof of unique ergodicity for the hyperviscous Navier–Stokes system. We show how different strengthened hyperviscosity produce varying levels of equivalence.     

Boundary Effects on Exact Solutions of the Lagrangian-Averaged Navier–Stokes-α Equations

In order to clarify the behavior of solutions of the Lagrangian-averaged Navier–Stokes- (LANS-) equations in the presence of solid walls, we identify a variety of exact solutions of the full equations and their boundary layer approximations. The solutions demonstrate that boundary conditions suggested for the LANS- equations in the literature⁽¹⁾ for a bounded domain do not apply in a semi-infinite domain. The convergence to solutions of the Navier–Stokes equations as 0 is elucidated for infinite-energy solutions in a semi-infinite domain, and non-uniqueness of these solutions is discussed. We also study the boundary layer approximation of LANS- equations, denoted the Prandtl- equations, and report solutions for turbulent jets and wakes. Our version of the Prandtl- equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of Cheskidov.⁽²⁾     

Algebraic Splitting for Incompressible Navier–Stokes Equations

Fully discretized incompressible Navier–Stokes equations are solved by splitting the algebraic system with an approximate factorization. This splitting affects the temporal convergence order of velocity and pressure. The splitting error is proportional to the pressure variable, and a simple analysis shows that the original convergence order of the time-integration scheme can be retained by solving for incremental pressure. The combination of splitting and incremental pressure is shown to be equivalent to an error-correcting method using the full pressure. In numerical experiments employing a third-order time-integration scheme and various orders for the pressure increment, the splitting error is shown to control the convergence order, and the full order of the scheme is recaptured for both velocity and pressure. The difference between perturbing the momentum or the continuity equation is also explored.     

Ergodicity for the Randomly Forced 2D Navier–Stokes Equations

We study space-periodic 2D Navier–Stokes equations perturbed by an unbounded random kick-force. It is assumed that Fourier coefficients of the kicks are independent random variables all of whose moments are bounded and that the distributions of the first N ₀ coefficients (where N ₀ is a sufficiently large integer) have positive densities against the Lebesgue measure. We treat the equation as a random dynamical system in the space of square integrable divergence-free vector fields. We prove that this dynamical system has a unique stationary measure and study its ergodic properties.