Partial and full hyper-viscosity for Navier-Stokes and primitive equations

Abstract

The 3-D primitive equations and incompressible Navier-Stokes equations with full hyperviscosity and only horizontal hyper-viscosity are considered on the torus, i.e., the diffusion term −∆ isHyper-viscosity is applied in many numerical schemes, and in particular horizontal hyper-viscosity appears in meteorological models. A classical result by Lions states that for the Navier-Stokes equations uniqueness of global weak solutions for initial data in L 2 holds if −∆ is replaced by (−∆) 5/4 . Here, for the primitive equations the corresponding result is proven for (−∆) 8/5 . For the case of horizontal hyperviscosity l = 2 is sufficient in both cases. Strong convergence for ε → 0 of hyper-viscous solutions to a weak solution of the Navier-Stokes and primitive equations, respectively, is proven as well. The approach presented here is based on the construction of strong solutions via an evolution equation approach for initial data in L 2 and weak-strong uniqueness.2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 35K25, 35K30, 35Q30, 76D03, 86A05.

Gaussian invariant measures and stationary solutions of 2D primitive equations

“…The trick is now to sum the martingale identities ( 17), (18) for ω m andω m : in doing so the nonlinear skew symmetric part, together with boundary terms, is canceled, leaving us with martingales term and the symmetric Ornstein-Uhlenbeck generator,M…”

“…In this work we present a solution theory of 2-dimensional Primitive Equations in the hyperviscous setting D(∆) = −(−∆) θ , for large enough θ and a suitable stochastic forcing. The regularising effect of hyperviscosity for Navier-Stokes and Primitive Equations is well-understood in the deterministic setting, and it is often used in numerical simulations [20]; we refer to [19,22,23] and, more recently, [18] for a thorough discussion. The main contribution of the present paper is thus to introduce a Gaussian invariant measure in the context of 2-dimensional Primitive Equations, and then to exploit the techniques of [15] to provide a first well-posedness result for this singular SPDE in a hyperviscous setting.…”

Martingale Solutions of the Stochastic 2D Primitive Equations with Anisotropic Viscosity

“…where σ is the random external forces, f is an external force term and W is a cylindrical Wiener process, the definitions of which will be introduced in Section 2. As in [27], note that the vertical periodicity and parity conditions in (1.1) correspond in to an equivalent set of equations on (0, 1) × (−1, 0) with lateral periodicity and…”

“…Furthermore, if ∂ z v 0 ∈ L q for q > 2, the local z-weak solutions extended to a global strong solution. For the case of full hyper-viscosity or only horizontal hyper-viscosity, i.e., the diffusion term −∆ is replaced by −∆ + ε(−∆) l or by −∆ + ε(−∆ H ) l , respectively, where ε > 0, l > 1, strong convergence for ε → 0 of hyper-viscous solutions to a weak solution of the 3D deterministic Navier-Stokes and primitive equations, respectively, was obtained by Hussein in [27].…”

Gaussian invariant measures and stationary solutions of 2D Primitive Equations

“…In this work we present a solution theory of 2-dimensional Primitive Equations in the hyperviscous setting D(∆) = −(−∆) θ , for large enough θ and a suitable stochastic forcing. The regularising effect of hyperviscosity for Navier-Stokes and Primitive Equations is well-understood in the deterministic setting, and it is often used in numerical simulations [19]; we refer to [18,21,22] and, more recently, [17] for a thorough discussion. The main contribution of the present paper is thus to introduce a Gaussian invariant measure in the context of 2-dimensional Primitive Equations, and then to exploit the techniques of [15] to provide a first well-posedness result for this singular SPDE in a hyperviscous setting.…”