Knowlpedia

Definition

Hypodissipative Navier–Stokes equations

Incompressible momentum with a positive fractional dissipation order less than the usual Laplacian.

The hypodissipative Navier–Stokes equations with exponent 0<α<10<\alpha<1 and ν>0\nu>0 are

∂tu+(u⋅∇)u+∇p=−ν(−Δ)αu+f,div⁡u=0.\partial_tu+(u\cdot\nabla)u+\nabla p =-\nu(-\Delta)^\alpha u+f,\qquad\operatorname{div}u=0.

The replaces the second-order dissipation of . Its differential order is 2α<22\alpha<2.

Exponent conventions

If the dissipative term is written −ν∣∇∣βu-\nu|\nabla|^\beta u, then β=2α\beta=2\alpha. A parameter range stated for β\beta cannot be read as the same range for α\alpha. The endpoint α=1\alpha=1 recovers ordinary viscosity; it is outside the strict hypodissipative range used here.

References