TY - CHAP

T1 - Existence, Uniqueness and Regularity for the Second-Gradient Navier-Stokes Equations in Exterior Domains

AU - Degiovanni, Marco

AU - Marzocchi, Alfredo

AU - Mastaglio, Sara

PY - 2021

Y1 - 2021

N2 - We study the well-posedness of the problem
⎧
⎪
⎨
⎪
⎩
∂u
∂t
+ (Du)u + ∇p = νΔu − τΔΔu in ]0,+∞[×Ω,
divu = 0 in ]0,+∞[×Ω,
u(t,x) =
∂u
∂n (t,x) = 0
on ]0,+∞[×∂Ω,
u(0,x) = u 0 (x) in Ω,
where u :]0,+∞[×Ω → R n is the velocity field, p :]0,+∞[×Ω → R is the pressure,
ν is the kinematical viscosity, τ the so-called hyperviscosity and Ω is a general
domain as for existence and uniqueness of the solution, and an exterior domain as
for regularity results.
This problem has been physically well motivated in the recent years as the
simplest case of an isotropic second-order fluid, i.e. a fluid whose power expended
depends on second derivatives of the velocity field.

AB - We study the well-posedness of the problem
⎧
⎪
⎨
⎪
⎩
∂u
∂t
+ (Du)u + ∇p = νΔu − τΔΔu in ]0,+∞[×Ω,
divu = 0 in ]0,+∞[×Ω,
u(t,x) =
∂u
∂n (t,x) = 0
on ]0,+∞[×∂Ω,
u(0,x) = u 0 (x) in Ω,
where u :]0,+∞[×Ω → R n is the velocity field, p :]0,+∞[×Ω → R is the pressure,
ν is the kinematical viscosity, τ the so-called hyperviscosity and Ω is a general
domain as for existence and uniqueness of the solution, and an exterior domain as
for regularity results.
This problem has been physically well motivated in the recent years as the
simplest case of an isotropic second-order fluid, i.e. a fluid whose power expended
depends on second derivatives of the velocity field.

KW - Fluid Mechanics

KW - Navier-Stokes equations

KW - Fluid Mechanics

KW - Navier-Stokes equations

UR - http://hdl.handle.net/10807/201701

U2 - 10.1007/978-3-030-68144-9

DO - 10.1007/978-3-030-68144-9

M3 - Chapter

SN - 978-3-030-68143-2

SP - 181

EP - 202

BT - Waves in Flows

A2 - Bodnar, Tomaš

A2 - Galdi, Giovanni P.

A2 - Nečasová, Šarka

ER -