Dynamics for Generalized Incompressible Navier--Stokes Equations in ℝ²

Let ν,γ>0, α∈(12,1) and f∈(W1,2⁢(ℝ2))2. Then for each initial data u0∈V and any T>0, there exists a unique u∈L∞⁢(0,T;V) such that (2.2) and (2.3) hold. Moreover, u′∈L2⁢(0,T;H) and u∈C⁢([0,T);V).

Proof.

The existence can be obtained by the well-known viscosity solution method (see [16]): consider the following equation, ε>0:

By the classical result for Navier–Stokes equations we know that for each ε>0 and u0∈V, there exists a unique solution uε∈L∞⁢(0,T;V)∩L2⁢(0,T;H2⁢(ℝ2)) which satisfies (2.4)1 almost everywhere in ℝ2×(0,T).

We will get a solution for the weak formulation (2.2) by taking ε→0+.

For this, we need to show that {uε} is bounded in V with the bounds independent of ε, which will imply u∈L∞⁢(0,T;V). Indeed, this can be done by considering the corresponding vorticity ωε (the curl of the incompressible two-dimensional velocity):

ω ε = ∇ ⊥ ⋅ u ε = ∂ 1 ⁡ u ε ⁢ 2 - ∂ 2 ⁡ u ε ⁢ 1 ,

which obeys

∂ t ⁡ ω ε + u ε ⋅ ∇ ⁡ ω ε + γ ⁢ ω ε + ν ⁢ Λ 2 ⁢ α ⁢ ω ε - ε ⁢ Δ ⁢ ω ε = g

with g∈L2⁢(ℝ2), g=∇⊥⋅f. It is easy to see that ∥ωε⁢(⋅,t)∥L2⁢(ℝ2) can be controlled by the bounds which depend only on ∥ωε⁢(⋅,0)∥L2⁢(ℝ2) and the coefficients in the equations. Then the uniformly (with respect to ε) boundedness of uε in V can be deduced immediately by noting uε=K⋆ωε (see, e.g., Lemma 3.4; where K=12⁢π⁢x⊥|x|2 is the Biot–Savart kernel). Note that here we used crucially the fact that u0∈V and f∈(W1,2⁢(ℝ2))2.

To see u∈L2⁢(0,T;H2⁢α), we multiply (1.1) (this can be justified by multiplying (2.4) with Λ2⁢α⁢uε and then taking the limitation) by Λ2⁢α⁢u and integrate in space to deduce that

d d ⁢ t ⁢ ∥ Λ α ⁢ u ∥ 2 + 2 ⁢ γ ⁢ ∥ Λ α ⁢ u ∥ 2 + 2 ⁢ ν ⁢ ∥ Λ 2 ⁢ α ⁢ u ∥ 2 ≤ 2 ⁢ ∫ ℝ 2 | u ⋅ ∇ ⁡ u | ⁢ | Λ 2 ⁢ α ⁢ u | ⁢ 𝑑 x .

We will use the fact u∈L∞⁢(0,T;V) to deal with the nonlinear term as follows:

where p∈(2,∞) is large enough (e.g., H(2⁢α-1)-↪L2⁢pp-2⁢(ℝ2) since α∈(12,1)) such that

for any η>0; where (2⁢α-1)- denotes the number smaller than 2⁢α-1 but approaching 2⁢α-1. Consequently, combining with H1↪Lp⁢(ℝ2) for any p∈[2,∞), we have

Hence, by taking η small enough such that 3⁢η⁢∥u∥L∞⁢(0,T;V)≤ν, we integrate the above inequality over (0,T) and obtain that

ν ⁢ ∫ 0 T ∥ Λ 2 ⁢ α ⁢ u ⁢ ( s ) ∥ 2 ⁢ 𝑑 s ≤ ∥ Λ α ⁢ u ⁢ ( 0 ) ∥ 2 + C η ′ ⁢ ∥ u ∥ L ∞ ⁢ ( 0 , T ; V ) ⁢ ∫ 0 T ∥ u ⁢ ( s ) ∥ 2 ⁢ 𝑑 s ,

which, combined with the fact u⁢(0)∈V and the a priori bounds about ∥u∥L∞⁢(0,T;V) again, implies that u∈L2⁢(0,T;H2⁢α).

We can see from (1.1) that u′∈L2⁢(0,T;H), directly after we obtained u∈L∞⁢(0,T;V)∩L2⁢(0,T;H2⁢α). Then u∈C⁢([0,T);V) follows immediately.

The uniqueness is obvious by noticing the fact that the solution belongs to L∞⁢(0,T;V): let (ui,pi) be the solutions corresponding to initial data u0i, i=1,2, and write w=u1-u2, then w solves the following equation:

Multiplying (2.7) by w and integrating in space, we get

Then, noting that u1∈L∞⁢(0,T;V) and using the fact (since α>12)

∥ w ⁢ ( t ) ∥ L 4 ⁢ ( ℝ 2 ) ≤ C ⁢ ∥ w ⁢ ( t ) ∥ r ⁢ ∥ w ⁢ ( t ) ∥ H α 1 - r

for some constant r∈(0,1), we can deduce from (2.8) that

d d ⁢ t ⁢ ∥ w ⁢ ( t ) ∥ 2 ≤ C 1 ⁢ ∥ w ⁢ ( t ) ∥ 2 ,

where the constant C1 depends on ∥u1∥L∞⁢(0,T;V) and γ,ν; which immediately implies that

This finishes the proof. ∎

Let 0<α<2, x∈ℝ2 and θ∈𝒮, the Schwartz class. Then

Λ α ⁢ θ ⁢ ( x ) = C α ⁢ P . V . ∫ ℝ 2 [ θ ⁢ ( x ) - θ ⁢ ( y ) ] | x - y | 2 + α ⁢ 𝑑 y ,

where Cα>0.

Proposition 2.3 (Pointwise identity)

Let 0≤α≤2, x∈ℝ2 and θ∈𝒮. Then

2 ⁢ θ ⁢ Λ α ⁢ θ ⁢ ( x ) = Λ α ⁢ ( θ 2 ⁢ ( x ) ) + D α ⁢ [ θ ] ⁢ ( x ) ,

where

D α ⁢ [ θ ] ⁢ ( x ) = P . V . ∫ ℝ 2 [ θ ⁢ ( y ) - θ ⁢ ( x ) ] 2 | x - y | 2 + α ⁢ 𝑑 y ≥ 0 .

Proposition 2.4 (Positivity lemma)

Let 0≤α≤2, x∈ℝ2 and θ,Λα⁢θ∈Lp with 1≤p<∞. Then

∫ ℝ 2 | θ | p - 2 ⁢ θ ⁢ Λ α ⁢ θ ⁢ 𝑑 x ≥ 0 .

Let u0∈V. Then the solution of (1.1) with initial datum u0 exists for all time, is unique and satisfies the energy equation

The kinetic energy is bounded uniformly in time, with bounds independent of viscosity ν:

The vorticity

ω = ∇ ⊥ ⋅ u = ∂ 1 ⁡ u 2 - ∂ 2 ⁡ u 1

obeys

with g∈L1⁢(ℝ2)∩L∞⁢(ℝ2), g=∇⊥⋅f. The map

[ 0 , + ∞ ) → L 2 ⁢ ( ℝ 2 ) , t ↦ ω ⁢ ( t )

is continuous. If the initial vorticity ω0 is in Lp⁢(ℝ2),p>1, then the p-enstrophy is bounded uniformly in time, with bounds independent of viscosity:

for p≥1. Moreover, the positive semi-orbit

O + ⁢ ( ω ) = { ω = ω ⁢ ( ⋅ , t ) ∣ t ≥ 0 } ⊂ L 2 ⁢ ( ℝ 2 )

is equi-integrable in L2⁢(ℝ2): for every ϵ>0, there exists R>0 such that

for all t≥0, where the radius R depends on the coefficients γ,ν,α and ω⁢(x,0).

Proof.

See Lemma 2.1 for the existence and uniqueness of solutions.

The energy equation (3.1) follows from the incompressibility of u and integration by parts. The bounds (3.2) and (3.4) follow from the positivity lemma and an application of the Gronwall inequality; see, for example, Section 4.

In the following, we only prove the equi-integrability (3.5). As in [5], we consider the function

Y R ⁢ ( t ) = ∫ ℝ 2 χ ⁢ ( x R ) ⁢ ω 2 ⁢ ( x , t ) ⁢ 𝑑 x ,

where χ⁢(⋅) is a nonnegative smooth function supported in {x∈ℝ2:|x|≥12} and identically equal to 1 for |x|≥1. We multiply equation (3.3) by 2⁢χ⁢(xR)⁢ω⁢(x,t) and integrate in space. The only challenging term we encounter is

2 ⁢ ν ⁢ ∫ ℝ 2 Λ 2 ⁢ α ⁢ ω ⁢ ( x ) ⁢ χ ⁢ ( x R ) ⁢ ω ⁢ ( x , t ) ⁢ 𝑑 x .

Using Proposition 2.3, we have

where we have used

∫ ℝ 2 Λ 2 ⁢ α ⁢ ( ω 2 ) ⁢ 𝑑 x = Λ 2 ⁢ α ⁢ ( ω 2 ) ^ ⁢ ( 0 ) = 0

since α>0, and the fact that Λ2⁢α⁢(1-χ⁢(xR)) is well defined because 1-χ⁢(xR)∈Cc∞⁢(ℝ2). Moreover, it is easy to see that

1 - χ ( x R ) = : ϕ ( x R ) .

In view of

Λ 2 ⁢ α ⁢ ( ϕ ⁢ ( x R ) ) = 1 R 2 ⁢ α ⁢ ( Λ 2 ⁢ α ⁢ ϕ ) ⁢ ( x R )

and

| ( Λ 2 ⁢ α ⁢ ϕ ) ⁢ ( x ) | ≤ ∫ ℝ 2 | Λ 2 ⁢ α ⁢ ϕ ^ ⁢ ( ξ ) | ⁢ 𝑑 ξ = ∫ ℝ 2 | ξ | 2 ⁢ α ⁢ | ϕ ^ ⁢ ( ξ ) | ⁢ 𝑑 ξ ≤ C 0 ,

we have

So we can obtain that

d d ⁢ t ⁢ Y R ⁢ ( t ) + 2 ⁢ γ ⁢ Y R ⁢ ( t ) ≤ C R ⁢ ∥ u ⁢ ( ⋅ , t ) ∥ L 2 ⁢ ( ℝ 2 ) ⁢ ∥ ω ⁢ ( ⋅ , t ) ∥ L 4 ⁢ ( ℝ 2 ) 2 + C 0 ⁢ ν R 2 ⁢ α ⁢ ∥ ω ⁢ ( ⋅ , t ) ∥ L 2 ⁢ ( ℝ 2 ) 2 + C ⁢ ( Y R ⁢ ( t ) ⁢ ∫ | x | ≥ R 2 | g ⁢ ( x ) | 2 ⁢ 𝑑 x ) 1 2 ,

that is,

On the other hand, multiplying (3.3) by ω⁢(x,t) and integrating in space, we obtain that

d d ⁢ t ⁢ ∫ ℝ 2 | ω ⁢ ( x , t ) | 2 ⁢ 𝑑 x + 2 ⁢ γ ⁢ ∫ ℝ 2 | ω ⁢ ( x , t ) | 2 ⁢ 𝑑 x + 2 ⁢ ν ⁢ ∫ ℝ 2 | Λ α ⁢ ω ⁢ ( x , t ) | 2 ⁢ 𝑑 x ≤ 2 ⁢ ∫ ℝ 2 | ω ⁢ ( x , t ) | ⁢ | g ⁢ ( x ) | ⁢ 𝑑 x ,

which implies that, for any t≥0,

∥ ω ⁢ ( x , t ) ∥ L 2 ⁢ ( ℝ 2 ) 2 + 2 ⁢ ν ⁢ ∫ 0 t e γ ⁢ ( s - t ) ⁢ ∥ Λ α ⁢ ω ⁢ ( x , s ) ∥ L 2 ⁢ ( ℝ 2 ) 2 ⁢ 𝑑 s ≤ F < ∞

with a positive constant F which is bounded in terms of γ, ∥g∥L2⁢(ℝ2)2 and ∥ω⁢(x,0)∥L2⁢(ℝ2)2 (but is independent of t). Hence, combining with the embedding Hα↪L4⁢(ℝ2) for α>12, we have

Therefore, from (3.2), (3.4), (3.8) and the fact that |g|2 is integrable, the right-hand side of (3.7) will be arbitrary small if we take R large enough. Then the equi-integrability (3.5) follows from the fact that YR⁢(0) is small for large R. ∎

The vorticity equation (3.3) with a different draft term is a special case of the viscous surface quasi-geostrophic equation, which has been studied by many authors (e.g., see [1, 6, 7, 8] and the references therein). Especially, in [8], Dlotko, Kania and Sun obtained some approximation estimates that may be helpful in considering the inviscid limit problem (as what we do in this section for (1.1)) for the surface quasi-geostrophic equation.

Let ω0∈L2⁢(ℝ2), g∈L2⁢(ℝ2) and u∈L∞⁢(0,∞;V). Then for any t0>0, the positive semi-orbit

O + ⁢ ( t 0 , ω 0 ) = { ω ⁢ ( ⋅ , t ) ∣ t ≥ t 0 }

is relatively compact in L2⁢(ℝ2).

Let s≥0 and Λs⁢ω∈L2⁢(ℝ2). Then for u=12⁢π⁢x⊥|x|2⋆ω, we have Λ1+s⁢u∈(L2⁢(ℝ2))2 and

∥ Λ 1 + s ⁢ u ∥ ( L 2 ⁢ ( ℝ 2 ) ) 2 ≤ C ⁢ ∥ Λ s ⁢ ω ∥ L 2 ⁢ ( ℝ 2 ) ,

Proof.

Note that u=12⁢π⁢x⊥|x|2⋆ω=Λ-1⁢ℛ⊥⁢ω, hence

∥ Λ 1 + s ⁢ u ∥ ( L 2 ⁢ ( ℝ 2 ) ) 2 = ∥ Λ s ⁢ ℛ ⊥ ⁢ ω ∥ ( L 2 ⁢ ( ℝ 2 ) ) 2 ≤ C ⁢ ∥ Λ s ⁢ ω ∥ L 2 ⁢ ( ℝ 2 ) ,

where the constant C depends only on the L2-bounds of the Riesz transform ℛ. ∎

Let ω0∈L2⁢(ℝ2), g∈L2⁢(ℝ2) and u∈L∞⁢(0,∞;V). Then the solution ω⁢(⋅,t) of (3.3) is uniformly (with respect to t≥t0 ) bounded in Wα,2⁢(ℝ2) for any t0>0.

Proof.

At first, we multiply (3.3) by ω to deduce that

where the constant M~0 depends only on ν, ∥g∥L2⁢(ℝ2) and ∥ω⁢(⋅,0)∥L2⁢(ℝ2).

Secondly, we multiply again the vorticity equation (3.3) by Λ2⁢α⁢ω and integrate in space, to get

d 2 ⁢ d ⁢ t ⁢ ∫ ℝ 2 | Λ α ⁢ ω | 2 ⁢ 𝑑 x + ∫ ℝ 2 ( u ⋅ ∇ ⁡ ω ) ⁢ Λ 2 ⁢ α ⁢ ω ⁢ 𝑑 x + γ ⁢ ∫ ℝ 2 | Λ α ⁢ ω | 2 ⁢ 𝑑 x + ν ⁢ ∫ ℝ 2 | Λ 2 ⁢ α ⁢ ω | 2 ⁢ 𝑑 x = ∫ ℝ 2 g ⁢ Λ 2 ⁢ α ⁢ ω ⁢ 𝑑 x .

The nonlinear term ∫ℝ2(u⋅∇⁡ω)⁢Λ2⁢α⁢ω⁢𝑑x can be estimated as (2.5) and (2.6):

| ∫ ℝ 2 ( u ⋅ ∇ ⁡ ω ) ⁢ Λ 2 ⁢ α ⁢ ω ⁢ 𝑑 x | ≤ C ⁢ ∥ u ∥ V ⋅ ∥ ω ∥ L 2 ⁢ ( ℝ 2 ) 1 - r ⋅ ∥ Λ 2 ⁢ α ⁢ ω ∥ L 2 ⁢ ( ℝ 2 ) 1 + r

with some positive constant r∈(0,1). Then, applying the Young inequality, we obtain that

where the constant M~1 depends on ∥u⁢(⋅,t)∥V, ∥ω⁢(⋅,t)∥L2⁢(ℝ2), ∥g∥L2⁢(ℝ2) and ν.

Then, combining with (3.4), (3.9), (3.10) and Lemma 3.4, we can finish the proof by applying a Gronwall’s inequality. ∎

Let ω0∈L2⁢(ℝ2)∩L∞⁢(ℝ2), g∈L1⁢(ℝ2)∩L∞⁢(ℝ2) and u∈L∞⁢(0,∞;V). Then the solution ω⁢(⋅,t) of (3.3) satisfies

ω ∈ L ∞ ⁢ ( ℝ 2 × ( t 0 , ∞ ) )

for any t0>0.

Proof.

The proof is based on an idea of Caffarelli and Vasseur [1]. The details for our case are exactly as in [2, Lemmas 2.2 and 2.3]. Note that our assumptions here are stronger than that required in [2]. ∎

A stationary statistical solution of damped and driven generalized incompressible Navier–Stokes equations in the vorticity phase space is a Borel probability measure μ(ν) on L2⁢(ℝ2) such that

and