2.6 Dissipative solutions of compressible fluids
2.6.3 A possible adaptation for a special domain
In the previous sections we have given the definition of a weak and adissipative solution in the bounded domain and the full domain. Now we consider the domain Ω =R2×(0,1), an infinite slab.
Navier stokes system:
For the compressible Navier–Stokes system (2.2.1)-(2.2.3) with a monotone isen-tropic pressure law (2.1.35) and finite energy initial data, we assume a far field condition,
|ϱ−ϱ˜| →0, u→0as|xh| → ∞, (2.6.19) where(ϱ˜,0)is a static solution, and, a boundary condition
u·n= 0 and [S·n]tan = 0 on∂Ω. (2.6.20) In the presence of an external forcef in the momentum equation we observe that the static solutionsϱ˜satisfy
∇xp(ϱ˜) =ϱ˜f, for a time independent functionf.
Weak solution: First we give the definition of weak solution in this domain from Feireisl and Novotný [75, Section 2.2]. We consider a finite energy initial data, i.e., ϱ0≥0,ϱ0 ∈L1loc(Rd) and
E0 = ˆ
Ω
(︃1 2
|(ϱu)0|2 ϱ0
+ (P(ϱ0)−(ϱ0−ϱ˜)P′(ϱ˜))−P(ϱ˜) )︃
dx <∞. (2.6.21) Definition 2.6.7. Let γ ≥1 and(ϱ0,(ϱu)0) be a finite energy initial data. We say that(ϱ,u) is a weak solution of the Navier–Stokes system with pressure law(2.1.35) inΩ =R2×(0,1), if the following is true.
• Regularity class: We have 0≤ϱ,ϱ−ϱ˜∈Cweak(0, T;L2+Lγ(Ω)), u∈L2(0, T;W1,2(Ω;Rd))and ϱu∈Cweak(0, T;L2+L
2γ γ+1(Ω)).
• The renormalized continuity equation holds in weak sense for the class of test functions is Cc∞([0, T)×Ω). The momentum equation remains true in weak sense for text function class {ϕϕϕ∈Cc∞([0, T]×Ω;Rd)|ϕϕϕ·n= 0 inΩ}.
• The far field conditions are incorporated through the energy inequality. The total energyE is defined in[0, T) as,
E(τ) = ˆ
Ω
(︃1
2ϱ|u|2+ (P(ϱ)−(ϱ−ϱ˜)P′(ϱ˜))−P(ϱ˜) )︃
(τ,·) dx
It satisfies, example isG(xh, x3) =−x3 which resemblances the simplest form of thegravitational potential, as a consequence of such choice of G, one can choose ϱ˜∈C2(Ω)∩L∞(Ω). Later In our application we consider this particular form of G. Hence we investigate on this particularG.
Next, we give the definition of a dissipative solution.
Definition 2.6.9. Let f ∈ L∞(Ω) and Let 0 < ϱ˜ ∈ W1,∞(Ω) and it satisfies
∇xp(ϱ˜) =ϱ˜f. We say that(ϱ,u) with
ϱ−ϱ˜∈Cweak([0, T];L2+Lγ(Ω)), ϱ≥0, ϱu∈Cweak([0, T];L2+Lγ+12γ (Ω)), andu∈L2(0, T;W1,2(Ω)),
is adissipative solution to (2.2.1)-(2.2.3) with boundary condition (2.6.20), initial data(ϱ0,(ϱu)0)and far field condition (2.6.19) satisfying
ϱ0 ≥0, E0 = if there exist theturbulent defect measures
Rm∈L∞(0, T;M+(Ω;Rd×dsym)), Re ∈L∞(0, T;M+(Ω)), satisfying the compatibility condition
λ1Tr(Rm)≤Re≤λ2Tr(Rm), λ1, λ2>0, (2.6.24) such that the following holds:
• Equation of continuity: For any φ∈Cc∞([0, T]×Ω), it holds
• Energy inequality: The total energyE is defined in [0, T)as
In the Definition 2.6.7 as well as in Definition 2.6.9, we notice a different form of the energy inequality, (2.6.22) and (2.6.27). In the sub-section 2.2.2, we discussed informally the invading domain technique forRd, and how the far field conditions are incorporated through energy inequality.
In this case, the situation is a bit more delicate. We again try to justify informally how we obtain such energy inequality as described in (2.6.22) and (2.6.27).
First,we assume f = ∇xG with G ∈ W1,∞(Ω) and the initial data (ϱ0,(ϱu)0) satisfies (2.6.21). We consider ΩR=B(0, R)×(0,1)whereB(0, R)is a ball of radius R in R2, also assume the system is provided by no-slip boundary condition, i.e., uR = 0 on ∂ΩR, where (ϱR,R) denotes a finite energy weak solution in ΩR. We recall the energy inequality in ΩR:
ER(τ) + from the continuity equation we have
[︃ˆ
This motivates to consider E˜R(τ) =
Using this we rewrite energy inequality as E˜R(τ) +
ˆ τ
0
ˆ
ΩR
S(∇xuR) :∇xuR dxdt ≤E˜0,R (2.6.28) Next, we consider a possible extension inRdas
ϱ
`R=
{︄ϱR inB(0, R)
ϱ˜otherwise andu`R=
{︄uR inB(0, R) 0 otherwise .
This helps us to extend the inequality (2.6.28) in Rd. We haveE0,R≤E0, whereE0 is independent ofR. Using some structural property of kinetic energy and pressure, we obtain an uniform bound forϱ`R−ϱ˜, and similarly for other variables. Finally a suitable limiting process gives the precise energy inequality as in (2.6.22) or (2.6.27).
The above discussion is too informal, it is just to give an idea how we get the energy inequality. It is mathematically incorrect formulation for our problem as we consider no-slip boundary condition on ∂(B(0, R)×(0,1)) instead of proposed Navier slip boundary. One can consider the weak solutions with Navier-slip boundary condition in bounded domain but this leads some other problem of possible zero extensions ofuR. Although there is a standard approach to deal this difficulty is by introducing a suitable symmetry class.
Symmetry Class: Ebin[46] described that the slip boundary condition (imper-meability boundary condition) inR2×(0,1)can be transformed into periodic ones by considering the space of symmetric functions. Here ϱ,uh(=u1, u2) were extended as even functions in thex3-variable defined on R2×T1, whileu3 is extended as an odd function inx3 on the same set, i.e.,
ϱ(t, xh,−x3) =ϱ(t, xh, x3), uh(t, xh,−x3) =uh(t, xh, x3),
u3(t, xh,−x3) =−u3(t, xh, x3). (2.6.29) for allt∈(0, T), xh ∈R2, x3 ∈T1. A similar convention is adopted for the initial data.
Hence, the consideration of the domain R2×(0,1)with slip boundary condition is equivalent toR2×T1. We have to consider solutions in the class (2.6.29). Just a small remark that, now we can justify the consideration of no-slip boundary condition on∂(B(0, R)×T1) in the informal justification of the energy inequality above and a possible extension of uR in wholeΩby zero outsideB(0, R)×T1. Here also we have a similar definition of weak solution in domain R2×T1
Definition 2.6.10. Let γ ≥1, (ϱ0,(ϱu)0)is a finite energy initial data, then we say (ϱ,u) solves the Navier–Stokes system with pressure law in R2×T1, if
• Regularity class: 0≤ϱ,ϱ−ϱ˜∈Cweak(0, T;L2+Lγ(Ω)), u∈L2(0, T;W1,2(Ω;Rd))and ϱu∈Cweak(0, T;L2+L
2γ γ+1(Ω)).
• The renormalized continuity equation holds for the class of test functions is Cc1([0, T]×R2×T1). The momentum equation remains true for text functions in Cc1([0, T]×R2×T1;Rd).
• The energy inequality is similar to that in (2.6.27).
Dissipative solution: Similarly, we can provide the definition of a dissipative solution in Ω =R2×T1. The definition is similar to the Definition 2.6.9,
Definition 2.6.11. Letγ ≥1, (ϱ0,(ϱu)0) is a finite energy initial data, then we say (ϱ,u)solves the Navier–Stokes system with pressure law in R2×T1, if
• Regularity class: 0≤ϱ, ϱ−ϱ˜∈Cweak(0, T;L2+Lγ(Ω)), u∈L2(0, T;W1,2(Ω;Rd))and ϱu∈Cweak(0, T;L2+Lγ+12γ (Ω)).
• The renormalized continuity equation holds for the class of test functions is Cc1([0, T]×R2×T1). The momentum equation remains true for text functions in Cc1([0, T]×R2×T1;Rd).
• The far field conditions are incorporated through the energy inequality. The total energyE is defined in[0, T) as,
E(τ) = ˆ
Ω
(︃1
2ϱ|u|2+ (P(ϱ)−(ϱ−ϱ˜)P′(ϱ˜))−P(ϱ˜) )︃
(τ,·) dx It satisfies,
E(τ) + ˆ τ
0
ˆ
Ω
S(∇xu) :∇xu dx dt ≤E0 (2.6.30) for a.e. τ >0.
Euler System
For the compressible Euler system (2.2.1)-(2.2.3) with a monotone isentropic pressure law (2.1.35) and finite energy initial data, we assume a far field and boundary condition as follows:
• Boundary condition: The impermeability or slip boundary condition on ∂Ωis given by
m·n= 0 on ∂Ω.
• Far field condition: Consideringx= (xh, x3), the conditions read as
|ϱ−ϱ˜| →0, u→0 as|xh| → ∞, (2.6.31) where a static solution(ϱ˜,0)satisfies∇xp(ϱ˜) =ϱ˜f inΩ. Now we provide the definition.
Definition 2.6.12. We say functions ϱ,u with
ϱ−ϱ˜∈Cweak([0, T];L2+Lγ(Ω)), ϱ≥0, m∈Cweak([0, T];L2+L
2γ γ+1(Ω)), are adissipative solution to the compressible Euler equation (2.3.1)-(2.3.2) with initial data(ϱ0,(ϱu)0)satisfying, if there exist theturbulent defect measures
Rm∈L∞(0, T;M+(Ω;Rd×dsym)), Re ∈L∞(0, T;M+(Ω)), satisfying compatibility condition
λ1Tr(Rm)≤Re≤λ2Tr(Rm), λ1, λ2>0, (2.6.33) such that the following holds:
• Equation of continuity: For any τ ∈(0, T) and anyφ∈Cc1([0, T)×Ω) it holds
• Energy inequality: The total energyE is defined in[0, T) as E(τ) =
Remark 2.6.13. If we want to provide the definition in R2×T1. Then we have to mentionϱ,mbelong to certain symmetry class along with f. The test function in momentum equation belongs to the class{φφφ∈Cc1([0, T)×Ω;Rd)}.
Remark 2.6.14. Euler system is equipped with impermeability boundary condition.
Existence of s dissipative solution of Euler system can be proved by taking a vanishing viscosity limit of the Navier–Stokes system equipped with Navier slip boundary condition.
Remark 2.6.15. We can repeat our ‘informal justification’ in this context to legit-imize the choice of total energy and energy inequality (2.6.36).
Remark 2.6.16. The purpose of this definition is to use it in the case of rotating fluids which we consider in the Chapter 4.