SLIDE 1 Transition Regime Models
A Low Mach Number Limit of a Dispersive Navier-Stokes System
Konstantina Trivisa
Joint work with C.D. Levermore and Weiran Sun
HYP-2012, Padova
SLIDE 2 Transition Regime Models
Outline What is our objective? Transition regime models. Main result. Local well-posedness results for the DNS system and the ghost effect system. Strategy for the rigorous proof of the low Mach number limit
Take into account the anti-symmetric structure of the high-order dispersive term. Establish uniform bounds for the solution by considering the interaction of its fast and slow motion parts.
A priori estimates for the slow motion ψǫ = (ǫpǫ, ǫuǫ, θǫ). A priori estimates for the fast motion (pǫ, uǫ). Local decay of the energy of the fast motion.
SLIDE 3 Transition Regime Models
Our objective Establish a low Mach number limit for classical solutions over the whole space of a compressible fluid dynamic system that includes dispersive corrections to the Navier-Stokes equations. ⇓ The limiting system is a so-called ghost effect system (Sone 2002) which is not derivable from the Navier-Stokes system of gas dynamics but is derivable from kinetic equations.
SLIDE 4 Transition Regime Models
The long term objective This work is part of a research program that aims to identify fluid dynamic regimes and to construct a unified model that captures them all. Such a model can also be useful in transition regimes when classical fluid equations are inadequate to describe the dynamics of fluids while computations using kinetic models are expensive.
SLIDE 5
Kinetic equations
Evolution of systems consisting of a large number of particles. Mathematical formulation: partial differential equations for the particle distribution function f(t, x, v):
t: time; x: position; v: velocity; f ≥ 0.
SLIDE 6 Classical examples of kinetic equations:
Boltzmann equation (gas dynamics): ∂tf + v · ∇
xf = C[f];
f = f(t, x, v): particle distribution function; C[f]: collision among particles; Example: C[f] =
1) − f(v)f(v1)
SLIDE 7
Classical macroscopic regime:
Many collisions, systems close to local equilibrium. Measurement: Kn = mean free path macroscopic length .
Kn: Knudsen number; Macroscopic regime: small Knudsen number regime.
SLIDE 8 Classical example of fluid equations:
Navier-Stokes equation ∂tρ + ∇
x · (ρu) = 0 ,
conservation of mass
∂t(ρu) + ∇
x · (ρu ⊗ u) + ∇ xp = ∇ x · Σ ,
conservation of momentum
∂t(ρe) + ∇
x · (ρeu) + ∇ x · (pu) = ∇ x · (Σu) − ∇ x · q , conservation of energy
ρ: mass density ρu: momentum density ρe = 1
2ρ|u|2 + D 2 ρθ: energy density
p: pressure Σ: viscous stress tensor q: heat flux (p, Σ, q) = (p, Σ, q)(ρ, u, θ, ∇
xu, ∇ xθ): constitutive relation
SLIDE 9 Kinetic equation =⇒ fluid equations?
Boltzmann equation ∂tf + v · ∇
xf = 1
Kn C[f]
ρ = f, ρu = vf, ρe = 1
2|v|2f.
→ ⇓ ← Kn → 0
Navier-Stokes equation ∂tρ + ∇
x · (ρu) = 0 ,
∂t(ρu) + ∇
x · (ρu ⊗ u) + ∇ xp = ∇ x · Σ ,
∂t(ρe) + ∇
x · (ρeu) + ∇ x · (pu) = ∇ x · (Σu) − ∇ x · q ,
SLIDE 10 Transition Regime Models
Transition Regime Plenty collisions but not enough to drive the system very close to local equilibrium; Computations using full kinetic equations are expensive. Classical macroscopic systems are inaccurate. Purpose of developing transition regime models Bridge the gap between kinetic equations and classical macroscopic equations.
SLIDE 11 Transition Regime Models
Transition regime models Kinetic equations ⇓ Transition regime models ⇓ Classical macroscopic equations
SLIDE 12 Transition Regime Models
Construction and analysis of interior equations:
Higher order equations: adding higher order correction terms; Larger moment systems: including more moments of the density function.
Construction of appropriate boundary conditions:
Boundary conditions of macroscopic equations should match the given boundary conditions of the underlying kinetic equations. Boundary layer analysis.
SLIDE 13 Transition Regime Models Thermal induced flow
Thermal induced flow:
Maxwell: 1879 ρ∂tu + ∇
xp − ∇ x · Σ + τ∇ x∆xθ = 0.
Kogan, Galkin and Fridlender: nonlinear thermal stress, 1976. Aoki, Sone, Sugimoto, Takata ...
SLIDE 14 Transition Regime Models Ghost effect regime
Ghost effect regime:
small bulk velocity: U = ǫu small fluctuations in pressure field: p = p0 + ǫp1 large variation in temperature/density field: ∇
xρ, ∇ xθ ∼ O(1)
Classical fluid equations like NS are not accurate in the ghost effect regime.
SLIDE 15 Transition Regime Models Ghost-effect system
Ghost effect system ∇
x(ρθ) = 0,
∂tρ + ∇
x · (ρu) = 0,
∂t(ρu) + ∇
x · (ρu ⊗ u) + ∇ xP = ∇ x · Σ + ∇ x · ˜
Σ, ∂t(ρθ) + ∇
x · (ρθu) = −∇ x · q,
(1) Constitutive relation Σ = µ(θ)
xu + (∇ xu)⊤ − 2
3(∇
x · u)I
˜ Σ = τ1(θ)
x θ − 1
3(∆xθ)I
xθ ⊗ ∇ xθ − 1
3|∇
xθ|2I
q = −k(θ)∇
xθ.
SLIDE 16 Transition Regime Models Main Question
Question: Can one in some sense unify the classical fluid equations and the ghost-effect system? ⇓ Higher order equations: dispersive Navier-Stokes. Reference:
Levermore: Gas Dynamics Beyond Navier-Stokes .
SLIDE 17 Transition Regime Models Main Question
Main questions:
Well-posedness of the DNS system? Recovery of the ghost effect system in the ghost effect regime?
SLIDE 18 Transition Regime Models Main Question
Main results:
Well-posedness of the DNS system?
Result: local well-posedness in Sobolev spaces; Reference: Ph.D. Thesis of Weiran Sun (2009).
Recovery of a ghost effect system in the ghost effect regime?
Result: DNS converges to a ghost effect system in the low Mach number limit. Reference: Levermore, Sun, Trivisa SIMA (2012).
SLIDE 19 Transition Regime Models The model
Dispersive Navier-Stoke system (DNS): ∂tρ + ∇
x · (ρu) = 0 ,
ρ∂tu + ρu · ∇
xu + ∇ xp(ρ, θ) = ∇ x · Σ + ∇ x · ˜
Σ
3 2ρ∂tθ + 3 2ρu · +∇ xθ + p∇ x · u = (Σ + ˜
Σ) : ∇
xu + ∇ x · ˜
q − ∇
x · q ,
(ρ, u, θ)(x, 0) = (ρin, uin, θin)(x) , (2) time variable t ∈ [0, ∞), space variable x ∈ R3. Density: ρ, bulk velocity: u ∈ R3, temperature: θ.
SLIDE 20 Transition Regime Models Constitutive relations
Constitutive relations Ideal gas law: p(ρ, θ) = ρθ. Viscous stress tensor: Σ = µ(θ)(∇
xu + (∇ xu)⊤ − 2
3(∇
x · u)I), µ(θ) > 0.
Heat flux: q(θ) = −κ(θ)∇
xθ, κ(θ) > 0.
The total energy density: ρe = 1 2ρ|u|2 + 3 2ρθ.
SLIDE 21 Transition Regime Models Constitutive relations
The quantities ˜ Σ and ˜ q denote dispersive corrections to the stress tensor and heat flux respectively and are given by
SLIDE 22 Transition Regime Models Constitutive relations
Dispersive term in the velocity equation ∇
x · ˜
Σ where ˜ Σ = τ1(ρ, θ)(∇2
x θ − 1
3(∆xθ)I) + τ2(ρ, θ)(∇
xθ ⊗ ∇ xθ − 1
3|∇
xθ|2I)
+ τ3(ρ, θ)(∇
xu(∇ xu)⊤ − (∇ xu)⊤∇ xu)
(3) Dispersive term in the temperature equation ∇
x · ˜
q where ˜ q = τ4(ρ, θ)(∆xu + 1 3∇
x∇ x · u)
+ τ5(ρ, θ)∇
xθ · (∇ xu + (∇ xu)⊤ − 2
3(∇
x · u)I)
+ τ6(ρ, θ)
xu − (∇ xu)⊤
· ∇
xθ.
(4) τ1, . . . , τ6 are C ∞ functions of their variables.
SLIDE 23 Transition Regime Models Constitutive relations
One feature of the DNS system is that it possesses an entropy structure provided the transport coefficients in ˜ Σ and ˜ q satisfy τ4 = θ 2τ1, τ2 θ + 2τ5 θ2 = ∂θ τ4 θ2
(5) such that ˜ Σ : ∇
xu
θ +˜ q·∇
xθ
θ2 = ∇
x·
τ1 2θ∇
xθ ·
xu + (∇ xu)⊤ − 2
3(∇
x · u)I
SLIDE 24 Transition Regime Models Entropy equation
Entropy density η = ρ log ρ θ3/2
Entropy equation ∂η + ∇
x ·
q θ
− Σ θ : ∇
xu − q
θ2 · ∇
xθ
˜ Σ θ : ∇
xu − ˜
q θ2 · ∇
xθ
Total entropy is formally dissipated by the dispersive NS system in the same way as in the NS system over domains without boundaries.
SLIDE 25 Transition Regime Models Local well-posedness
The proof of the local well-posedness of the DNS system follows using the classical energy method for hyperbolic-parabolic systems. Although we have third-order dispersive terms, the leading
- rders of these terms form an anti-symmetric structure.
⇓ Therefore they do not hamper the usual L2-Hs estimates. The rest of the dispersive terms are of orders up to two. Although this is the same order as the dissipation, they do not introduce extra difficulties because they are of order O(ǫ2) while the dissipative terms are of order O(ǫ). Here we do need the viscosity coefficient µ(θ) and κ(θ) to be bounded away from zero when θ is bounded from below.
SLIDE 26 Transition Regime Models Local well-posedness for the ghost effect system
Local well-posedness for the ghost effect system Classical energy estimates for hyperbolic-parabolic equations. Difficulty: There is a third-order term in θ in the GES and there is no anti-symmetric structure to balance this term. Key observation: The leading order of this term is in the form
- f a gradient, which can be incorporated into the pressure
- term. By doing so the rest of the terms can be treated as
- perturbations. Similar proofs can be found for combustion
models and Kazhikhov-Smagulov type models.
SLIDE 27 Transition Regime Models Validity of DNS
Main Question:
Can the dispersive Navier-Stokes system recover the ghost effect system in the ghost effect regime?
Result
The DNS system converges to the ghost effect system in a low Mach number limit.
SLIDE 28 Transition Regime Models Scaling
The ghost effect system can be formally derived from kinetic equations using a Hilbert expansion method (cf. Sone (2002)). This is a system beyond classical fluid equations that describes the phenomenon in which the temperature field of the fluid has finite variations, and the flow is driven by the gradient of the temperature field. Scaling
Kundsen number: ǫ. Transport coefficients µ = ǫˆ µ, κ = ǫˆ κ, τi = ǫ2ˆ τi, i = 1, · · · , 6 .
SLIDE 29 Transition Regime Models Scaling
Ghost effect regime pǫ = eǫpǫ , Uǫ = ǫuǫ , Θǫ = eθǫ , ρǫ = eρǫ. Long time scale: t = 1
ǫτ.
Low Mach number limit: Von K´ armen relation: Re ∝ Kn Ma. Here Ma denotes the Mach number which is typically used to compare a typical flow velocity with a characteristic speed of sound c.
SLIDE 30 Transition Regime Models Dispersive System in terms of Fluctuations
Flunctuation Equations: ∂ρǫ + ∇
x · (ρǫuǫ) = 0,
e−θǫ(∂t + uǫ · ∇
x)uǫ + 1 ǫ∇ xpǫ = e−ǫpǫ(∇ x · Σ + ∇ x · ˜
Σ),
3 2(∂t + uǫ · ∇ x)θǫ + ∇ x · uǫ = ǫ2e−ǫpǫ
Σ + ˜ Σ
xuǫ + ∇ x · ˜
q
- (ρǫ, uǫ, θǫ)(x, 0) = (ρin
ǫ , uin ǫ , θin ǫ )(x).
fast motion: (uǫ, pǫ) vary on the time scale O(ǫ); slow motion: (ǫpǫ, ǫuǫ, θǫ) vary on the time scale O(1).
SLIDE 31 Function Space
Define the operator Λǫ and norm wHs+1
ǫ
:
Λǫ = (I − ǫ2∆x)1/2 , wHs+1
ǫ
= ǫs+1wHs+1 + wHs Λs+1
ǫ
wL2 .
Define the norms:
|(p, u, θ)|ǫ,s,t := sup
[0,t]
ǫ
(ǫp, ǫu, θ)Hs+1
t
xu2 Hs + ∇ xΛs+1 ǫ
(ǫu, θ)2
Hs+1
1/2 , |(pin, uin, θin)|ǫ,s,0 := (pin, uin)Hs + Λs+1
ǫ
(ǫpin, ǫuin, θin)Hs+1 ,
SLIDE 32
Main Theorem
Theorem (Levermore-Sun-Trivisa) Let s ≥ 6. Suppose that there exist positive constants M0, θ, c0, σ such that the initial data of the fluctuation equations satisfy
|(pin
ǫ , uin ǫ , θin ǫ − θ)|ǫ,s,0 ≤ M0 ,
|θin
ǫ − θ| ≤ c0|x|−1−σ ,
|∇
xθin ǫ | ≤ c0|x|−2−σ .
(uin
ǫ , θin ǫ − θ) → (uin, θin − θ)
in Hs(R3) . Then there exists a T > 0 such that for any ǫ ∈ (0, 1], the Cauchy problem for the fluctuation equations has a unique solution
(pǫ, uǫ, θǫ − θ) ∈ C0([0, T]; H2s+1(R3)) ∩ L∞(0, T; H2s+2(R3)) ∩ L2(0, T; H2s+3(R3)) .
SLIDE 33 Main Theorem Continued
Theorem (Continued) Furthermore, there exists a positive constant M depending only on T, M0 such that
|(pǫ, uǫ, θǫ − θ)|ǫ,s,t ≤ M .
The sequence of solutions (pǫ, uǫ, θǫ) converges weakly ∗ in L∞(0, T; Hs(R3)) and strongly in L2(0, T; Hs′
loc(R3)) for all s′ < s to
the limit (0, u, θ), where (u, Θ) = (u, eθ) satisfies
̺Θ = 1 , ∂t̺ + ∇
x · (̺u) = 0 ,
∂t(̺u) + ∇
x · (̺u ⊗ u) + ∇ xP∗ = ∇ x · Σ + ∇ x · ˜
Σ , ∇
x ·
5
2u − κ(Θ)∇ xΘ
SLIDE 34 Main Theorem Continued
Theorem (Continued) The initial of the limiting ghost effect system satisfies
(u, θ)|t=0 = (vin, θin) ,
where
(uin
ǫ , θin ǫ − θ) → (vin, θin − θ)
in Hs(R3) ,
∇
x ·
5
2vin − κ(Θin)∇ xΘin
= 0 , Π 1 Θin uin
1 Θin vin
with Π being the projection operator onto the divergence free part in the Hodge decomposition.
SLIDE 35 Transition Regime Models
Strategy Our analysis builds on the framework of M´ etivier and Schochet (2001) and Alazard (2005). M´ etivier and Schochet proved the incompressible limit for the non-isentropic Euler equations for classical solutions with general initial data. Alazard proved the low Mach number limit for the compressible Navier-Stokes for classical solutions with general initial data.
- Due to the presence of the high-order dispersive terms, our result
does not follow from their analysis. We need to take into account
- f the anti-symmetric structure of those terms.
SLIDE 36 Transition Regime Models
Let uǫ = (pǫ, uǫ, θǫ) be the solution to the DNS system and ψǫ = (ǫpǫ, ǫuǫ, θǫ), then DNS can be reformulated as A1(ψǫ)(∂t + u · ∇
x)uǫ + 1
ǫ A2(ψǫ)uǫ = A3(ψǫ)uǫ + R , where A1(ψǫ) is a diagonal matrix, A2(ψǫ)uǫ are formed by certain combinations of the singular terms with the leading orders from the dispersive terms, A3(ψǫ)uǫ are the dissipative terms, and R includes the rest of the dispersive terms.
SLIDE 37 Transition Regime Models
Remark:
In the case of compressible Navier-Stokes A2(ψǫ) is anti-symmetric. Here it is not readily anti-symmetric but it is anti-symmetrizable by a certain symmetrizer matrix composed of symmetric positive operators. We establish uniform bounds for the solution by considering the interactions of its fast and slow motion parts.
SLIDE 38 Transition Regime Models Dispersive System in terms of Fluctuations
Theorem For each fixed ǫ > 0, let (pǫ, uǫ, θǫ) ∈ C([0, T]; Hs(R3)) be the solution to the scaled DNS system. Let Ω = | | |(pǫ, uǫ, θǫ−θ)| | |ǫ,s,T , Ω0 = | | |(pin
ǫ , uin ǫ , θin ǫ −θ)|
| |ǫ,s,0 . Then there exists an increasing function C(·) such that | | |(pǫ, uǫ, θǫ−θ)| | |ǫ,s,T ≤ C(Ω0)e(
√ T+ǫ)C(Ω) ,
which further indicates that there exists T0 > 0 independent of ǫ such that |(pǫ, uǫ, θǫ−θ)|ǫ,s,T are uniformly bounded in ǫ over [0, T0].
SLIDE 39 Transition Regime Models
Future program Global well-posed ness of DNS. Boundary conditions for transition regime models. Well-posedness of transition regime models on bounded domains. Comparison of transition regime models with classical systems.
