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INTRODUCTION BACKGROUND RESULTS METHODOLOGY

The Doi Model for the Suspensions of Rod-like Molecules in a Compressible Fluid

Hantaek Bae

Center for Scientific Computation and Mathematical Modeling, University of Maryland Joint work with K. Trivisa, University of Maryland HYP 2012, Padova, Italy 6/28, 2012

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

DOI MODEL

The Doi model describes the interaction between

• 1. the orientation of molecules at the microscopic scale and;
• 2. the macroscopic properties of the fluid in which these molecules

are contained. Here, we consider the Doi model for suspensions of rod-like molecules in a dilute regime. Outline of the Talk

• 1. Introducing a compressible model;
• 2. Existence of a weak solution.

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

SYSTEM OF EQUATIONS

• 1. Conservation of mass: ρt + ∇ · (uρ) = 0.
• 2. Equation of the particle distribution

ft + ∇ · (uf) + ∇τ · (Pτ ⊥∇uτf) − ∆τf − ∆f = 0, τ ∈ S2, (1) ∇τ · (Pτ ⊥∇uτf) : a drift-term on S2 representing the shear forces acting on the rods, (2) Pτ ⊥∇uτ : the projection of the vector ∇uτ on S2, (3) ∆τf : the rotational diffusion = ⇒ change the orientation of rods spontaneously.

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

• 3. Equation of Motion: (ρu)t + ∇ · (ρu ⊗ u) = ∇ · T

T = S − pI3×3 (Stokes’ Law), S = Sf + Sp, p = pf + pp. (1) Sf =

• ∇u + (∇u)t

+ (∇ · u)I3×3, (2) Sp = σ − ηI3×3

• =

⇒Energy Dissipation

, (3) σ(t, x) =

• S2 (3τ ⊗ τ − I3×3) f(t, x, τ)dτ

(thermodynamic consistency), (4) η(t, x) =

• S2 f(t, x, τ)dτ (particle density),

(5) p = ργ + η2

• =

⇒Regularity of η

, γ > 3 2.

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

ρt + ∇ · (ρu) = 0, (ρu)t + ∇ · (ρu ⊗ u) − ∆u − ∇(∇ · u) + ∇ργ + ∇η2 = ∇ · σ − ∇η, ft + ∇ · (uf) + ∇τ · (Pτ ⊥(∇xuτ)f) − ∆τf − ∆xf = 0, ηt + ∇ · (ηu) − ∆η = 0. x ∈ Ω ⊂ R3: bounded domain with Dirichlet boundary condition u = 0, f = 0, η = 0 on ∂Ω. Known Results (Incomplete)

• 1. Constantin et al (2005, 2007, 2008), Lions - Masmoudi (2000, 2007,

2012), Otto - Tzavaras (2008), B - Trivisa (2011).

• 2. Carrillo et al (2006, 2008, 2011), Mellet - Vasseur (2007, 2008)

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

WEAK SOLUTION

The notion of weak solution usually follows from the energy identity.

• 1. Energy

d dt

• Ω

ρ|u|2 2 + ργ γ − 1 + η2

• dx +
• Ω
• |∇u|2 + |∇ · u|2 + 2|∇η|2

dx = −

• Ω

∇u : σdx +

• Ω

(∇ · u)ηdx.

• 2. Entropy: ψ(t, x) =
• S2(f ln f)(t, x, τ)dτ

ψt + ∇ · (uψ) − ∆ψ + 4

• S2
• ∇τ
• f
• 2

dτ + 4

• S2
• ∇
• f
• 2

dτ = ∇u : σ

Otto - Tzavaras

−(∇ · u)η.

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

The energy-entropy dissipation d dt

• Ω

ρ|u|2 2 + ργ γ − 1 + η2 + ψ

• dx +
• Ω
• |∇u|2 + |∇ · u|2 + 2|∇η|2

dx + 4

• Ω
• S2
• ∇τ
• f
• 2

dτdx + 4

• Ω
• S2
• ∇
• f
• 2

dτdx = 0. Definition: We say {ρ, u, f, η} is a weak solution if

• 1. ρ is a renormalized solution,

b(ρ)t + ∇ · (b(ρ)u) +

• b

′(ρ)ρ − b(ρ)

• ∇ · u = 0,
• 2. {u, f, η} is a distributional solution,
• 3. {ρ, u, f, η} satisfies the energy-entropy dissipation inequality.

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

THEOREM

Let γ > 3

2 and Ω be a smooth bounded domain. Assume that initial

data {ρ0, u0, f0, η0} satisfy ρ0 ∈ L1 ∩ Lγ(Ω), ρ0u0 = m0 ∈ L

2γ γ+1 (Ω),

m2 ρ0 ∈ L1(Ω) for ρ0 = 0, m2 ρ0 = 0 for ρ0 = 0, f0, f0| log f0| ∈ L1(Ω × S2), η0 ∈ L2(Ω). Then, there exists a weak solution {ρ, u, f, η} such that ρ ∈ Lp(Ω × (0, T)), p = 5γ/3 − 1.

H.B and K. Trivisa, To appear in Mathematical Models and Methods in Applied Sciences (M3AS), 2012

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

PROOF OF THEOREM

• 1. Construction of an approximate sequence of solutions via

regularization (P.L.Lions) ρt + ∇ · (ρu) = 0, (ρǫu)t + ∇ · ((ρu)ǫ ⊗ u) − ∆u − ∇(∇ · u) + ∇ργ + ∇η2 = ∇ · σǫ − ∇ηǫ, ft + ∇ · (uǫf) + ∇τ · (Pτ ⊥(∇xuǫτ)f) − ∆τf − ∆f = 0, ηt + ∇ · (uǫη) − ∆η = 0. = ⇒ d dt

• Ω

ρǫ|u|2 2 + ργ γ − 1 + η2 + ψ

• dx +
• Ω
• |∇u|2 + |∇ · u|2 + 2|∇η|2

dx + 4

• Ω
• S2
• ∇τ
• f
• 2

dτdx + 4

• Ω
• S2
• ∇
• f
• 2

dτdx = 0.

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

• 2. Compactness of an approximate sequence

(1) ρ ∈ L∞(0, T; Lγ(Ω)) is not enough to pass to the limit in ργ = ⇒ need to show ρ satisfies a better integrability (E.Feireisl) (2) Nonlinear terms in the weak formulation of f:

• Ω

∂u(n)

i

∂xj

• S2 τjf (n) ∂χ

∂τi dτdx, χ ∈ D(Ω × S2). = ⇒ need to show

• S2 τjf (n) ∂χ

∂τi dτ converges strongly in L2(Ω × (0, T)).

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

COMPACTNESS

Suppose an approximate sequence of solutions {ρn, un, f n, ηn, σn}n≥1 satisfies the energy/entropy inequality. Then,

• 1. ηn and σn converges strongly in L2(Ω × (0, T)),
• 2. ρn(ηn)2 converges weakly to ρη2 in L1+(Ω × (0, T)),
• 3. If in addition we assume that ρn

0 converges to ρ0 in L1(Ω), then

ρn → ρ in L1(Ω × (0, T)). Lemma (Simon): Let X, B, and Y be Banach spaces such that X ⊂comp B ⊂ Y. Then, {v; v ∈ Lp(0, T; X), vt ∈ L1(0, T; Y)} is compactly embedded in Lp(0, T; B).

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

• 1. Convergence of σ

σt =

• S2 (3τ ⊗ τ − I) ftdτ ∈ L1(0, T; W−1,1),

∇σ =

• S2 (3τ ⊗ τ − I) ∇fdτ ∈ L3/2(0, T; L18/11).

W1, 18

11 ⊂comp L2 ⊂ W−1,1 =

⇒ σn → σ ∈ L

3 2 (0, T; L2).

|σ| ≤ 3η ∈ L∞(0, T; L2) = ⇒ σn → σ ∈ L2(Ω × (0, T)).

• 2. Convergence of ρη2

H1 ⊂comp Lr ∀r < 6 = ⇒ (ηn)2 → η2 ∈ L1+(0, T; Lq), ∀q < 3, 1/γ < 2/3 = ⇒ 1/q + 1/γ < 1 = ⇒ ρn(ηn)2 → ρη2 ∈ L1+(Ω × (0, T)).

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

STRONG CONVERGENCE OF ρ IN L1(Ω × (0, T))

We need to show the weak convergence of {ρn ln ρn}. (ρ ln ρ)t + ∇ · (uρ ln ρ) + (∇ · u)ρ = 0, [ργ − 2(∇ · u)] ρ = −ρη2 + · · ·

• 1. Higher Integrability: θ > 0, depending only γ, such that

ρLγ+θ(Ω×(0,T)) ≤ C(T). (Best possible θ is 2γ/3 − 1) = ⇒ can pass to the limit to ργ

• 2. Limit of Effective Viscous Flux

lim

n→∞

T

• Ω

[(ρn)γ − 2∇ · un] Tk(ρn)dxdt = T

• Ω

[ργ − 2∇ · u] Tk(ρ)dxdt

INTRODUCTION BACKGROUND RESULTS METHODOLOGY

• 3. Let ρ be a weak limit of the sequence {ρn}. Then,

lim sup

n→∞

Tk(ρn) − Tk(ρ)Lγ+1(Ω×(0,T)) ≤ C(T). Note: γ + 1 > 2.

• 4. Strong Convergence of ρ: Lk ≃ z ln z.
• Ω
• Lk(ρ) − Lk(ρ)
• dx ≤

t

• Ω
• Tk(ρ) − Tk(ρ)
• (∇ · u)dxds.

= ⇒ ρ ln ρ = ρ ln ρ, for all t ∈ [0, T]. = ⇒ the strong convergence of {ρn} in L1(Ω × (0, T)).