The Doi Model for the Suspensions of Rod-like Molecules in a - PowerPoint PPT Presentation

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY 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

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY D OI 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.

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY S YSTEM OF EQUATIONS 1. Conservation of mass: ρ t + ∇ · ( u ρ ) = 0 . 2. Equation of the particle distribution τ ∈ S 2 , f t + ∇ · ( uf ) + ∇ τ · ( P τ ⊥ ∇ u τ f ) − ∆ τ f − ∆ f = 0 , (1) ∇ τ · ( P τ ⊥ ∇ u τ f ) : a drift-term on S 2 representing the shear forces acting on the rods, (2) P τ ⊥ ∇ u τ : the projection of the vector ∇ u τ on S 2 , (3) ∆ τ f : the rotational diffusion = ⇒ change the orientation of rods spontaneously.

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY 3. Equation of Motion: ( ρ u ) t + ∇ · ( ρ u ⊗ u ) = ∇ · T T = S − p I 3 × 3 (Stokes’ Law) , S = S f + S p , p = p f + p p . � ∇ u + ( ∇ u ) t � (1) S f = + ( ∇ · u ) I 3 × 3 , (2) S p = σ − η I 3 × 3 , � �� � = ⇒ Energy Dissipation � (3) σ ( t , x ) = S 2 ( 3 τ ⊗ τ − I 3 × 3 ) f ( t , x , τ ) d τ (thermodynamic consistency), � (4) η ( t , x ) = S 2 f ( t , x , τ ) d τ (particle density), γ > 3 (5) p = ρ γ + η 2 , 2. ���� = ⇒ Regularity of η

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY ρ t + ∇ · ( ρ u ) = 0 , ( ρ u ) t + ∇ · ( ρ u ⊗ u ) − ∆ u − ∇ ( ∇ · u ) + ∇ ρ γ + ∇ η 2 = ∇ · σ − ∇ η, f t + ∇ · ( uf ) + ∇ τ · ( P τ ⊥ ( ∇ x u τ ) f ) − ∆ τ f − ∆ x f = 0 , η t + ∇ · ( η u ) − ∆ η = 0 . x ∈ Ω ⊂ R 3 : 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)

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY W EAK S OLUTION The notion of weak solution usually follows from the energy identity. 1. Energy � ρ | u | 2 � � � ρ γ d � |∇ u | 2 + |∇ · u | 2 + 2 |∇ η | 2 � γ − 1 + η 2 + dx + dx dt 2 Ω Ω � � = − ∇ u : σ dx + ( ∇ · u ) η dx . Ω Ω � 2. Entropy : ψ ( t , x ) = S 2 ( f ln f )( t , x , τ ) d τ � � � � 2 � � 2 � � � � � � ψ t + ∇ · ( u ψ ) − ∆ ψ + 4 � ∇ τ f d τ + 4 � ∇ f d τ � � S 2 S 2 = ∇ u : σ − ( ∇ · u ) η. � �� � Otto - Tzavaras

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY The energy-entropy dissipation � � ρ | u | 2 � � ρ γ d γ − 1 + η 2 + ψ |∇ u | 2 + |∇ · u | 2 + 2 |∇ η | 2 � � + dx + dx dt 2 Ω Ω � � � � � � � � 2 2 � � � � � � + 4 � ∇ τ f d τ dx + 4 � ∇ f d τ dx = 0 . � � Ω S 2 Ω S 2 Definition: We say { ρ, u , f , η } is a weak solution if 1. ρ is a renormalized solution, � � ′ ( ρ ) ρ − b ( ρ ) b ( ρ ) t + ∇ · ( b ( ρ ) u ) + b ∇ · u = 0 , 2. { u , f , η } is a distributional solution, 3. { ρ, u , f , η } satisfies the energy-entropy dissipation inequality.

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY T HEOREM Let γ > 3 2 and Ω be a smooth bounded domain. Assume that initial data { ρ 0 , u 0 , f 0 , η 0 } satisfy ρ 0 ∈ L 1 ∩ L γ (Ω) , 2 γ γ + 1 (Ω) , ρ 0 u 0 = m 0 ∈ L m 2 m 2 0 ∈ L 1 (Ω) for ρ 0 � = 0 , 0 = 0 for ρ 0 = 0 , ρ 0 ρ 0 f 0 , f 0 | log f 0 | ∈ L 1 (Ω × S 2 ) , η 0 ∈ L 2 (Ω) . Then, there exists a weak solution { ρ, u , f , η } such that ρ ∈ L p (Ω × ( 0 , T )) , p = 5 γ/ 3 − 1 . H.B and K. Trivisa, To appear in Mathematical Models and Methods in Applied Sciences (M3AS), 2012

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY P ROOF OF T HEOREM 1. Construction of an approximate sequence of solutions via regularization ( P.L.Lions ) ρ t + ∇ · ( ρ u ) = 0 , ( ρ ǫ u ) t + ∇ · (( ρ u ) ǫ ⊗ u ) − ∆ u − ∇ ( ∇ · u ) + ∇ ρ γ + ∇ η 2 = ∇ · σ ǫ − ∇ η ǫ , f t + ∇ · ( u ǫ f ) + ∇ τ · ( P τ ⊥ ( ∇ x u ǫ τ ) f ) − ∆ τ f − ∆ f = 0 , η t + ∇ · ( u ǫ η ) − ∆ η = 0 . = ⇒ � � ρ ǫ | u | 2 � � ρ γ d γ − 1 + η 2 + ψ � |∇ u | 2 + |∇ · u | 2 + 2 |∇ η | 2 � + dx + dx dt 2 Ω Ω � � � � � � 2 � � 2 � � � � � � + 4 � ∇ τ f d τ dx + 4 � ∇ f d τ dx = 0 . � � Ω S 2 Ω S 2

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY 2. Compactness of an approximate sequence (1) ρ ∈ L ∞ ( 0 , T ; L γ (Ω)) is not enough to pass to the limit in ρ γ = ⇒ need to show ρ satisfies a better integrability ( E.Feireisl ) (2) Nonlinear terms in the weak formulation of f : ∂ u ( n ) � � S 2 τ j f ( n ) ∂χ χ ∈ D (Ω × S 2 ) . i d τ dx , ∂ x j ∂τ i Ω � S 2 τ j f ( n ) ∂χ d τ converges strongly in L 2 (Ω × ( 0 , T )) . = ⇒ need to show ∂τ i

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY C OMPACTNESS Suppose an approximate sequence of solutions { ρ n , u n , f n , η n , σ n } n ≥ 1 satisfies the energy/entropy inequality. Then, 1. η n and σ n converges strongly in L 2 (Ω × ( 0 , T )) , 2. ρ n ( η n ) 2 converges weakly to ρη 2 in L 1 + (Ω × ( 0 , T )) , 3. If in addition we assume that ρ n 0 converges to ρ 0 in L 1 (Ω) , then ρ n → ρ in L 1 (Ω × ( 0 , T )) . Lemma (Simon): Let X , B , and Y be Banach spaces such that X ⊂ comp B ⊂ Y . Then, { v ; v ∈ L p ( 0 , T ; X ) , v t ∈ L 1 ( 0 , T ; Y ) } is compactly embedded in L p ( 0 , T ; B ) .

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY 1. Convergence of σ � S 2 ( 3 τ ⊗ τ − I ) f t d τ ∈ L 1 ( 0 , T ; W − 1 , 1 ) , σ t = � S 2 ( 3 τ ⊗ τ − I ) ∇ fd τ ∈ L 3 / 2 ( 0 , T ; L 18 / 11 ) . ∇ σ = 11 ⊂ comp L 2 ⊂ W − 1 , 1 = ⇒ σ n → σ ∈ L W 1 , 18 3 2 ( 0 , T ; L 2 ) . ⇒ σ n → σ ∈ L 2 (Ω × ( 0 , T )) . | σ | ≤ 3 η ∈ L ∞ ( 0 , T ; L 2 ) = 2. Convergence of ρη 2 H 1 ⊂ comp L r ⇒ ( η n ) 2 → η 2 ∈ L 1 + ( 0 , T ; L q ) , ∀ r < 6 = ∀ q < 3 , ⇒ ρ n ( η n ) 2 → ρη 2 ∈ L 1 + (Ω × ( 0 , T )) . 1 /γ < 2 / 3 = ⇒ 1 / q + 1 /γ < 1 =

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY S TRONG C ONVERGENCE OF ρ IN L 1 (Ω × ( 0 , T )) We need to show the weak convergence of { ρ n ln ρ n } . ( ρ ln ρ ) t + ∇ · ( u ρ ln ρ ) + ( ∇ · u ) ρ = 0 , [ ρ γ − 2 ( ∇ · u )] ρ = − ρη 2 + · · · 1. Higher Integrability: θ > 0, depending only γ , such that � ρ � L γ + θ (Ω × ( 0 , T )) ≤ C ( T ) . (Best possible θ is 2 γ/ 3 − 1) ⇒ can pass to the limit to ρ γ = 2. Limit of Effective Viscous Flux � T � T � � [( ρ n ) γ − 2 ∇ · u n ] T k ( ρ n ) dxdt = [ ρ γ − 2 ∇ · u ] T k ( ρ ) dxdt lim n →∞ 0 Ω 0 Ω

I NTRODUCTION B ACKGROUND R ESULTS M ETHODOLOGY 3. Let ρ be a weak limit of the sequence { ρ n } . Then, � T k ( ρ n ) − T k ( ρ ) � L γ + 1 (Ω × ( 0 , T )) ≤ C ( T ) . lim sup n →∞ Note: γ + 1 > 2 . 4. Strong Convergence of ρ : L k ≃ z ln z . � t � � � � � � L k ( ρ ) − L k ( ρ ) dx ≤ T k ( ρ ) − T k ( ρ ) ( ∇ · u ) dxds . Ω 0 Ω = ⇒ ρ ln ρ = ρ ln ρ, t ∈ [ 0 , T ] . for all ⇒ the strong convergence of { ρ n } in L 1 (Ω × ( 0 , T )) . =