A Journey through the World of Incompressible Viscous Flows : an - - PowerPoint PPT Presentation

A Journey through the World of Incompressible Viscous Flows : an Evolution Equation Perspective

Matthias Hieber

TU Darmstadt, Germany

Mathematics for Nonlinear Phenomena, Conference in Honor of Prof. Yoshikazu Giga on his 60th Birthday Sapporo August 17, 2015

Balance Laws for Incompressible Fluids

Incompressible fluids are subject to balance laws of momentum, mass and energy ̺(∂t + u · ∇)u + ∇π = div S in Ω, div u = 0 in Ω, ̺(∂t + u · ∇)ǫ + div q = S : ∇u in Ω, u = q · ν = 0

• n ∂Ω,

u, ̺, π velocity, density, pressure of fluid S stress tensor ǫ internal energy q heat flux Ω ⊂ Rn bounded domain with smooth boundary conservation of energy since for total energy e := |u|2/2 + ǫ ρ(∂t + u · ∇)e + div(q + πu − Su) = 0 in Ω integrating yields ∂tE(t) = 0 where E(t) =

• Ω ρedx provided

q · ν = u = 0

• n ∂Ω

Hence, above boundary conditions imply : total energy is preserved, independent of choice of S and q

Entropy Considerations

define free energy ψ as ψ = ψ(ρ, θ), θ temperature define entropy η by η = −∂θψ Clausius-Duhem equation yields : entropy production given by θr := S : ∇u − q · ∇θ/θ + (ρ2∂ρψ − π)(div u) total entropy N =

• Ω ρη is non-decreasing provided r ≥ 0

div u has no sign : require π = ρ2∂ρψ, Maxwell’s relation Further, S : ∇u ≥ 0 and q · ∇θ ≤ 0, classical conditions Summary : For all choices of S and q, there is conservation of energy and total entropy is non-decreasing provided classical, Maxwell’s conditions and boundary conditions hold we say : model is thermodynamically consistent Example : classical laws due to Newton and Fourier : S := SNewton := 2µsD + µbdiv u I, 2D = (∇u + [∇u]T), q = −α0∇θ thermodynamically consistent if µs ≥ 0, 2µs + nµb ≥ 0 and α0 ≥ 0

Examples

Isothermal and Incompressible Situation In the special case where θ = const, ̺ = const, we obtain S = 0 : Euler’s equations, S = SNewton = 2µD(u) : equations of Navier-Stokes S = SNewton + Selastic, fluids of Oldroyd-B type S = Snon−Newton, non-Newtonian fluids hydrostatic approximation of pressure : primitive equations in geophysical sciences Non-Isothermal Situation S = SEricksen + SLeslie, Ericksen-Leslie model for liquid crystals

Stage 1 : The equations of Navier-Stokes

Setting ̺ = µ = 1, we have ut − ∆u + (u · ∇)u + ∇p = f , in [0, T] × Ω div u = 0, in [0, T] × Ω u = 0, in [0, T] × ∂Ω u(0) = u0, in Ω Strategy for strong well-posedness : write equations of Navier-Stokes as Evolution Equation u′(t) − Au(t) = −P[u(t) · ∇)u(t) in Banach space Lp

σ(Ω), where

◮ A = P∆, Stokes operator ◮ P, Helmholtz projection

rewrite evolution equation as integral equation u(t) = etAu0 − t e(t−s)AP[(u(s) · ∇)u(s)]ds solve integral equation via fixed point methods central importance : properties of Stokes operator and Stokes semigroup

The Stokes Operator, 1 < p < ∞

Define Stokes operator Apu = Pp∆u for 1 < p < ∞ with domain D(Ap) = W 2,p(Ω) ∩ W 1,p (Ω) ∩ Lp

σ(Ω) provided ∂Ω smooth

Then A2 selfadjoint and generator of bounded analytic semigroup on L2

σ(Ω)

Ω bounded : A2 invertible and A−1

2

has compact resolvent. Lp-setting, much more difficult : key result by Y. Giga ’81 : Ap generates analytic semigroup on Lp

σ(Ω)

Hence : above strategy can be made rigorous and solving Navier-Stokes is equivalent to u(t) = etAu0 − t e(t−s)AP[(u(s) · ∇)u(s)]ds Find function space F in which iteration scheme

◮ u1(t) = etAu0 ◮ un+1(t) = etAu0 −

t

0 e(t−s)AP[(un(s)∇)un(s)]ds converges.

Unique, Strong solutions for Equations of Navier-Stokes

Assume Ω ⊂ R3 bounded, ∂Ω smooth Fujita-Kato : if either u0 ∈ D(A)1/4 or interval of existence for T is sufficiently small, then there exists a unique, strong solution on [0, T). in particular : L2-situation : u0 ∈ ˙ H1/2 Extension of iteration schema on scaling invariant function spaces key results by Y. Giga ’86, T. Kato : u0 ∈ Lp

σ(Ω) for p ≥ 3

Cannone-Meyer : Well-posedness for u0 ∈ B−1+3/p

p,∞

(R3) Koch-Tataru : Well-posedness for u0 ∈ BMO−1(R3) Bourgain-Pavlovic : Ill-posedness for u0 ∈ B−1

∞,∞(R3), i.e. solution

• perator u0 → u(t) is not continuous with respect to · B−1

∞,∞

global strong solution provided n = 2

The role of the Stokes semigroup

Following Y. Giga’s approach : 2 main ingredients needed in proof for u0 ∈ L3(R3) : global Lp − Lq-smoothing : etAf q ≤ Ct−n/2(1/p−1/q)f p, t > 0, 1 < p ≤ q < ∞ global gradient estimates : ∇etAf p ≤ Ct−1/2f p, t > 0 Then : (tn/2(1/p−1/q)uj) is Cauchy sequence in BC([0, T); Lq

σ)

and (t1/2∇uj) is Cauchy sequnece in BC([0, T); Lq) Hence : global solution provided u0 small further : analyticity of etA implies classical solution

Navier-Stokes in Geophysical Setting

Geophysical situation : rotating coordinate system yields additional term : Coriolis force ut − ∆u + (u · ∇)u + ωe3 × u + ∇p = f , in [0, T] × R3 div u = 0, in [0, T] × Ω u(0) = u0, in Ω Babenko, Mahalov, Nikolenco : pioneering result on global well-posedness for large data provided ω is large enough global well-posedness result Chemin, Desjardins, Gallagher, Grenier : let u0 ∈ L2(R2)2 + H1/2(R3) with div u0 = 0. Then exists ω0 > 0 such that for all ω ≥ ω0 the (NSC)-equation admits a unique, global mild solution surprising : no smallness condition for u0 proof relies on dispersive estimates for linear part.

Stage 2 : The Stokes equation on L∞

σ (Ω)

Aim : well-posedness for Navier-Stokes equations for non-decaying data P = Id − ∇(−∆)−1div is not bounded in L∞(Rn) even for Ω = Rn positive result on half space : Ω = Rn

+ : Desch, H., Pr¨

uss, 2001 : A generates analytic semigroup on L∞

σ (Ω)

Here : L∞

σ (Ω) = {u ∈ L∞(Ω) :

• u · ∇ϕ = 0, ϕ ∈

W 1,1(Ω)} negative result for layers : Ω = R2 × (0, 1) negative results for L1(Ω) for Ω= half space or layer How to treat situation of domains ? Difficulties :

◮ localization procedure does not work ◮ pressure estimate for ∇q of the form

∇qp ≤ C∆up does not hold for p = ∞.

Approaches

Masuda, 1972, Stewart, 1974 : A priori estimates for various elliptic

• perators on L∞(Ω)

breakthrough result by Abe-Giga, 2013 : L∞-estimates for Stokes system via blow-up argument for admissable domains

◮ N(u, p)(t, x) =

|u(t, x)| + t1/2|∇u(t, x)| + +t|∂tu(t, x)| + t|∇2u(t, x)| + t|∇p(t, x)|

◮ Then sup0<t<T0N(u, p)∞(t) ≤ Cu0∞ ◮ Idea : suppose false. ◮ rescaled blow-up sequence (um, pm) on rescaled domain converning to

solution (v, q) with v0 ≡ 0.

◮ if convergence strong enough : N(v, q)(0, 0) ≥ 1/2 ◮ if limit unique, then v ≡ 0 ≡ ∇q yielding contraction

approach jointly with K. Abe and Y. Giga : extend Masuda-Stewart approach to Stokes

Resolvent Estimates for Stokes equation for p = ∞

Consider λv − ∆v + ∇q = f in Ω div v = g in Ω v = h

• n ∂Ω

Aim : For p > n set Mp(v, q)(x, λ) = |λ||v(x)| + |λ|1/2|∇v(x)| + |λ|n/2p∇2vLp(Ωx,|λ|−1/2) +|λ|n/2p∇qLp(Ωx,|λ|−1/2) and show sup

λ∈ΣΘ

Mp(v, q)∞(λ) ≤ Cf ∞ Here Ωx0,r = Bx0(r) ∩ Ω ΣΘ = sector in complex plane of angle Θ ∈ (π/2, π)

Idea of Approach

Step 1 : Localize Localize equation in Ω′ = Bx0((η + 1)r) ∩ Ω by setting u = θv, p = (q − qc)θ with cutoff function θ. Then (u, q) solves resolvent equation in Ω′ with error terms on right hand side h and g 2 scaling parameters : η > 0, r > 0 to be determined later Step 2 : Apply Lp-estimates in Ω′ |λ|up + |λ|1/2∇up + ∇2up + ∇pp ≤ Cp(hp + ∇gp + |λ|gW −1,p ) Note : ∂Ω′ is not smooth. Step 3 : Estimates for error terms Recall : h = f θ − 2∇v∇θ − v∆θ + (q − qc)∇θ estimate for the first three terms as in elliptic situation

Key estimate for pressure

Step 4 : Handle pressure term by Poincar´ e-Sobolev type inequality |ϕ − (ϕ)|Lp(Ωx0,s) ≤ Csn/p|∇ϕ|L∞

d (Ω) for ϕ ∈

W 1,∞

d

(Ω) (ϕ) mean value of ϕ |f |L∞

d (Ω) = supx∈Ω dΩ(x)|f (x)| and d distance function from boundary

Step 5 : We call a domain Ω strictly admissible if pressure term can be estimated by the velocity, i.e. more precisely if sup

x∈Ω

dΩ(x)|∇q(x)| ≤ CΩ|W (v)|L∞(∂Ω) with W (v) = (∇v −(∇v)T)nΩ bounded or exterior domains with smooth boundary are admissible Combining this estimate with Poincar´ e-Sobolev type estimate in Step 4 yields |h|Lp(Ω′) ≤ Cr n/p (η+1)n/p|f |∞+(η+1)−(1−n/p)(r −2|v|∞+r −1|∇v|∞)

Interpolation and Adjusting Parameters

Step 6 : Apply interpolation inequality |ϕ|L∞(Ωx0,r ) ≤ Cr −n/p |ϕ|Lp(Ωx0,r ) + r|∇ϕ|Lp(Ωx0,r )

• for u and ∇u.

Choose r = |λ|−1/2 to obtain Mp(v, q)(x0, λ) ≤ C

• (η+1)n/p|f |∞+(η + 1)−(1−n/p)|Mp(v, q)|∞(λ)
• Choosing η large and p > n, the second term on right hand side can

be absorbed into left hand side. Hence sup

λ∈ΣΘ

Mp(v, q)∞(λ) ≤ Cf ∞ where Mp(v, q)(x, λ) = |λ||v(x)| + |λ|1/2|∇v(x)| + |λ|n/2p∇2vLp(Ωx,|λ|−1/2) +|λ|n/2p∇qLp(Ωx,|λ|−1/2)

Working with Y. Giga on the Black Board

Results

Abe, Giga, H. ’15 : A generates analytic semigroup on C0,σ(Ω) and L∞

σ (Ω) of angle π/2 for Ω for large class of domains

method extends to other boundary conditions

◮ B(v) = 0,

v · n = 0 on ∂Ω, where B(v) = αvtan + (D(v)n)tan

method does not imply information on large time behaviour of T∞(t) = etA Maremonti, ’12 : exterior domains : maximum modulus theorem yields |T∞(t)| ≤ M for all t > 0 Bolkart, H. ’15 : solution u(t) = T∞(t)u0 admits a pointwise upper bound similar to heat equation, but not a heat-kernel bound Consequence : Stokes operator generates bounded analytic semigroup on L∞

σ of angle π/2

T∞ admits L∞ − C 2+α- smoothing effect uniform estimates for ∇T∞(t) and ∇2T∞(t)

Stage 3 : Viscoelastic Fluids of Oldroyd-B type

stress tensor S is subdivided into S = SNewton + Selastic Newtonian fluids : Selastic = 0 generalized Newtonian fluids : SNN = 2µ(|D(u)|2)D(u) viscoelastic fluids : need information on entire time history of D(u) : Selastic satisfies differential equation Oldroyd-B model : τ + λ1 Daτ Dt = 2ν[D(u) + λ2 DaD(u) Dt ], where Da

Dt the objective derivative for a ∈ [−1, 1] is given by

Daτ Dt = ∂τ ∂t + (u · ∇)τ + τW (u) − W (u)τ − a(D(u)τ + τD(u)). Fluids of this type have elastic and viscous properties yielding a parabolic-hyperbolic system

Orientation of Rigid Body

Fall of rigid body in Oldroyd Fluid subject to Gravitation

Oldroyd-B Fluids as Parabolic-Hyperbolic System

               (ut + (u · ∇)u) − (1 − α)∆u + ∇p = div τ in Ω, div u = 0 in Ω, (τ ′ + (u · ∇)τ) + τ = 2αD(u) − ga(τ, ∇u) in Ω, u = 0

• n ∂Ω

u(0, ·) = u0, in Ω, τ(0, ·) = τ0 in Ω, (1) Here Ω ⊂ Rn is exterior domain with ∂Ω of class C 3 Applying Helmholtz projection P yields ∂t u τ

• +

−(1 − α)P∆ −P div −2αD Id u τ

• =
• −Pu · ∇u

−u · ∇τ − ga(τ, ∇u)

• with A = −P∆ Stokes operator, D deformation tensor

Question : global weak or strong well-posedness ? Do all global, strong or weak solutions to Oldroyd-system tend to 0 as t → ∞ ? Is there a decay rate ?

Linearized equation and the Oldroyd Semigroup

Linearized equation reads as ∂ ∂t u τ

• =

(1 − α)P∆ Pdiv 2αD −Id u τ

• .

choose state space as X := Lp

σ(Ω) × W 1,p(Ω)n

define Oldroyd operator Bp by Bp := (1 − α)Ap −P div −2αD Id

• D(Bp)

:= W 2,p(Ω) ∩ W 1,p (Ω) ∩ Lp

σ(Ω) × W 1,p(Ω)n.

with Ap := −P∆ Stokes operator on Lp

σ(Ω)

Strong stability of the Oldroyd semigroup

Results on the Oldroyd semigroup on exterior domains : −Bp generates a bounded analytic semigroup (e−tBp)t≥0 on Lp

σ(Ω) × W 1,p(Ω)n of angle ϕ(α).

0 ∈ σ(−Bp), but 0 / ∈ σp(−Bp) (e−tBp)t≥0 is strongly stable, i.e. e−tBp(u, τ)T → 0 as t → ∞ for all (u, τ) ∈ Lp

σ(Ω) × W 1,p(Ω)n.

Global Well-Posedness and Asymptotic Stability of Trivial Solution Existence Result : Existence of a unique, global strong solution for small data. Precisely : If u0D(A2) + τ0H2 are small enough, then Oldroyd system admits a unique, global strong solution (u, p, τ) for all t ≥ 0 Main Stability Theorem : Under a small conditions all global strong solutions of Oldroyd system tend to 0 as t → ∞.

Stage 4 : Geophysical Flows in Oberwolfach, Winter 2013

Isothermal Primitive Equations of Geophysics

Primitive equations are given by ∂tv + u · ∇v − ∆v + ∇Hπ = f in Ω × (0, T), ∂zπ = in Ω × (0, T), (2) div u = in Ω × (0, T), v(0) = a. Ω = G × (−h, 0), where G = (0, 1)2, h > 0 velocity u is written as u = (v, w) with v = (v1, v2) v and w denote the horizontal and vertical components of u, π pressure, f external force ∇H = (∂x, ∂y)T, ∆, ∇, div three dimensional operators. System is complemented by the set of boundary conditions ∂zv = 0, w = 0

• n Γu × (0, T),

v = 0, w = 0

• n Γb × (0, T),

u, π are periodic

• n Γl × (0, T).

(3) Γu := G × {0}, Γb := G × {−h}, Γl := ∂G × [−h, 0]

Primitive Equations

’92-’95 : full primitive equations introduced by Lions, Temam and Wang, existence of a global weak solution for a ∈ L2. ’01 : Guill´ en-Gonz´ alez, Masmoudi, Rodiguez-Bellido : existence of a unique, local, strong solution for a ∈ H1 ’07, Cao and Titi : breakthrough result : existence of a unique, global strong solution for arbitrary initial data a ∈ H1 aim : develop an Lp-approach via evolution equations show existence of a unique, global strong solution to primitive equations for arbitrary large data a having less differentiability properties than H1.

Strategy of our Lp-Approach

solution of the linearized equation is governed by an analytic semigroup Tp on the space Xp Xp is defined as the range of the hydrostatic Helmholtz projection Pp : Lp(Ω)2 → Lp(Ω)2 This space corresponds to solenoidal space Lp

σ(Ω) for Navier-Stokes

equations generator of Tp is −Ap called the hydrostatic Stokes operator. rewrite primitive equations as

• v ′(t) + Apv(t) = Ppf (t) − Pp(v · ∇Hv + w∂zv),

t > 0, v(0) = a. consider integral equation v(t) = e−tApa + t e−(t−s)Ap Ppf (s) + Fpv(s)

• ds,

t ≥ 0, where Fpv := −Pp(v · ∇Hv + w∂zv)

Strategy of Lp-approach, continued

show that v is unique, local, strong solution, i.e. v ∈ C 1((0, T ∗]; Xp) ∩ C((0, T ∗]; D(Ap)), p ∈ (1, ∞) Note D(Ap) ֒ → W 2,p(Ω)2 ֒ → H1(Ω)2 for p ≥ 6/5

• btain existence of a unique, global, strong solution for

a ∈ [Xp, D(Ap)]1/p for p ≥ 6/5 provided sup0≤t≤T v(t)H2(Ω) is bounded by some constant B = B(aH2(Ω), T) for any T > 0. key estimate : global H2-bound

Main Result

Theorem (H., Kashiwabara ’14) : Let p ∈ [6/5, ∞), a ∈ V1/p,p and f ≡ 0. Then there exists a unique, strong global solution (v, π) to primitive equations within the regularity class v ∈ C 1((0, ∞); Lp(Ω)2)∩C((0, ∞); W 2,p(Ω)2), π ∈ C((0, ∞); W 1,p(G)∩Lp

0(G)).

Moreover, the solution (v, π) decays exponentially, i.e. there exist constants M, c, ˜ c > 0 such that ∂tv(t)Lp(Ω) + v(t)W 2,p(Ω) + πW 1,p(G) ≤ Mt−˜

ce−ct,

t > 0. Remarks : Vθ,p := [Xp, D(Ap)]θ, 0 ≤ θ ≤ 1 and 1 < p < ∞, is complex interpolation space between Xp and D(Ap) of order θ note that V1/p,p ֒ → H2/p,p(Ω)2 for all p ∈ (1, ∞) if p = 2, then V1/2,2 coincides with H1 subject to bc.,

• pen problems : case p = ∞ ?, transfer Giga’s iteration scheme for

(NS) to this setting ?

Final Stage : Nematic Liquid Crystals

Nematic versus crystal : Nematic phase : molecules align along a direction

The general Ericksen-Leslie Model in R3 : original form

ut + (u · ∇u) = div σ

• n (0, T) × Ω,

div u = 0

• n (0, T) × Ω

d × (g + div( ∂W

∂(∇d)) − ∂W ∂d )

= 0

• n (0, T) × Ω,

|d| = 1 in (0, T) × Ω (u, d)(0) = (u0, d0) in Ω u velocity, σ stress tensor, d director describing orientation stress tensor σ = −pI − ∂W

∂dki dkj + σLeslie

W = W (d, ∇d) Oseen-Frank energy functional given by W = 1

2[k1(div d)2 + k2|d × (∇ × d)|2 + k3|d(∇ × d)|2+

(k2 + k4)(tr (∇d)2 − (div d)2)] with elasticity constants ki σLeslie = α1(dd : D)dd + α2dN + α3Nd + α4D + α5ddD + α6Ddd D = D(u) = 1

2[(∇u) + (∇u)T ]

N = dt + (u · ∇)d + V (u)d with V (u) = 1

2[(∇u) − (∇u)T]

g = λ1N + λ2Dd

The simplified Ericksen-Leslie model

For a bounded domain Ω ⊂ Rn, n ≥ 2, consider ut − ∆u + (u · ∇)u + ∇π = −λdiv ([∇d]T∇d) in (0, T) × Ω, dt + (u · ∇)d) = γ(∆d + |∇d|2d) in (0, T) × Ω, div u = 0 in (0, T) × Ω, |d| = 1 in (0, T) × Ω (u, ∂νd) = (0, 0)

• n (0, T) × ∂Ω,

(u, d)(0) = (u0, d0) in Ω u velocity, π pressure d : (0, T) × Ω → Rn : macroscopic molecular orientation Two type of approaches : I Fluid-type approach : couple equation for d to methods for Navier-Stokes II Geometric approach by harmonic maps on spheres : couple fluid equation to this geometric approach Rigorous Analysis was started by I Lin, Lin-Liu ’95, fluid type approach II Lin, Wang : existence results via heat flow of harmonic maps

The Quasilinear Approach

Main idea : incorporate the term div ([∇d]T∇d) into the quasilinear

• perator A representing the left hand side of equation. More precisely, we

rewrite ut − ν∆u + (u · ∇)u + ∇π = −λdiv ([∇d]T ∇d) in (0, T) × Ω, dt + (u · ∇)d) = γ(∆d + |∇d|2d) in (0, T) × Ω, div u = 0 in (0, T) × Ω, (u, ∂νd) = (0, 0)

• n (0, T) × ∂Ω,

as ∂t u d

• +

Aq PBq(d) Dq u d

• =
• −Pu · ∇u

−u · ∇d + |∇d|2d

• where

Aq Stokes operator Dq Neumannn-Laplacian operator P Helmholtz projection [Bq(d)h]i := ∂idl∆hl + ∂kdl∂k∂ihl thus : Bq(d)d = div([∇d]T∇d)

Liquid Crystals as Quasilinear Evolution Equation

We rewrite the (simplified) Ericksen-Leslie system as ˙ z(t) + A(z(t))z(t) = F(z(t)), t ∈ J, z(0) = z0, (4) with state space X0 := Lq,σ(Ω) × Lq(Ω)n, 1 < q < ∞ Ω ⊂ Rd bounded domain with boundary ∂Ω ∈ C 2 the quasilinear part A(z) given by the tri-diagonal matrix A(z) = Aq PBq(d) Dq

• ,

Stokes operator Aq = −P∆ in Lq,σ(Ω) Neumann-Laplacian Dq in Lq(Ω) Bq given by [Bq(d)h]i := ∂idl∆hl + ∂kdl∂k∂ihl F(z) = (−Pu · ∇u, −u · ∇d + |∇d|2d)

Local Wellposedness by maximal Lp-regularity

Summarizing, we obtain Let 2/p + n/q < 1, z0 = (u0, d0) ∈ Xγ,µ. i.e. u0, d0 ∈ B2µ−2/p

q,p

(Ω)n with div u0 = 0 in Ω Then there is a unique local solution z ∈ H1

p,µ(J, X0) ∩ Lp,µ(J; X1) on

J. Moreover,z ∈ C([0, a]; Xγ,µ) ∩ C((0, a]; Xγ), i.e. the solution regularizes instantly in time. For each k ∈ N, tk[ d

dt ]kz ∈ H1 p,µ(J; X0) ∩ Lp,µ(J; X1) and

z ∈ C ω((0, a); X1). Condition |d| = 1 is preserved by the flow induced by the Ericksen-Leslie model

Global solutions, dynamics and convergence to equilibria

E0 = {0} × Rn is obviously an equilibria for (LCE) linearization of (LCE) at z∗ ∈ E0 is given by ˙ z + A∗z = f , z(0) = z0 u∗ ∈ E is normally stable Theorem : Let p, q as above. Then for each equilibrium z∗ ∈ {0} × Rn there exists ǫ > 0 such that a solution z(t) of (LCE) with initial data z0 ∈ Xγ, |z0 − z∗|Xγ ≤ ǫ exists globally and converges exponentially to z∞ ∈ {0} × Rn in Xγ, as t → ∞

Determination of Equilibria and Lyapunov Functionals

Define energy by E := 1

2

• Ω[|u|2 + |∇d|2]dx = Ekin + Epot

Calculation yields d dt E(t) = −

• Ω

[|∇u|2 + |∆d + |∇d|2d|2]dx Hence, E(t) is non-increasing along solutions E is even a strict Ljapunov functional, i.e. strictly decreasing along constant solutions. In fact : if dE(t)/dt = 0 at some time, then ∇u = 0 and ∆d + |∇d|2d = 0 in Ω. Hence u = 0 and d satisfies the nonlinear eigenvalue problem    ∆d + |∇d|2d = 0 in Ω, |d|2 = 1 in Ω, ∂νd = 0

• n ∂Ω.

(5) Lemma : if d ∈ H2

2(Ω; Rn) satisfies eigenvalue problem, then d is

constant in Ω.

Determination of Equilibria

Hence : energy functional E is strict Ljapunov functional for (LCE). Equilibria are given by E = {z∗ = (u∗, d∗) : u∗ = 0, d∗ ∈ Rn, |d∗| = 1} Summary : rather complete understanding of dynamics of simplified model Back to Full Model how to understand the model and the many terms involved ? how to proceed with the analysis ? basic idea : try to understand the model from a thermodynamical point of view, develop a thermodynamically consistent extension of the model this understanding is also the key for analytical investigations

Free Energy of Liquid Crystals : back to Stage 1

now : free energy ψ = ψ(ρ, θ, τ) with τ = 1

2|∇d|2 2

energy flux is now given by Φe := q + πu − Su − ΠDtd, Dt = ∂t + u · ∇d, where Π has to be modeled by entropy principle constitutive laws S = SN + SE + SL, SE = −θλ∇d[∇d]T, q = −α0∇θ − α1(d · ∇θ)d. SN means Newton stress, SE the Ericksen stress and SL the Leslie stress the balance of entropy reads as ρ(∂t + u · ∇)η + div Φη = r, with Φη = q/θ and

Evolution of director d by entropy principle

θr = − q · ∇θ/θ + 2µs|D|2

2 + µb|div u|2 + (ρ2∂ρψ − π)div u

+ (ρ∂τψ − λ)∇d[∇d]T : ∇u + (Π − ρ∂τψ∇d) : ∇Dtd + SL : ∇u + (divΠ + βd) · Dtd. for some scalar function β. entropy production r nonnegative provided µs ≥ 0, 2µs + nµb ≥ 0, α0 ≥ 0, α0 + α1 ≥ 0. The next five blue terms r have no sign, hence we require π = ρ2∂ρψ, λ = ρ∂τψ/θ, Π = ρ∂τψ∇d argument for Leslie stress SL more involved γDtd = div[(ρ∂τψ)∇]d + βd for some γ = γ(ρ, θ, τ) ≥ 0 condition |d| = 1 requires β = λ|∇d|2 this leads to the equation for d γ(∂t + u · ∇)d = div[λ∇]d + λ|∇d|2d, This is the basic equation governing the evolution of the director field d

The Complete Ericksen-Leslie Model, non-isothermal

∂tρ + div(ρu) = 0 in Ω ρ(∂t + u · ∇)u + ∇π = div S in Ω ρ(∂t + u · ∇)ǫ + div q = S : ∇u − πdiv u + div(λ∇dDtd) in Ω γ(∂t + u · ∇)d − µV Vd − div[λ∇]d = λ|∇d|2d + µDPDDd in Ω u = 0, q · ν = 0, ∂νd = 0

• n ∂Ω

supplemented by the thermodynamical laws ǫ = ψ + θη, η = −∂θψ, κ = ∂θǫ, π = ρ2∂ρψ, λ = ρ∂τψ, and the constitutive laws for S = SN + SE + SL SN = 2µsD + µbdiv u I, SE = −ρ ∂ψ ∂∇d [∇d]T, Sstretch

L

= µD + µV 2γ n ⊗ d + µD − µV 2γ d ⊗ n, n = µV Vd + µDPdDd − γDtd, Sdiss

L

= µP γ (n ⊗ d + d ⊗ n) + γµL + µ2

P

2γ (PdDd ⊗ d + d ⊗ PdDd) + µ0(Dd|d)d q = −α0∇θ − α1(d|∇θ)d.

Maximal Regularity of the Principal part of the Linearization : incompressible case

We concentrate on the system associated with w := (θ, d). The principal part of the linearization becomes ∂tw + A(w0, ∇)w = f in Ω, ∂νw =

• n ∂Ω,

w(0) = w0 in Ω. where A = A(w0, ∇) is given by A = −a0∆ − a1∇dT

0 ∇d0 : ∇2,

−b0∇d0 : (λ0∆ + ∂τλ0[∇d0]T∇d0 : ∇2)∇ b1[∇d0]T∇, −γ−1

0 (λ0∆ + ∂τλ0[∇d0]T ⊗ ∇d0 : ∇2).

• .

Here κ0 = κ(θ0, τ0) etc., and a0 = α0 ρκ0 , a1 = [∂τǫ0]2 θ0γ0κ0 , b0 = ∂τǫ0 γ0κ0 , b1 = ∂τǫ0 γ0θ0 . A(w0, ∇) : second order diagonal, but third and first order

• ff-diagonal

This is a mixed-order problem

Maximal Regularity in Y0 := Lq(Ω) × H1

q(Ω; Rn)

Schur reduction : reduced variable wred = [θ, dred]T where dred = c(ξ) · d yields reduced symbol Ared(ξ) Ared(ξ) = a0|ξ|2 + a1|c(ξ)|2 −ib0(λ0|ξ|2 + ∂τλ0|c(ξ)|2) ib1|c(ξ)|2

λ0 γ0 |ξ|2 + ∂τ λ0 γ0 |c(ξ)|2

• .

now : reduced symbol is homogeneous of second order and normally elliptic in the sense that σ(Ared(ξ)) ⊂ (0, ∞) for each ξ = 0. hence : reduced equation has maximal regularity regain d by solving ∂td−λ0 γ0 ∆d = f 1

d := fd+i ∂τλ0

γ0 c(∇)dred−b1c(∇)θ, t > 0, d(0) = 0, with d ∈ 0H1

p(J; H1 q(Rn; Rn)) ∩ Lp(J; H3 q(Rn; Rn))

for f 1

d ∈ Lp(J; H1 q(Rn; Rn)).

Then half space result and localization procedure yields following result

Global Well-posedness of Complete Ericksen-Leslie System

rewrite problem as quasilinear evolution equation ˙ z + A(z)z = F(z), t > 0, z(0) = z0, z = (u, w) = (u, θ, d) base space X0 := Lq,σ(Ω) × Lq(Ω) × H1

q(Ω; Rn)

A(z) is coupled 3 × 3 operator matrix Regularity assumptions (R) : µ, α, γ ∈ C 2, ψ ∈ C 4 Theorem (H., Pruess, ’15) : Assume (R), let p, q, µ as above and z0 ∈ Xγ,µ. There exists a unique, local, strong solution z ∈ H1

p,µ(J, X0) ∩ Lp,µ(J; X1) on J

solution regularizes instantly in time and |d|2 ≡ 1, E(t) ≡ E0, and −N is a strict Lyapunov functional Then any equilibrium z∗ ∈ ¯ E of EL-system is stable in Xγ. For each z∗ ∈ ¯ E there is ε > 0 such that if |z0 − z∗|Xγ,µ ≤ ε, then the solution z of (EL)-system with initial data z0 exists globally in time and converges at an exponential rate in Xγ to some z∞ ∈ ¯ E.