On three-dimensional flows of activated fluids Josef Mlek Neas - - PowerPoint PPT Presentation

On three-dimensional flows of activated fluids

Josef Málek

Nečas Center for Mathematical Modeling and Mathematical institute Charles University, Faculty of Mathematics and Physics

• Anna Abbatiello, Tomáš Los and Ondřej Souček

Jan Blechta and K.R. Rajagopal Miroslav Bulíček September 3, 2018

Section 1 Foreword

Soil liquefaction occurs when a saturated or partially saturated soil substantially loses strength and stiffness in response to an applied stress such as shaking during an earthquake or other sudden change in stress condition, in which material that is ordinarily a solid behaves like a liquid. Source: Wikipedia.

Photo: Niigata earthquake 1964.

Geometry and structure of material

Typical problem geometry and zoom into the structure of the granular material composed of solid grain matrix filled with an interstitial fluid. Mixture or single continuum model ?

Formulation of the problem

PROBLEM

div v = 0 ∂tv + div(v ⊗ v) − div S = −∇p ∂tpf + v · ∇pf − ∆pf = 0

• in QT

|S| ≤ τ(pf) ⇐ ⇒ D = O, |S| > τ(pf) ⇐ ⇒ S = τ(pf) D |D| + 2ν∗D.

• in QT

v · n = 0 g(s, vτ) = 0 ∇pf · n = 0

• n ΣT

v(0, ·) = v0 pf = p0 in Ω DATA ◮ Ω ⊂ R3, T > 0, v0, p0, ps and τ(pf) := (ps − pf)+

• L. Chupin and J. Mathé, Existence theorem for homogeneous incompressible Navier-Stokes

equation with variable theology, European Journal of Mechanics. B. Fluids 61 (2017) 135-143.

Two characterizations of the Bingham fluids

(I) Dichotomy |S| ≤ τ∗ ⇐ ⇒ D = O, |S| > τ∗ ⇐ ⇒ S = τ∗ D |D| + 2ν∗D. (II) Implicit constitutive tensorial relation 2ν∗D = (|S| − τ∗)+ |S| S (III) Two scalar constraints |Z| ≤ τ∗ and Z : D ≥ τ∗|D| Z := S − 2ν∗D

Earlier works & Tools: Duvaut & Lions (1976), Aubin & Frankowska (1995), Fuchs & Seregin (2000), Shelukhin (2002) - variational inequalities, calculus of multivalued functions, calculus of variations, regularity theory

Two characterizations of the Bingham fluids

(I) Dichotomy |S| ≤ τ∗ ⇐ ⇒ D = O, |S| > τ∗ ⇐ ⇒ S = τ∗ D |D| + 2ν∗D. (II) Implicit constitutive tensorial relation 2ν∗D = (|S| − τ∗)+ |S| S (III) Two scalar constraints |Z| ≤ τ∗ and Z : D ≥ τ∗|D| Z := S − 2ν∗D

Earlier works & Tools: Duvaut & Lions (1976), Aubin & Frankowska (1995), Fuchs & Seregin (2000), Shelukhin (2002) - variational inequalities, calculus of multivalued functions, calculus of variations, regularity theory

Questions

Chupin and Mathé

• considered Z : D = τ(pf)|D|, but there is one problem with their

convergence argument that can be removed if Z : D ≥ τ(pf)|D| is used instead of Z : D = τ(pf)|D|

• established the existence result in two dimensions. Energy equality
• available. Critical problem as 2d NS but with an additional nonlinearity.

Questions

• Is the model suitable to describe the “liquefaction"?
• Tensorial response of Bingham fluid characterized by two scalar

constraints (one inequality, one equality). How one can exploit it?

• Is it possible to develop mathematical theory in three

dimensions? .... to be continued later ....

Questions

Chupin and Mathé

• considered Z : D = τ(pf)|D|, but there is one problem with their

convergence argument that can be removed if Z : D ≥ τ(pf)|D| is used instead of Z : D = τ(pf)|D|

• established the existence result in two dimensions. Energy equality
• available. Critical problem as 2d NS but with an additional nonlinearity.

Questions

• Is the model suitable to describe the “liquefaction"?
• Tensorial response of Bingham fluid characterized by two scalar

constraints (one inequality, one equality). How one can exploit it?

• Is it possible to develop mathematical theory in three

dimensions? .... to be continued later ....

Section 2 Viscous fluids and visco-elastic fluids

Unsteady flows of incompressible fluids

Governing equations Ω ⊂ R3 div v = 0 ∂v ∂t + div(v ⊗ v) = −∇p + div S S = ST

• in (0, T) × Ω

v · n = 0 } on (0, T) × ∂Ω v(0, ·) = v0 } in Ω Energy balance A : B := 3

i,j=1 AijBij 1 2 ∂|v|2 ∂t

+ div

• |v|2

2 v + pv − Sv

• + S : ∇v = 0

d dt ˆ

Ω

|v|2 + 2 ˆ

Ω

S : ∇v + ˆ

∂Ω

(|v|2 + 2p)(v · n) − 2S : (v ⊗ n) = 0

Unsteady flows of incompressible fluids

Governing equations Ω ⊂ R3 div v = 0 ∂v ∂t + div(v ⊗ v) = −∇p + div S S = ST

• in (0, T) × Ω

v · n = 0 } on (0, T) × ∂Ω v(0, ·) = v0 } in Ω Energy balance A : B := 3

i,j=1 AijBij 1 2 ∂|v|2 ∂t

+ div

• |v|2

2 v + pv − Sv

• + S : ∇v = 0

d dt ˆ

Ω

|v|2 + 2 ˆ

Ω

S : ∇v + ˆ

∂Ω

(|v|2 + 2p)(v · n) − 2S : (v ⊗ n) = 0

Unsteady flows of incompressible fluids

Governing equations Ω ⊂ R3 div v = 0 ∂v ∂t + div(v ⊗ v) = −∇p + div S S = ST

• in (0, T) × Ω

v · n = 0 } on (0, T) × ∂Ω v(0, ·) = v0 } in Ω Energy balance A : B := 3

i,j=1 AijBij 1 2 ∂|v|2 ∂t

+ div

• |v|2

2 v + pv − Sv

• + S : ∇v = 0

d dt ˆ

Ω

|v|2 + 2 ˆ

Ω

S : ∇v + ˆ

∂Ω

(|v|2 + 2p)(v · n) − 2S : (v ⊗ n) = 0

Internal flows

ˆ

∂Ω

(−S) : (v ⊗ n) = ˆ

∂Ω

(−S)n · v = ˆ

∂Ω

• (−S)v
• τ · vτ

Boundary conditions

• v · n = 0 on ∂Ω
• constitutive equation involving vτ and/or (−Sn)τ

s := (−Sn)τ zτ := z − (z · n)n

n s (Sn)τ Sn Ω ∂Ω

ˆ

∂Ω

(−S) : (v ⊗ n) = ˆ

∂Ω

(−S)n · v = ˆ

∂Ω

• (−Sn
• τ · vτ

vτ = 0 no slip boundary condition s = γ∗vτ with γ∗ > 0 Navier’s slip boundary condition s = 0 (perfect) slip boundary condition

Energy estimates and constitutive equations

• Governing equations

Ω ⊂ R3 div v = 0 ∂v ∂t + div(v ⊗ v) = −∇p + div S, S = ST

• in (0, T) × Ω

v · n = 0 } on (0, T) × ∂Ω v(0, ·) = v0 } in Ω

• Energy equality valid for t ∈ (0, T]

D := 1

2

• ∇v + (∇v)T

v(t)2

2 + 2

ˆ t ˆ

Ω

S : D + 2 ˆ t ˆ

∂Ω

s · vτ = v02

2

• To close the system

we add a material dependent relation involving S and D we add a material dependent relation involving s and vτ Constitutive equations

Energy estimates and constitutive equations

• Governing equations

Ω ⊂ R3 div v = 0 ∂v ∂t + div(v ⊗ v) = −∇p + div S, S = ST

• in (0, T) × Ω

v · n = 0 } on (0, T) × ∂Ω v(0, ·) = v0 } in Ω

• Energy equality valid for t ∈ (0, T]

D := 1

2

• ∇v + (∇v)T

v(t)2

2 + 2

ˆ t ˆ

Ω

S : D + 2 ˆ t ˆ

∂Ω

s · vτ = v02

2

• To close the system

we add a material dependent relation involving S and D we add a material dependent relation involving s and vτ Constitutive equations

Energy estimates and constitutive equations

• Governing equations

Ω ⊂ R3 div v = 0 ∂v ∂t + div(v ⊗ v) = −∇p + div S, S = ST

• in (0, T) × Ω

v · n = 0 } on (0, T) × ∂Ω v(0, ·) = v0 } in Ω

• Energy equality valid for t ∈ (0, T]

D := 1

2

• ∇v + (∇v)T

v(t)2

2 + 2

ˆ t ˆ

Ω

S : D + 2 ˆ t ˆ

∂Ω

s · vτ = v02

2

• To close the system

we add a material dependent relation involving S and D we add a material dependent relation involving s and vτ Constitutive equations

Classes of constitutive equations

div v = 0 ∂v ∂t + div(v ⊗ v) = −∇p + div S, S = ST (1) G(S, D) = O implicit algebraic equations (2) G(

∗

S, S,

∗

D, D) = O

∗

A an objective time derivative rate type viscoelastic fluids (3) G(

∗

S, S,

∗

D, D) − ∆S = O rate type viscoelastic fluids with stress diffusion (4) G(

∗∗

S,

∗

S, S,

∗∗

D,

∗

D, D) = O rate type viscoelastic fluids of higher order

Classes of constitutive equations

div v = 0 ∂v ∂t + div(v ⊗ v) = −∇p + div S, S = ST (1) G(S, D) = O implicit algebraic equations (2) G(

∗

S, S,

∗

D, D) = O

∗

A an objective time derivative rate type viscoelastic fluids (3) G(

∗

S, S,

∗

D, D) − ∆S = O rate type viscoelastic fluids with stress diffusion (4) G(

∗∗

S,

∗

S, S,

∗∗

D,

∗

D, D) = O rate type viscoelastic fluids of higher order

Section 3 Implicit constitutive equations and implicitly stated boundary conditions

G(S, D) = O KR Rajagopal (2003)

S = 2νD Navier-Stokes 2ν(|S|2, |D|2)D = 2α(|S|2, |D|2)S generalized viscosity 2νD = (|S| − σ∗)+ |S| S Bingham 1 2ν S = (|D| − d∗)+ |D| D Euler/Navier-Stokes

• K. R. Rajagopal: On implicit constitutive theories. Appl. Math., 48 (2003)

279—319.

• J. Málek, V. Průša: Derivation of equations of continuum mechanics and

thermodynamics of fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, (eds.Y. Giga, A. Novotný), Springer available

• nline (2017)

Euler/limiting shear-rate limiting shear- rate rigid body Euler/shear- thickening shear- thickening rigid/shear- thickening Euler/Navier- Stokes Navier-Stokes Bingham = rigid/Navier- Stokes Euler/shear- thinning shear-thinning rigid/shear- thinning Euler limiting shear stress perfect plastic |D| ≤ δ∗ ⇐ ⇒ S = O no activation |S| ≤ σ∗ ⇐ ⇒ D = O Summary of systematic classification of fluid-like responses with corresponding |S| vs |D| diagrams.

no-slip slip/Navier’s slip Navier’s slip stick-slip slip |vτ| ≤ δ∗ ⇐ ⇒ s = 0 no activation |s| ≤ s∗ ⇐ ⇒ vτ = 0 Summary of systematic classification of boundary conditions with corresponding |s| vs |vτ| diagrams.

Robustness of G(S, D) = O

2νD = (|S|−σ∗)+

|S|

S

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y 1e − 06 1e − 05 1e − 04 1e − 03 1e − 02 1e − 01 1e + 00 1e + 01 |D| 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y 1e − 06 1e − 05 1e − 04 1e − 03 1e − 02 1e − 01 1e + 00 1e + 01 |D|

• J. Hron, J. Málek, J. Stebel, K. Touška: A novel view on computations of steady flows of

Bingham fluids using implicit constitutive relations, MORE/2017/08 (2017)

• J. Blechta, J. Málek, K.R. Rajagopal: On classification of fluids. Part 1: Incompressible fluids, to

be submitted (2018)

• J. Blechta: Ph.D. Thesis (2018)

Formulation of the problem

PROBLEM div v = 0 ∂tv + div(v ⊗ v) − div S = −∇p G(S, D) = O

• in QT

v · n = 0 s := −(Sn)τ g(s, vτ) = 0

• n ΣT

v(0, ·) = v0 in Ω

DATA ◮ Ω ⊂ Rd bounded, open set with ∂Ω ∈ C1,1 and n : ∂Ω → Rd ◮ T > 0 and QT := (0, T) × Ω, ΣT := (0, T) × ∂Ω ◮ v0 ◮ G and g - constitutive functions in the bulk and on the boundary

Main questions addressed (supercritical problems)

AIMS ◮ To develop theory in three dimensions - d = 3 ◮ To establish large data existence of solution for any set of data (Ω, T, v0) and for robust class of constitutive equations described by G and g ◮ To develop a theory with p ∈ L1(QT ) - important

• heat-conducting incompressible fluids
• one/two equation turbulence model
• incompressible fluids with pressure and shear-rate dependent viscosity
• corresponding numerical methods and their analysis

Large data and long time existence theory

Robust mathematical theory for a large class of constitutive equations and boundary conditions is available.

• M. Bulíček, P. Gwiazda, J. Málek, A. Świerczewska-Gwiazda, On unsteady flows of implicitly

constituted incompressible fluids, SIAM J. Math. Anal. 44 (2012) 2756–2801.

• M. Bulíček, P. Gwiazda, J. Málek, K. R. Rajagopal, A. Świerczewska-Gwiazda, On flows of fluids

described by an implicit constitutive equation characterized by a maximal monotone graph, Mathematical Aspects of Fluid Mechanics (Eds. J. C. Robinson, J. L. Rodrigo and W. Sadowski), London Mathematical Society Lecture Note Series (No. 402) (2012), Cambridge University Press, 23–51.

• M. Bulíček, J. Málek On unsteady internal fows of Bingham fuids subject to threshold slip on the

impermeable boundary, (Eds. H. Amann, Y. Giga, H. Okamoto, H. Kozono, M. Yamazaki), Recent Developments of Mathematical Fluid Mechanics, Birkhäuser/Springer, Basel, 2016, 135-156.

• M. Bulíček, J. Málek, Internal flows of incompressible fluids subject to stick-slip boundary

conditions, Vietnam Journal of Mathematics 45 (2017), 207–220.

• E. Maringová, J. Žabenský: On a Navier-Stokes-Fourier-like system capturing transitions between

viscous and inviscid fluid regimes and between no-slip and perfect-slip boundary conditions, Nonlinear Analysis: Real World Applications 41 (2018) 152-178.

Key earlier theoretical results

• Existence of WS to NSEs in 2d and 3d (Leray (1929-1934), Oseen (1922))
• Existence of WS to NSEs in bounded domains, its 2d uniqueness and 3d

conditional uniqueness and existence (Hopf (1952), Kiselev & Ladyzhenkaya (1959),

Prodi (1959), Serrin (1963))

• Existence of WS to S = 2(ν0 + ν1|D|r−2)D for r ≥ 11

5 and its uniqueness

if r ≥ 5

2 (Ladyzhenskaya (1967-1972), J.-L. Lions (1969))

• Nečas, Bellout, Bloom, Málek, R ‌

užička (1993-2000): r ≥ 9

5

• DalMaso, Murat (1996), Frehse, Málek, Steinhauer, R ‌

užička (1996-2000), Bulíček, Málek, Rajagopal (2007), Wolf (2009): r ≥ 8

5

• Diening, R ‌

užička, Wolf (2010), Breit, Diening, Schwarzacher (2013): r > 6

5

• Bulíček, Ettwein, Kaplický, Pražák (2010): uniqueness for r > 11

5

• Existence of WS to the class of monotone responses G(S, D) = O, Orlicz

function-type response (Bulíček, Gwiazda, Málek, Świerczewska-Gwiazda (2012): r > 6

5 )

• Existence of WS to activated fluids with activated boundary conditions

(Bulíček, Málek (2016), Blechta, Málek, Rajagopal (2018): r > 6

5 )

Section 4 Structure of implicit relations

Basic information

A PRIORI ESTIMATES Multiplying the 2nd Eq. by v

1 2 ∂|v|2 ∂t

+ div( 1

2|v|2v) − div(Sv) + S : D = − div(pv)

Since v · n = 0, integrating it over Ω leads to 1 2 d dtv2

2 +

ˆ

Ω

S : D dx + ˆ

∂Ω

s · vτ dS = 0 For the power-law fluids S = |D|r−2D ⇐ ⇒ D = |S|r′−2S r′ = r/(r − 1) : S : D = 1 r + 1 r′

• S : D = 1

r |D|r + 1 r′ |S|r′ For Navier’s slip s = γ∗vτ ⇐ ⇒ vτ =

1 γ∗ s :

s · vτ = ( 1

2 + 1 2)s · vτ = γ∗ 2 |vτ|2 + 1 2γ∗ |s|2

Maximal monotone r-responses G(S, D) = O

(A1) G(O, O) = O (A2) Monotone response. For any couples (Si, Di), i = 1, 2, satisfying G(Si, Di) = O: (S1 − S2) : (D1 − D2) ≥ 0 (A3) Maximal monotone response. Let (S, D) ∈ Rd×d

sym × Rd×d sym.

If (S − S∗) : (D − D∗) ≥ 0 ∀ (S∗, D∗) such that G(S∗, D∗) = O then G(S, D) = O (A4) r-response. There are α∗ > 0 and c∗ ≥ 0 so that for any (S, D) such that G(S, D) = O S : D ≥ α∗

• |D|r + |S|r′

− c∗ Similar assumptions on g(s, vτ) = 0: maximal monotone 2-responses.

Function spaces - Stick-slip versus No-slip

W 1,q

n

:= {v ∈ W 1,q(Ω; Rd); v · n = 0 on ∂Ω}, W 1,q

n,div := {v ∈ W 1,q(Ω; Rd); div v = 0; v · n = 0 on ∂Ω},

versus W 1,q := {v ∈ W 1,q(Ω; Rd); v = 0 on ∂Ω}, W 1,q

0,div := {v ∈ W 1,q(Ω; Rd); div v = 0; v = 0 on ∂Ω},

Section 5 Activated fluids with threshold slip - existence of unsteady flows for large data

G(S, D) := 2ν∗D − (|S| − τ∗)+ |S| S Bingham fluid g(s, v) := v − (|s| − s∗)+ |s| s stick/slip bc Theorem Let Ω ⊂ Rd be a C1,1 domain. Then for any v0 ∈ L2

0,div there exists

v ∈ L∞(0, T; L2(Ω)d) ∩ L2(0, T; W 1,2

n,div)

S ∈ L2(Q)d×d

sym,

s ∈ L2(0, T; L2(∂Ω)d) p1 ∈ L2(Q), p2 ∈ L

d+2 d+1 (0, T; W 1, d+2 d+1 (Ω))

solving for almost all time t ∈ (0, T) and for all w ∈ W 1,∞

n

∂tv, w − ˆ

Ω

(v ⊗ v) · ∇w + ˆ

Ω

S : D(w) + ˆ

∂Ω

s · w = ˆ

Ω

(p1 + p2) div w and fulfilling G(S, Dv) = O a.e. in QT and g(s, vτ) = 0 a.e. in ΣT

• M. Bulíček, J. Málek: On unsteady internal flows of Bingham fluids subject to threshold slip on

the impermeable boundary, in Recent Developments of Mathematical Fluid Mechanics (eds. H. Amann et al.), pp. 135-156 (2016)

G(S, D) := S − (|S| − d∗)+ |D| S(|D|)D Euler/power-law fluid g(s, v) := v − (|s| − s∗)+ |s| s stick/slip bc Theorem Let Ω ⊂ Rd be a C1,1 domain and r >

2d d+2. Then for any v0 ∈ L2 0,div there

exists v ∈ L∞(0, T; L2(Ω)d) ∩ Lr(0, T; W 1,r

n,div)

S ∈ Lr′(Q)d×d

sym,

s ∈ L2(0, T; L2(∂Ω)d) p1 ∈ Lr′(Q), p2 ∈ L

(d+2)r 2d

(Q) solving for almost all time t ∈ (0, T) and for all w ∈ W 1,∞

n

∂tv, w − ˆ

Ω

(v ⊗ v) · ∇w + ˆ

Ω

S : D(w) + ˆ

∂Ω

s · w = ˆ

Ω

(p1 + p2) div w and fulfilling G(S, Dv) = O a.e. in QT and g(s, vτ) = 0 a.e. in ΣT

• J. Blechta, J. Málek, K.R. Rajagopal: On classification of fluids. Part 1: Incompressible fluids, to

be submitted (2018)

Function spaces - Stick-slip versus Slip

W 1,q

n

:= {v ∈ W 1,q(Ω; Rd); v · n = 0 on ∂Ω}, W 1,q

n,div := {v ∈ W 1,q(Ω; Rd); div v = 0; v · n = 0 on ∂Ω},

versus W 1,q := {v ∈ W 1,q(Ω; Rd); v = 0 on ∂Ω}, W 1,q

0,div := {v ∈ W 1,q(Ω; Rd); div v = 0; v = 0 on ∂Ω},

By the Helmholtz decomposition, for q ∈ (1, ∞): W 1,q

n

= W 1,q

n,div ⊕ {∇ϕ; ϕ ∈ W 2,q, ∇ϕ · n = 0 on ∂Ω}.

Similar decomposition for W 1,q

0 (Ω)d is open.

• Essential difference in the weak formulation
• s∗ can be artificial (big enough) so that it is never active
• in analysis if v ∈ L∞(0, T; C(Ω))
• in computer simulations

Theorem

Theorem Let Ω ⊂ R3 and the assumptions (A1)–(A4) are satisfied with r > 6

• 5. Then for

any Ω ∈ C1,1 and T ∈ (0, ∞) and for arbitrary v0 ∈ L2

n,div, and γ∗ ≥ 0 ,

(1) there exists weak solution to Problem. Steps: ◮ Structural assumptions (A1)–(A4) on G(S, D) = O ◮ Interplay between the chosen boundary conditions and global integrability of p ◮ Convergence lemma to fulfil G(S, D) = O and g(s, vτ) = 0 ◮ L∞- and/or W 1,∞ approximations of Lr(0, T; W 1,r(Ω))-functions ◮ Extension to Orlicz functions setting - duality ◮ Structural assumptions (A1)–(A5) on G(t, x, S, D) = O

• M. Bulíček, P. Gwiazda, J. Málek, A. Świerczewska-Gwiazda: On Unsteady Flows of Implicitly

Constituted Incompressible Fluids, SIAM J. Math. Anal., Vol. 44, No. 4, pp. 2756–2801 (2012)

Attainment of the constitutve equation(s) - Convergence lemma

Lemma Let U ⊂ QT be arbitrary (measurable) and r ∈ (1, ∞). Assume that

• the response G(S, D) = O maximal monotone (satisfying (A2)–(A3))
• {Sn}∞

n=1 and {Dn}∞ n=1 satisfy

G(Sn, Dn) = O for a.a. (t, x) ∈ U Dn ⇀ D weakly in Lr(U)d×d Sn ⇀ S weakly in Lr′(U)d×d lim sup

n→∞

ˆ

U

Sn · Dn dx dt ≤ ˆ

U

S · D dx dt. Then G(S, D) = O almost everywhere in U.

Step 1. Sn : Dn ⇀ S : D weakly in L1(U) From (A2) 0 ≤ (Sn − Sm) : (Dn − Dm) a.e. in U Hence, by the assumptions, lim

n→∞ lim m→∞ (Sn − Sm) : (Dn − Dm)1 ≤ 0

which implies lim

n→∞ lim m→∞

ˆ

U

(Sn − Sm) : (Dn − Dm)ϕ = 0 ∀ϕ ∈ L∞(U) Setting L := limℓ→∞ ´

U(Sℓ : Dℓ)ϕ we conclude that

0 = lim

n→∞ lim m→∞

ˆ

U

Sn · Dn ϕ − ˆ

U

Sn · Dm ϕ − ˆ

U

Sm : Dn ϕ + ˆ

U

Sm : Dm ϕ

• = 2
• L −

ˆ

U

S : D ϕ

Step 2. G(S, D) = O a.e. in U Take arbitrarily a nonnegative ϕ ∈ L∞(U) and (S∗, D∗) such that G(S∗, D∗) = O Then from (A2) and Step 1 0 ≤ lim

n→∞

ˆ

U

(Sn − S∗) : (Dn − D∗)ϕ = ˆ

U

(S − S∗) : (D − D∗)ϕ. Since ϕ ≥ 0 arbitrary we get 0 ≤ (S − S∗) : (D − D∗) a.e. in U Since (S∗, D∗) satisfying G(S∗, D∗) = O is arbitrary, the maximality of the graph implies G(S, D) = O a.e. in U

Methods

lim sup

n→∞

ˆ

U

Sn · Dn dx dt ≤ ˆ

U

S · D dx dt.

• subcritical case r ≥ 11

5

• Minty’s method
• energy equality - v is an admissible test function
• supercritical case 6

5 < r < 11 5

• Generalized Minty’s method - Convergence lemma
• L∞-truncation r ≥ 8

5

• Lipschitz truncation

Identification of the limit for boundary terms

Assume that sn ⇀ s weakly in L2(0, T; L2(∂Ω)3), vn ⇀ v weakly in L2(0, T; L2(∂Ω)3) and g(sn, vn) = 0

• it is enough to show that

lim sup

n→∞

ˆ

∂Ω

sn · vn ≤ ˆ

∂Ω

s · v

• however we also have

vn → v strongly in L1(0, T; L1(∂Ω)3) By Egorov theorem, for any ε > 0 there exists Uε ⊂ ΣT such that |ΣT \ Uε| ≤ ε and vn → v strongly in L∞(Uε)3 = ⇒ lim sup

n→∞

ˆ

Uε

sn · vn ≤ ˆ

Uε

s · v and g(s, v) = 0 a.e. in Uε. But ε is arbitrary and g(s, v) = 0 a.e. on ΣT

References - Lipschitz trucation method/1

• Stationary problems
• E. Acerbi, N. Fusco, An approximation lemma for W 1,p-functions, in Material Instabilities in

Continuum Mechanics (Oxford Univ. Press), pp. 1-5 (1988).

• J. Frehse, J. Málek, M. Steinhauer, On analysis of steady flows of fluids with shear-dependent

viscosity based on the Lipschitz truncation method, SIAM J. Math. Anal. 34 (2003) 1064-1083.

• L. Diening, J. Málek, M. Steinhauer, On Lipschitz truncations of Sobolev functions (with variable

exponent) and their selected applications, ESAIM Control Optim. Calc. Var. 14 (2008) 211-232.

• D. Breit, L. Diening, M. Fuchs: Solenoidal Lipschitz truncation and applications in fluid

mechanics, J. Differential Equations 253 (2012) 1910-1942.

• Novelty:

∇uλχ{u=uλ}r ≤ δ(λ) with δ(λ) → 0 as λ → ∞ which implies uλ → u in W 1,r as λ → ∞

References - Lipschitz trucation method/2

• Evolutionary problems
• J. Kinnunen and J. L. Lewis, Very weak solutions of parabolic systems of p-Laplacian type, Ark.
• Mat. 40 (2002) 105-132.
• L. Diening, , M. R‌užička, J. Wolf, Existence of weak solutions for unsteady motions of generalized

Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. 9 (2010) 1-46.

• D. Breit, L. Diening, S. Schwarzacher, Solenoidal Lipschitz truncation for parabolic PDEs, Math.

Methods Mod. Appl. Sci 23 (2013) 2671-2700.

• M. Bulíček, P. Gwiazda, J. Málek, A. Świerczewska-Gwiazda: On Unsteady Flows of Implicitly

Constituted Incompressible Fluids, SIAM J. Math. Anal., Vol. 44, No. 4, pp. 2756–2801 (2012)

Section 6 Pore pressure activated fluids

Formulation of the problem

PROBLEM div v = 0 ∂tv + div(v ⊗ v) − div S = −∇p for τ(pf) = (ps − pf)+ 2ν∗D = (|S| − τ(pf))+ |S| S ∂tpf + v · ∇pf − ∆pf = 0

• in QT

v · n = 0 g(s, vτ) = 0 ∇pf · n = 0

• n ΣT

v(0, ·) = v0 pf = p0 in Ω

DATA ◮ Ω ⊂ R3, T > 0, v0, p0 and ps

• L. Chupin and J. Mathé, Existence theorem for homogeneous incompressible Navier-Stokes

equation with variable theology, European Journal of Machanics. B. Fluids 61 (2017) 135-143.

Convergence lemma modified

Lemma Let U ⊂ QT be arbitrary (measurable). Assume that

• {Sn = 2ν∗Dn + Zn}∞

n=1, {Dn}∞ n=1 and {pn f }∞ n=1 satisfy

Zn = τ(pn

f )

Dn |Dn| + 1

n

with τ(pn

f ) = (ps − pn f )+

Dn ⇀ D weakly in L2(U)d×d Zn ⇀ Z weakly in L2(U)d×d pn

f → pf

a.e. in U and strongly in L2(U) sup

n pn f ∞ < ∞

lim sup

n→∞

ˆ

U

Zn · Dn dx dt ≤ ˆ

U

Z · D dx dt. Then G(S, D, pf) = O almost everywhere in U.

Three steps of the proof

Step 1. |Z| ≤ τ(pf) Step 2. Zn : Dn ⇀ Z : D weakly in L1(U) Step 3. Z : D ≥ τ(pf)|D|

Formulation of the problem

PROBLEM div v = 0 ∂tv + div(v ⊗ v) − div S = −∇p + b for τ(pf) = (ps − pf)+ 2ν∗D = (|S| − τ(pf))+ |S| S ∂tpf + v · ∇pf − ∆pf = g + v · ∇ps

• in QT

v · n = 0 g(s, vτ) = 0 ∇pf · n = 0

• n ΣT

v(0, ·) = v0 pf = p0 in Ω

DATA ◮ Ω ⊂ R3, T > 0, v0, p0, ps and g := div b

Result

Theorem Let Ω ⊂ R3 such that ∂Ω ∈ C1,1, T > 0 and for arbitrary v0 ∈ L2

n,div, p0 ∈ L∞(Ω), b ∈ (L2(0, T; W 1,2 n ))∗, g ∈ Lq(QT ) with q > 5

2 ps ∈ Lq(0, T; W 1,q(Ω) with q > 10. Then there exists weak solution to our problem.

• A. Abbatiello, T.Los, J. Málek, O. Souček: On three-dimensional flows of pore pressure activated

Bingham fluids, to be submitted (2018)

Proof - n-approximations

Consider Zn(pf, D) := (ps − pf)+ Dvn |Dvn| + 1

n

(Bn) zn(v) := s∗ vτ |vτ| + 1

n

(Tn) and smooth Gn, |G′

n| ≤ 1 n

Gn(s) := 1 for s ≤ n, Gn(s) = 0 for s > 2n. Take approximation vn

,t + div(vn ⊗ vn)Gn(|vn|2) − div Zn + div Dvn = −∇pn

with constitutive equations (Bn)–(Tn). Since (Bn)–(Tn) implies Z = Zn(pf, D), z = ζ(vτ) with Zn and ζn being continuous monotone with linear growth (at infinity), the existence follows from standard monotone operator theory (due to the presence

• f Gn)

Pressure for n fixed

vn

,t, ˜

w + (Sn, D( ˜ w)) + (div(vn ⊗ vn)G(|vn|), ˜ w) + (sn, ˜ wτ)∂Ω − b, ˜ w = 0 ¯ for all ˜ w ∈ W 1,2

n,div and a.a. t ∈ (0, T)

Define pn as the solution of the following problem (∇pn, ∇z) + (Sn, ∇(2)z) + (div(vn ⊗ vn)G(|vn|), ∇z) + (sn, (∇z)τ)∂Ω − b, ∇z = 0 for all z ∈ W 2,2(Ω) with ∇z · n = 0 on ∂Ω and a.a. t ∈ (0, T) w = ˜ w + ∇z (Sn,D(w))) + (div(vn ⊗ vn)G(|vn|), w) + (sn, wτ)∂Ω − b, w = vn

,t, ˜

w + (∇p, ∇z) = vn

,t, ˜

w + ∇z + (∇p, ˜ w + ∇z) which finally leads to: vn

,t, w + (Sn, D(w)) + (div(vn ⊗ vn)G(|vn|), w) + (sn, wτ)∂Ω

= (pn, div w) + b, w for all w ∈ W 1,2

n

and a.a. t ∈ (0, T)

Apriori estimates

• I. Test by vn (convective term) vanishes to get

1 2 d dtvn2

2 +

ˆ

Ω

Sn · D(vn) + ˆ

∂Ω

sn · vn = 0 sup

t∈(0,T )

vn(t)2

2+

ˆ

QT

|Sn|2+|∇vn|2+|vn|

10 3 +

ˆ

(0,T )×∂Ω

|sn|2+|vn|2 ≤ C(v0)

• II. Find pn

2 with zero mean value solving at each time level

ˆ

Ω

∇pn

2 · ∇ϕ = −

ˆ

Ω

(div vn ⊗ vn)Gn(|v|) · ϕ But ˆ

QT

| div(vn ⊗ vn)Gn(|v|)|

5 4 ≤ C =

⇒ ˆ T pn

2

5 4

1, 5

4 ≤ C

Define pn

1 := pn − pn 2 .

Apriori estimates - continuation

• III. For pn

1 := pn − pn 2 find ϕ with zero mean value such that ∇ϕ · n = 0 on ∂Ω

solving ∆ϕ = pn

1 =

⇒ ˆ

QT

|∇2ϕ|2 + ˆ

(0,T )×∂Ω

|∇ϕ|2 ≤ ˆ

QT

|pn

1 |2

Test by ∇ϕ and integrate over QT ˆ

QT

|pn

1 |2 = −

ˆ

QT

∇pn

1 · ∇ϕ =

ˆ

QT

(∇pn

2 − div(vn ⊗ vnGn(|v|))) · ∇ϕ

+ ˆ

QT

Sn · ∇2ϕ + ˆ

(0,T )×∂Ω

sn · ∇ϕ = ˆ

QT

Sn · ∇2ϕ + ˆ

(0,T )×∂Ω

sn · ∇ϕ ≤ C ˆ

QT

|pn

1 |2

1

2

IV. vn

,t(L2(0,T ;W 1,2

n

∩L5(Ω)))∗ ≤ C

Convergences

Aubin-Lions and apriori estimates: vn ⇀ v weakly in L2(0, T; W 1,2

n ),

Sn ⇀ S weakly in L2(Q)d×d, sn ⇀ s weakly in L2(0, T; L2(∂Ω)), vn → v strongly in Lq(Q), q ∈ [1, 10/3) vn → v strongly in Lq(0, T; Lq(∂Ω)), q ∈ [1, 8/3) pn

1 ⇀ p1

weakly in L2(Q), pn

2 ⇀ p2

weakly in L

5 4 (0, T; W 1, 5 4 (Ω)),

vn

,t ⇀ v,t

weakly in (L2(0, T; W 1,2

n ) ∩ L5(Q))∗

solving the original problem. It remains to show the validity of constitutive equations. In fact, g(s, vτ) = 0 is

• easy. We focus on G(S, Dv) = O.

Convergence III

Assume that {kn}∞

n=1 is such that 0 < A ≤ kn ≤ B < ∞. Test the n-th

approximation by wn := Tkn(vn − v) := (vn − v) min

• 1,

kn |vn − v|

• Note Tk(u) = u if |u| ≤ k.

Taking wn as a test function Sn = Zn + Dvn lim sup

n→∞

ˆ

QT

Sn · D(wn) − pn

1 div wn

= lim sup

n→∞

ˆ

QT

−vn

,t, wn − (div(vn ⊗ vn)Gn(|nn|))) + ∇pn 2 ) · wn

+ ˆ

ΣT

sn · wn ≤ 0

Find Z ∈ L10/3(Q) fulfilling Z = O if D = O, Z = τ(pf) D |D| if D = O. Then lim sup

n→∞

ˆ

QT

(Zn − Z) · D(wn) ≤ lim sup

n→∞

ˆ

|vn−v|≥kn

kn |vn − v||pn

1 |(|∇vn| + |∇v|) ≤ ε

where the last inequality follows from the proper choice of A, B and kn. Indeed, considering In := C∗(|pn

1 |2 + |∇vn|2 + |∇v|2 + |S|2 + 1)

sup

n

ˆ

QT

In < ∞ we observe that lim sup

n→∞

ˆ

|vn−v|<kn(Zn − Z) · D(vn − v) ≤ lim sup n→∞

ˆ

|vn−v|≥kn

kn |vn − v|In AIM: RHS should tend to zero.

For N ∈ N arbitrary, fix A := N and B := N N+1 and define Qn

i := {(t, x) ∈ QT ; N i ≤ |vn − v| ≤ N i+1}

i = 1, . . . , N. Since

N

• i=1

ˆ

Qn

i

In ≤ C∗, there is, for each n ∈ N, an index in ∈ {1, . . . , N} such that ˆ

Qn

in

In ≤ C∗ N Setting kn := N in+1, RHS is estimated in the following way: ˆ

|vn−v|≥Nin+1

kn |vn − v|In = ˆ

Nin+2≥|vn−v|≥Nin+1

· · · + ˆ

|vn−v|≥Nin+2

. . . ≤ ˆ

Qn

in

· · · + 1 N ˆ

Nin+2≥|vn−v|≥Nin+1

In ≤ C∗ N . (2) We have concluded that lim sup

n→∞

ˆ

|vn−v|≤kn

• Zn − Z
• · (Dvn − Dv) ≤ C∗

N ,

Thus, for W n :=

• Zn − Z
• · (Dvn − Dv) ≥ 0.

lim sup

n→∞

ˆ

|vn−v|≤kn W n ≤ C∗

N + lim sup 1 n ˆ

QT

|Sn||Sn − S| ≤ C∗ N Applying Egorov theorem, one then concludes that W n → 0 strongly in L1(QT \ Ej) Ej ⊂ QT : lim

j→∞ |Ej| = 0

= ⇒ lim sup

n→∞

ˆ

QT \Ej

Zn · Dvn = ˆ

QT \Ej

S · Dv . Convergence lemma and the properties of Ej: = ⇒ Dv = |S| − τ(pf)+ |S| S a.e. in QT Thank you

Thus, for W n :=

• Zn − Z
• · (Dvn − Dv) ≥ 0.

lim sup

n→∞

ˆ

|vn−v|≤kn W n ≤ C∗

N + lim sup 1 n ˆ

QT

|Sn||Sn − S| ≤ C∗ N Applying Egorov theorem, one then concludes that W n → 0 strongly in L1(QT \ Ej) Ej ⊂ QT : lim

j→∞ |Ej| = 0

= ⇒ lim sup

n→∞

ˆ

QT \Ej

Zn · Dvn = ˆ

QT \Ej

S · Dv . Convergence lemma and the properties of Ej: = ⇒ Dv = |S| − τ(pf)+ |S| S a.e. in QT Thank you