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Regularity for systems of PDEs arising in continuum thermodynamics Miroslav Bul cek Mathematical Institute of the Charles University Sokolovsk a 83, 186 75 Prague 8, Czech Republic Challenges in analysis and modeling - K. R.

SLIDE 1

Regularity for systems of PDEs arising in continuum thermodynamics

Miroslav Bul´ ıˇ cek

Mathematical Institute of the Charles University Sokolovsk´ a 83, 186 75 Prague 8, Czech Republic Challenges in analysis and modeling - K. R. Rajagopal

March 31, 2012

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 1 / 15

SLIDE 2

Outline

Why to study regularity of PDEs

to have fun to learn something from physics what kind of regularity? weak & strong & classical solution to justify the model (in case regularity holds) to justify the numerical scheme and the error estimate (in case regularity holds) to show that the model is wrong

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 2 / 15

SLIDE 3

Outline

Why to study regularity of PDEs

to have fun to learn something from physics what kind of regularity? weak & strong & classical solution to justify the model (in case regularity holds) to justify the numerical scheme and the error estimate (in case regularity holds) to show that the model is wrong

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 2 / 15

SLIDE 4

Outline

Why to study regularity of PDEs

to have fun to learn something from physics what kind of regularity? weak & strong & classical solution to justify the model (in case regularity holds) to justify the numerical scheme and the error estimate (in case regularity holds) to show that the model is wrong

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 2 / 15

SLIDE 5

Outline

Why to study regularity of PDEs

to have fun to learn something from physics what kind of regularity? weak & strong & classical solution to justify the model (in case regularity holds) to justify the numerical scheme and the error estimate (in case regularity holds) to show that the model is wrong

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 2 / 15

SLIDE 6

Outline

Why to study regularity of PDEs

to have fun to learn something from physics what kind of regularity? weak & strong & classical solution to justify the model (in case regularity holds) to justify the numerical scheme and the error estimate (in case regularity holds) to show that the model is wrong

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 2 / 15

SLIDE 7

Outline

We demonstrate all results and open problems on the prototype: div v = 0 vt + div(v ⊗ v) − divT T T = f, · · · = · · · . Navier-Stokes equations (neglect coupling)

full problem unsteady vs. steady neglect inertia = ⇒ full regularity

power-law like models & more general situation (neglect coupling)

neglect inertia vs. full system regularity of stress vs. velocity (displacement gradient)

coupled problems (only with the equation for temperature/internal energy)

Newtonian fluid (with and without inertia) - non-Newtonian models -nonlinearity may help

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 3 / 15

SLIDE 8

Outline

We demonstrate all results and open problems on the prototype: div v = 0 vt + div(v ⊗ v) − divT T T = f, · · · = · · · . Navier-Stokes equations (neglect coupling)

full problem unsteady vs. steady neglect inertia = ⇒ full regularity

power-law like models & more general situation (neglect coupling)

neglect inertia vs. full system regularity of stress vs. velocity (displacement gradient)

coupled problems (only with the equation for temperature/internal energy)

Newtonian fluid (with and without inertia) - non-Newtonian models -nonlinearity may help

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 3 / 15

SLIDE 9

Navier-Stokes equations

T T T := −pI I I + 2ν0D D D(v), where D D D(v) is the symmetric part of ∇v Navier-Stokes equations vt − div(v ⊗ v) − ν0△v = −∇p + f div v = 0. (N-S) d = 2 - regularity, d = 3, . . . - regularity partial & conditional & special geometries & small data & short time Stokes equations vt − ν0△v = −∇p + f div v = 0. (S) maximal regularity in any d

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 4 / 15

SLIDE 10

Navier-Stokes equations

T T T := −pI I I + 2ν0D D D(v), where D D D(v) is the symmetric part of ∇v Navier-Stokes equations vt − div(v ⊗ v) − ν0△v = −∇p + f div v = 0. (N-S) d = 2 - regularity, d = 3, . . . - regularity partial & conditional & special geometries & small data & short time Stokes equations vt − ν0△v = −∇p + f div v = 0. (S) maximal regularity in any d

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 4 / 15

SLIDE 11

Navier-Stokes equations

T T T := −pI I I + 2ν0D D D(v), where D D D(v) is the symmetric part of ∇v Navier-Stokes equations vt − div(v ⊗ v) − ν0△v = −∇p + f div v = 0. (N-S) d = 2 - regularity, d = 3, . . . - regularity partial & conditional & special geometries & small data & short time Stokes equations vt − ν0△v = −∇p + f div v = 0. (S) maximal regularity in any d

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 4 / 15

SLIDE 12

Navier-Stokes equations

Navier-Stokes equations - steady case

steady Navier-Stokes equations − div(v ⊗ v) − ν0△v = −∇p + f div v = 0. d = 2, 3, 4 - regularity, d = 5 - the same scaling as for (N-S) in d = 3 - maybe a hint to (N-S)

J. Frehse & coauthors - existence of a regular solution for

d = 5, . . . , 10 but no hint to solve 3d (N-S)

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 5 / 15

SLIDE 13

Navier-Stokes equations

Navier-Stokes equations - steady case

steady Navier-Stokes equations − div(v ⊗ v) − ν0△v = −∇p + f div v = 0. d = 2, 3, 4 - regularity, d = 5 - the same scaling as for (N-S) in d = 3 - maybe a hint to (N-S)

J. Frehse & coauthors - existence of a regular solution for

d = 5, . . . , 10 but no hint to solve 3d (N-S)

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 5 / 15

SLIDE 14

Navier-Stokes equations

Navier-Stokes equations - steady case

steady Navier-Stokes equations − div(v ⊗ v) − ν0△v = −∇p + f div v = 0. d = 2, 3, 4 - regularity, d = 5 - the same scaling as for (N-S) in d = 3 - maybe a hint to (N-S)

J. Frehse & coauthors - existence of a regular solution for

d = 5, . . . , 10 but no hint to solve 3d (N-S)

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 5 / 15

SLIDE 15

Navier-Stokes equations

Navier-Stokes equations - steady case

steady Navier-Stokes equations − div(v ⊗ v) − ν0△v = −∇p + f div v = 0. d = 2, 3, 4 - regularity, d = 5 - the same scaling as for (N-S) in d = 3 - maybe a hint to (N-S)

J. Frehse & coauthors - existence of a regular solution for

d = 5, . . . , 10 but no hint to solve 3d (N-S)

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 5 / 15

SLIDE 16

Navier-Stokes equations

Challenge

Challenge If data are smooth, is there a smooth solution to (N-S)?

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 6 / 15

SLIDE 17

Power-law like & implicit models

Power-law like models

T T T := −pI I I + S S S, where G G G(S S S,D D D(v)) = 0 prototype: S S S ∼ ν0(1 + |D D D(v)|2)

r−2 2 D

D D(v) vt − div(v ⊗ v) − ν0 div

(1 + |D

D D(v)|2)

r−2 2 D

D D(v)

= −∇p + f

div v = 0. (Nr) r ≥ 3d+2

d+2 - strong solution & uniqueness (for smooth data)

d = 2 - full regularity Stokes equations vt − ν0 div

(1 + |D

D D(v)|2)

r−2 2 D

D D(v)

= −∇p + f

div v = 0. (Sr) for all r ∈ (1, ∞) the same as for (Nr), i.e., NO higher regularity for d = 3, . . .

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 7 / 15

SLIDE 18

Power-law like & implicit models

Power-law like models

T T T := −pI I I + S S S, where G G G(S S S,D D D(v)) = 0 prototype: S S S ∼ ν0(1 + |D D D(v)|2)

r−2 2 D

D D(v) vt − div(v ⊗ v) − ν0 div

(1 + |D

D D(v)|2)

r−2 2 D

D D(v)

= −∇p + f

div v = 0. (Nr) r ≥ 3d+2

d+2 - strong solution & uniqueness (for smooth data)

d = 2 - full regularity Stokes equations vt − ν0 div

(1 + |D

D D(v)|2)

r−2 2 D

D D(v)

= −∇p + f

div v = 0. (Sr) for all r ∈ (1, ∞) the same as for (Nr), i.e., NO higher regularity for d = 3, . . .

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 7 / 15

SLIDE 19

Power-law like & implicit models

Power-law like models

T T T := −pI I I + S S S, where G G G(S S S,D D D(v)) = 0 prototype: S S S ∼ ν0(1 + |D D D(v)|2)

r−2 2 D

D D(v) vt − div(v ⊗ v) − ν0 div

(1 + |D

D D(v)|2)

r−2 2 D

D D(v)

= −∇p + f

div v = 0. (Nr) r ≥ 3d+2

d+2 - strong solution & uniqueness (for smooth data)

d = 2 - full regularity Stokes equations vt − ν0 div

(1 + |D

D D(v)|2)

r−2 2 D

D D(v)

= −∇p + f

div v = 0. (Sr) for all r ∈ (1, ∞) the same as for (Nr), i.e., NO higher regularity for d = 3, . . .

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 7 / 15

SLIDE 20

Power-law like & implicit models

Power-law like models

T T T := −pI I I + S S S, where G G G(S S S,D D D(v)) = 0 prototype: S S S ∼ ν0(1 + |D D D(v)|2)

r−2 2 D

D D(v) vt − div(v ⊗ v) − ν0 div

(1 + |D

D D(v)|2)

r−2 2 D

D D(v)

= −∇p + f

div v = 0. (Nr) r ≥ 3d+2

d+2 - strong solution & uniqueness (for smooth data)

d = 2 - full regularity Stokes equations vt − ν0 div

(1 + |D

D D(v)|2)

r−2 2 D

D D(v)

= −∇p + f

div v = 0. (Sr) for all r ∈ (1, ∞) the same as for (Nr), i.e., NO higher regularity for d = 3, . . .

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 7 / 15

SLIDE 21

Power-law like & implicit models

Power-law like implicit

T T T := −pI I I + S S S, where G G G(S S S,D D D(v)) = 0 prototype I S S S = D D D |D D D| + ν(|D D D|)D D D prototype II D D D = S S S |S S S| + ˜ ν(|S S S|)S S S prototype I: higher regularity of ∇v prototype II: higher regularity of S S S

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 8 / 15

SLIDE 22

Power-law like & implicit models

Power-law like implicit

T T T := −pI I I + S S S, where G G G(S S S,D D D(v)) = 0 prototype I S S S = D D D |D D D| + ν(|D D D|)D D D prototype II D D D = S S S |S S S| + ˜ ν(|S S S|)S S S prototype I: higher regularity of ∇v prototype II: higher regularity of S S S

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 8 / 15

SLIDE 23

Power-law like & implicit models

No pressure - “no” advantage

Consider “nonlinear linearized elasticity”, very simplified i.e. T T T = ν(|ε|)ε ε := 1 2(∇u + (∇u)T) the resulting equations are similar − div(ν(|ε|)ε) = f |∇u| ≤ ε0, one can justify such a “model” - due to the work of prof. Rajagopal no regularity for d ≥ 3 |∇u| ≤ ε1 = ⇒ regularity ε0 ≤ ε1 = ⇒ perfect model:) ε1 ≪ ε0 singularity maybe be there, the model is incorrect

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 9 / 15

SLIDE 24

Power-law like & implicit models

No pressure - “no” advantage

Consider “nonlinear linearized elasticity”, very simplified i.e. T T T = ν(|ε|)ε ε := 1 2(∇u + (∇u)T) the resulting equations are similar − div(ν(|ε|)ε) = f |∇u| ≤ ε0, one can justify such a “model” - due to the work of prof. Rajagopal no regularity for d ≥ 3 |∇u| ≤ ε1 = ⇒ regularity ε0 ≤ ε1 = ⇒ perfect model:) ε1 ≪ ε0 singularity maybe be there, the model is incorrect

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 9 / 15

SLIDE 25

Power-law like & implicit models

No pressure - “no” advantage

Consider “nonlinear linearized elasticity”, very simplified i.e. T T T = ν(|ε|)ε ε := 1 2(∇u + (∇u)T) the resulting equations are similar − div(ν(|ε|)ε) = f |∇u| ≤ ε0, one can justify such a “model” - due to the work of prof. Rajagopal no regularity for d ≥ 3 |∇u| ≤ ε1 = ⇒ regularity ε0 ≤ ε1 = ⇒ perfect model:) ε1 ≪ ε0 singularity maybe be there, the model is incorrect

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 9 / 15

SLIDE 26

Power-law like & implicit models

No pressure - “no” advantage

Consider “nonlinear linearized elasticity”, very simplified i.e. T T T = ν(|ε|)ε ε := 1 2(∇u + (∇u)T) the resulting equations are similar − div(ν(|ε|)ε) = f |∇u| ≤ ε0, one can justify such a “model” - due to the work of prof. Rajagopal no regularity for d ≥ 3 |∇u| ≤ ε1 = ⇒ regularity ε0 ≤ ε1 = ⇒ perfect model:) ε1 ≪ ε0 singularity maybe be there, the model is incorrect

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 9 / 15

SLIDE 27

Power-law like & implicit models

Challenge

Challenge Are the solutions to (Sr) smooth (or at least C1,α)? Nightmare If S S S = ν(|∇v|)∇v then the solution is regular. |∇v| is sub-solution to an elliptic problem Challenge Is |D D D(v)| or any other relevant quantity a sub- or super-solution to something? Is there something behind the structure?

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 10 / 15

SLIDE 28

Power-law like & implicit models

Challenge

Challenge Are the solutions to (Sr) smooth (or at least C1,α)? Nightmare If S S S = ν(|∇v|)∇v then the solution is regular. |∇v| is sub-solution to an elliptic problem Challenge Is |D D D(v)| or any other relevant quantity a sub- or super-solution to something? Is there something behind the structure?

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 10 / 15

SLIDE 29

Power-law like & implicit models

Challenge

Challenge Are the solutions to (Sr) smooth (or at least C1,α)? Nightmare If S S S = ν(|∇v|)∇v then the solution is regular. |∇v| is sub-solution to an elliptic problem Challenge Is |D D D(v)| or any other relevant quantity a sub- or super-solution to something? Is there something behind the structure?

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 10 / 15

SLIDE 30

Power-law like & implicit models

Challenge

Challenge Are the solutions to (Sr) smooth (or at least C1,α)? Nightmare If S S S = ν(|∇v|)∇v then the solution is regular. |∇v| is sub-solution to an elliptic problem Challenge Is |D D D(v)| or any other relevant quantity a sub- or super-solution to something? Is there something behind the structure?

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 10 / 15

SLIDE 31

Coupling with internal energy

Coupled systems - basic framework

T T T := −pI I I + S S S, where G G G(e,S S S,D D D(v)) = 0 standard sets of equations (equation for internal energy e) vt − div(v ⊗ v) − divS S S = −∇p + f div v = 0 et − div(ev) − div(κ(e)∇e) = S S S · D D D(v) (N-S-Fe) “better” sets of equations (equation for global energy E := 1

2|v|2 + e)

vt − div(v ⊗ v) − divS S S = −∇p + f div v = 0 Et − div(v(E + p)) − div(κ(e)∇e + S S Sv) = f · v (N-S-FE)

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 11 / 15

SLIDE 32

Coupling with internal energy

Coupled systems - basic framework

T T T := −pI I I + S S S, where G G G(e,S S S,D D D(v)) = 0 standard sets of equations (equation for internal energy e) vt − div(v ⊗ v) − divS S S = −∇p + f div v = 0 et − div(ev) − div(κ(e)∇e) = S S S · D D D(v) (N-S-Fe) “better” sets of equations (equation for global energy E := 1

2|v|2 + e)

vt − div(v ⊗ v) − divS S S = −∇p + f div v = 0 Et − div(v(E + p)) − div(κ(e)∇e + S S Sv) = f · v (N-S-FE)

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 11 / 15

SLIDE 33

Coupling with internal energy

Coupled systems - basic framework

T T T := −pI I I + S S S, where G G G(e,S S S,D D D(v)) = 0 standard sets of equations (equation for internal energy e) vt − div(v ⊗ v) − divS S S = −∇p + f div v = 0 et − div(ev) − div(κ(e)∇e) = S S S · D D D(v) (N-S-Fe) “better” sets of equations (equation for global energy E := 1

2|v|2 + e)

vt − div(v ⊗ v) − divS S S = −∇p + f div v = 0 Et − div(v(E + p)) − div(κ(e)∇e + S S Sv) = f · v (N-S-FE)

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 11 / 15

SLIDE 34

Coupling with internal energy

Coupled systems - Newtonian models

S S S = ν(e)D D D(v) Navier-Stokes-Fourier vt − div(v ⊗ v) − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(ev) − div(κ(e)∇e) = ν(e)|D D D(v)|2 (N-S-F) no regularity known in any d; convective term, presence of D D D(v), no-H¨

lder

continuity of e, quadratic term on the right hand side Stokes-Fourier vt − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(κ(e)∇e) = ν(e)|D D D(v)|2 (S-F) no regularity known in general; quadratic term on the right hand side Lemma (M.B & Kaplick´ y & M´ alek) For very special ν, for solution to (S-F) we know that ∇2v ∈ L2 in any d; for d = 2 existence of classical solution.

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 12 / 15

SLIDE 35

Coupling with internal energy

Coupled systems - Newtonian models

S S S = ν(e)D D D(v) Navier-Stokes-Fourier vt − div(v ⊗ v) − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(ev) − div(κ(e)∇e) = ν(e)|D D D(v)|2 (N-S-F) no regularity known in any d; convective term, presence of D D D(v), no-H¨

lder

continuity of e, quadratic term on the right hand side Stokes-Fourier vt − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(κ(e)∇e) = ν(e)|D D D(v)|2 (S-F) no regularity known in general; quadratic term on the right hand side Lemma (M.B & Kaplick´ y & M´ alek) For very special ν, for solution to (S-F) we know that ∇2v ∈ L2 in any d; for d = 2 existence of classical solution.

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 12 / 15

SLIDE 36

Coupling with internal energy

Coupled systems - Newtonian models

S S S = ν(e)D D D(v) Navier-Stokes-Fourier vt − div(v ⊗ v) − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(ev) − div(κ(e)∇e) = ν(e)|D D D(v)|2 (N-S-F) no regularity known in any d; convective term, presence of D D D(v), no-H¨

lder

continuity of e, quadratic term on the right hand side Stokes-Fourier vt − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(κ(e)∇e) = ν(e)|D D D(v)|2 (S-F) no regularity known in general; quadratic term on the right hand side Lemma (M.B & Kaplick´ y & M´ alek) For very special ν, for solution to (S-F) we know that ∇2v ∈ L2 in any d; for d = 2 existence of classical solution.

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 12 / 15

SLIDE 37

Coupling with internal energy

Coupled systems - Newtonian models

S S S = ν(e)D D D(v) Navier-Stokes-Fourier vt − div(v ⊗ v) − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(ev) − div(κ(e)∇e) = ν(e)|D D D(v)|2 (N-S-F) no regularity known in any d; convective term, presence of D D D(v), no-H¨

lder

continuity of e, quadratic term on the right hand side Stokes-Fourier vt − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(κ(e)∇e) = ν(e)|D D D(v)|2 (S-F) no regularity known in general; quadratic term on the right hand side Lemma (M.B & Kaplick´ y & M´ alek) For very special ν, for solution to (S-F) we know that ∇2v ∈ L2 in any d; for d = 2 existence of classical solution.

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 12 / 15

SLIDE 38

Coupling with internal energy

Coupled systems - Newtonian models

S S S = ν(e)D D D(v) Navier-Stokes-Fourier vt − div(v ⊗ v) − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(ev) − div(κ(e)∇e) = ν(e)|D D D(v)|2 (N-S-F) no regularity known in any d; convective term, presence of D D D(v), no-H¨

lder

continuity of e, quadratic term on the right hand side Stokes-Fourier vt − div(ν(e)D D D(v)) = −∇p + f div v = 0 et − div(κ(e)∇e) = ν(e)|D D D(v)|2 (S-F) no regularity known in general; quadratic term on the right hand side Lemma (M.B & Kaplick´ y & M´ alek) For very special ν, for solution to (S-F) we know that ∇2v ∈ L2 in any d; for d = 2 existence of classical solution.

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 12 / 15

SLIDE 39

Coupling with internal energy

Navier-Stokes-Fourier

Challenge Are solution to (S-F) regular for more general ν’s? What with (N-S-F)?

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 13 / 15

SLIDE 40

Coupling with internal energy

Eddy viscosity - turbulence model - hope for regularity?

ν(e) = κ(e) ∼ ν0eα with some α ≥ 0 fluid dynamics - nonsense; TKE-models (simplified Kolmogorov model) with α = 1

different scaling than in Navier-Stokes Conjecture There exists ε0 > such that any solution to (N-S-F) satisfying ✂ 1 ✂

B1(0)

ν(e)|D D D(v)|2 ≤ ε0 is regular in ( 1

2, 1) × B 1

2 (0).

Lemma (M.B., Lewandowski, M´ alek) Let the conjecture hold. Then for any α ≥ 1

2 the solution to (N-S-F) is regular. Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 14 / 15

SLIDE 41

Coupling with internal energy

Eddy viscosity - turbulence model - hope for regularity?

ν(e) = κ(e) ∼ ν0eα with some α ≥ 0 fluid dynamics - nonsense; TKE-models (simplified Kolmogorov model) with α = 1

different scaling than in Navier-Stokes Conjecture There exists ε0 > such that any solution to (N-S-F) satisfying ✂ 1 ✂

B1(0)

ν(e)|D D D(v)|2 ≤ ε0 is regular in ( 1

2, 1) × B 1

2 (0).

Lemma (M.B., Lewandowski, M´ alek) Let the conjecture hold. Then for any α ≥ 1

2 the solution to (N-S-F) is regular. Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 14 / 15

SLIDE 42

Coupling with internal energy

Eddy viscosity - turbulence model - hope for regularity?

ν(e) = κ(e) ∼ ν0eα with some α ≥ 0 fluid dynamics - nonsense; TKE-models (simplified Kolmogorov model) with α = 1

different scaling than in Navier-Stokes Conjecture There exists ε0 > such that any solution to (N-S-F) satisfying ✂ 1 ✂

B1(0)

ν(e)|D D D(v)|2 ≤ ε0 is regular in ( 1

2, 1) × B 1

2 (0).

Lemma (M.B., Lewandowski, M´ alek) Let the conjecture hold. Then for any α ≥ 1

2 the solution to (N-S-F) is regular. Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 14 / 15

SLIDE 43

Coupling with internal energy

Eddy viscosity - turbulence model - hope for regularity?

ν(e) = κ(e) ∼ ν0eα with some α ≥ 0 fluid dynamics - nonsense; TKE-models (simplified Kolmogorov model) with α = 1

different scaling than in Navier-Stokes Conjecture There exists ε0 > such that any solution to (N-S-F) satisfying ✂ 1 ✂

B1(0)

ν(e)|D D D(v)|2 ≤ ε0 is regular in ( 1

2, 1) × B 1

2 (0).

Lemma (M.B., Lewandowski, M´ alek) Let the conjecture hold. Then for any α ≥ 1

2 the solution to (N-S-F) is regular. Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 14 / 15

SLIDE 44

Coupling with internal energy

Eddy viscosity - turbulence model - hope for regularity?

ν(e) = κ(e) ∼ ν0eα with some α ≥ 0 fluid dynamics - nonsense; TKE-models (simplified Kolmogorov model) with α = 1

different scaling than in Navier-Stokes Conjecture There exists ε0 > such that any solution to (N-S-F) satisfying ✂ 1 ✂

B1(0)

ν(e)|D D D(v)|2 ≤ ε0 is regular in ( 1

2, 1) × B 1

2 (0).

Lemma (M.B., Lewandowski, M´ alek) Let the conjecture hold. Then for any α ≥ 1

2 the solution to (N-S-F) is regular. Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 14 / 15

SLIDE 45

Coupling with internal energy

Coupled system - nonlinearity helps

S S S ∼ (ν(e) + |D D D(v)|2)

r−2 2 D

D D(v) Lemma (M.B., M´ alek, Shilkin) Let d = 2 then the solution to (N-S-F) are regular. Let d ≥ 3 and r ≥ 3d+2

d+2 then there exists a strong solution to (N-S-F).

Bul´ ıˇ cek (Charles University in Prague) Regularity for systems of PDEs March 31, 2012 15 / 15