Jump conditions across phase boundaries for the Navier-Stokes-

Korteweg equations

Dietmar Kröner, Freiburg

Paris, Nov.2, 2009

Co-workers:

• D. Diehl• A. Dressel• K. Hermsdörfer• C. Kraus

Outline• Introduction, numerical experiments for the NSK system• Jump conditions across the interface for NSK, static

case• The pressure jump for the incompressible Navier-Stokes

equations• Low Mach number limit for the compressible Navier-

Stokes system• Low Mach number limit for the NSK system• NSK system dynamical case• Phase field like scaling

Navier Stokes K orteweg system:

@t½+r ¢(½v) = 0@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

ρψ(ρ)

Double well

p

β2

Pressure

β1 β2

β1 ρρ

= ~W(½)

p(½) :=½2Ã0(½)

ρψ(ρ)

Double well

p

β2

Pressure

β1 β2

β1 ρρ

= ~W(½)

W(½)

p(½) :=½2Ã0(½)

MinimizeZ

­W(½) +"2jr ½j2dx

Navier Stokes K orteweg system:

@t½+r ¢(½v) = 0@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

Initial­data

Long timebehaviour of ½;v?

MinimizeZ

­W(½) +"2jr ½j2dx

Navier Stokes K orteweg system:

@t½+r ¢(½v) = 0@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

Theoretical results

Danchin, Desjardin :Existence of solutions for compressible °uid models ofKorteweg type. Annales de l'IHP, Analysenon lineaire, 18,(2001), 97-133.

Global existence result for initial data close to stableequilibrium, d=2,3;

local in timeexistence for ½0 ¸ c> 0:

Bresch, Desjardin, Lin 1: Existenceof global weak solutions in energy spacesif ¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3:

J .E. Dunn, J . Serrin: On the thermodynamics of interstitial work. Arch.Rat. Mech. Anal. 88 (1985), 95{133.

H. Hattori, D. Li: Theexistenceof global solutions toa °uid dynamicmodelfor materials for Korteweg type. J . Partial Di®erential Equations 9 (4) (1996)323-342.

H. Hattori, D. Li: Global solutions of a high-dimensional system for Ko-rtewegmaterials. J . Math. Anal. Appl. 198, No. 1, (1996), 84-97.

R. Danchin, B. Desjardin, : Existence of solutions for compressible °uidmodels of Korteweg type. Annales de l'IHP, Analyse non lineaire, 18,(2001),97-133. (­ = <n , isotherm)

D. Bresch, B. Desjardin, C.K.Lin: On some compressible °uid model: Ko-rteweg, lubrication and shallow water systems, 2001.

D. Bresch, B. Desjardins, C.-K. Lin. On some compressible °uid models:Korteweg, lubrication and shallow water systems. Commun. Partial Di®er.Equations, 28(3):843-868, Mars 2003.

S. Benzoni-Gavage, R.Danchin, S. Descombes: Well-posednessof one-dimensionalKortewegmodels, preprint 2004

S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness fortheEuler-Kortewegmodel in several spacedimensions, preprint 2005

M. Kotschote: Strong well-posedness for a Korteweg-type model for thedynamicsof a compressiblenon-isothermal °uid. Preprint Leipzig2006. (Initialboundary value )

1D. Bresch, B. Desjardin, C.K .Lin: On some compressible °uid model:K orteweg, lubrica-tion and shallow water systems, 2001

Danchin, Desjardin :Existence of solutions for compressible °uid models ofKorteweg type. Annales de l'IHP, Analysenon lineaire, 18,(2001), 97-133.

Global existence result for initial data close to stableequilibrium, d=2,3;

local in timeexistence for ½0 ¸ c> 0:

Bresch, Desjardin, Lin 1: Existenceof global weak solutions in energy spacesif ¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3:

J .E. Dunn, J . Serrin: On the thermodynamics of interstitial work. Arch.Rat. Mech. Anal. 88 (1985), 95{133.

H. Hattori, D. Li: Theexistenceof global solutions toa °uid dynamicmodelfor materials for Korteweg type. J . Partial Di®erential Equations 9 (4) (1996)323-342.

H. Hattori, D. Li: Global solutions of a high-dimensional system for Ko-rtewegmaterials. J . Math. Anal. Appl. 198, No. 1, (1996), 84-97.

R. Danchin, B. Desjardin, : Existence of solutions for compressible °uidmodels of Korteweg type. Annales de l'IHP, Analyse non lineaire, 18,(2001),97-133. (­ = <n , isotherm)

D. Bresch, B. Desjardin, C.K.Lin: On some compressible °uid model: Ko-rteweg, lubrication and shallow water systems, 2001.

D. Bresch, B. Desjardins, C.-K. Lin. On some compressible °uid models:Korteweg, lubrication and shallow water systems. Commun. Partial Di®er.Equations, 28(3):843-868, Mars 2003.

S. Benzoni-Gavage, R.Danchin, S. Descombes: Well-posednessof one-dimensionalKortewegmodels, preprint 2004

S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness fortheEuler-Kortewegmodel in several spacedimensions, preprint 2005

M. Kotschote: Strong well-posedness for a Korteweg-type model for thedynamicsof a compressiblenon-isothermal °uid. Preprint Leipzig2006. (Initialboundary value )

1D. Bresch, B. Desjardin, C.K .Lin: On some compressible °uid model:K orteweg, lubrica-tion and shallow water systems, 2001

Danchin, Desjardin :Existence of solutions for compressible °uid models ofKorteweg type. Annales de l'IHP, Analysenon lineaire, 18,(2001), 97-133.

Global existence result for initial data close to stableequilibrium, d=2,3;

local in timeexistence for ½0 ¸ c> 0:

Bresch, Desjardin, Lin 1: Existenceof global weak solutions in energy spacesif ¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3:

J .E. Dunn, J . Serrin: On the thermodynamics of interstitial work. Arch.Rat. Mech. Anal. 88 (1985), 95{133.

H. Hattori, D. Li: Theexistenceof global solutions toa °uid dynamicmodelfor materials for Korteweg type. J . Partial Di®erential Equations 9 (4) (1996)323-342.

H. Hattori, D. Li: Global solutions of a high-dimensional system for Ko-rtewegmaterials. J . Math. Anal. Appl. 198, No. 1, (1996), 84-97.

R. Danchin, B. Desjardin, : Existence of solutions for compressible °uidmodels of Korteweg type. Annales de l'IHP, Analyse non lineaire, 18,(2001),97-133. (­ = <n , isotherm)

D. Bresch, B. Desjardin, C.K.Lin: On some compressible °uid model: Ko-rteweg, lubrication and shallow water systems, 2001.

D. Bresch, B. Desjardins, C.-K. Lin. On some compressible °uid models:Korteweg, lubrication and shallow water systems. Commun. Partial Di®er.Equations, 28(3):843-868, Mars 2003.

S. Benzoni-Gavage, R.Danchin, S. Descombes: Well-posednessof one-dimensionalKortewegmodels, preprint 2004

S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness fortheEuler-Kortewegmodel in several spacedimensions, preprint 2005

M. Kotschote: Strong well-posedness for a Korteweg-type model for thedynamicsof a compressiblenon-isothermal °uid. Preprint Leipzig2006. (Initialboundary value )

1D. Bresch, B. Desjardin, C.K .Lin: On some compressible °uid model:K orteweg, lubrica-tion and shallow water systems, 2001

Danchin, Desjardin :Existence of solutions for compressible °uid models ofKorteweg type. Annales de l'IHP, Analysenon lineaire, 18,(2001), 97-133.

Global existence result for initial data close to stableequilibrium, d=2,3;

local in timeexistence for ½0 ¸ c> 0:

Bresch, Desjardin, Lin 1: Existenceof global weak solutions in energy spacesif ¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3:

J .E. Dunn, J . Serrin: On the thermodynamics of interstitial work. Arch.Rat. Mech. Anal. 88 (1985), 95{133.

H. Hattori, D. Li: Theexistenceof global solutions toa °uid dynamicmodelfor materials for Korteweg type. J . Partial Di®erential Equations 9 (4) (1996)323-342.

H. Hattori, D. Li: Global solutions of a high-dimensional system for Ko-rtewegmaterials. J . Math. Anal. Appl. 198, No. 1, (1996), 84-97.

R. Danchin, B. Desjardin, : Existence of solutions for compressible °uidmodels of Korteweg type. Annales de l'IHP, Analyse non lineaire, 18,(2001),97-133. (­ = <n , isotherm)

D. Bresch, B. Desjardin, C.K.Lin: On some compressible °uid model: Ko-rteweg, lubrication and shallow water systems, 2001.

D. Bresch, B. Desjardins, C.-K. Lin. On some compressible °uid models:Korteweg, lubrication and shallow water systems. Commun. Partial Di®er.Equations, 28(3):843-868, Mars 2003.

S. Benzoni-Gavage, R.Danchin, S. Descombes: Well-posednessof one-dimensionalKortewegmodels, preprint 2004

S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness fortheEuler-Kortewegmodel in several spacedimensions, preprint 2005

M. Kotschote: Strong well-posedness for a Korteweg-type model for thedynamicsof a compressiblenon-isothermal °uid. Preprint Leipzig2006. (Initialboundary value )

1D. Bresch, B. Desjardin, C.K .Lin: On some compressible °uid model:K orteweg, lubrica-tion and shallow water systems, 2001Danchin,­Desjardin:­Existence­of­solutions­for­compressible­fluid­

models­of­Korteweg­type.­Annales­de­l'IHP,­Analyse­non­lineaire,­18,(2001),­97-133.­

Global­existence­result­for­initial­data­close­to­stable­equilibrium,­d=2,3;­local­in­time­existence­for­

Bresch,­Desjardin,­Lin:­Existence­of­global­weak­solutions­in­energy­spaces­if­­­­­­­­­­­­­­­­­­Preprint­2002­(?)

H.­Hattori,­D.­Li:­The­existence­of­global­solutions­to­a­fluid­dynamic­model­for­materials­for­Korteweg­type.­J.­Partial­Differential­Equations­9­(4)­(1996)­323-342.

H. Hattori, D. Li: Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198, No. 1, (1996), 84-97.

R. Danchin, B. Desjardin, : Existence of solutions for compressible fluid models of Korteweg type. Annales de l'IHP, Analyse nonlineaire, 18,(2001), 97-133.

½0 ¸ c> 0:

¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3

(­ = <n ; isotherm)

D.­Bresch,­B.­Desjardins,­C.-K.­Lin.­On­some­compressible­fluidmodels:­Korteweg,­lubrication­and­shallow­water­systems.­Commun­Partial­Differ.­Equations,­28(3):843-868,­2003.

S.­Benzoni-Gavage,­R.­Danchin,­S.­Descombes:­Well-posedness­of­one-dimensional­Korteweg­models,­preprint­2004

S.­Benzoni-Gavage,­R.­Danchin,­S.­Descombes:­On­the­well-posedness­for­the­Euler-Korteweg­model­in­several­space­dimensions,­preprint­2005

M.­Kotschote:­Strong­well-posedness­for­a­Korteweg-type­model­for­the­dynamics­of­a­compressible­non-isothermal­fluid.­Preprint­Leipzig­2006.­(Initial­boundary­value)

Numerical results PhD Thesis Dennis Diehl

Navier Stokes K orteweg system:

@t½+r ¢(½v) = 0@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

Navier Stokes K orteweg system:

@t½+r ¢(½v) = 0@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

Stabilization:

@t½+r ¢(½v) = ±¢½@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

Navier Stokes K orteweg system:

@t½+r ¢(½v) = 0@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

Stabilization:

@t½+r ¢(½v) = ±¢½@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

MinimizeZ

­W(½) +"2jr ½j2dx

Euler Lagrange equation static case: ¡ W0(½) +°"2¢½= const:

· = ¡ W0(½) +°"2¢½= const:

Stabilization:

@t½+r ¢(½v) = ±¢½@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

@t½+r ¢(½v) = ±¢ ·@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

Jump conditions across the interface (static case)

pl ¡ pv = c· "

Navier Stokes K orteweg system:

@t½+r ¢(½v) = 0@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

(Luckhaus,­Modica,­Dreyer,­Kraus)

Stationary case:

Navier Stokes K orteweg system(dynamical case):

Jump­conditions:­??????

liquidvapor

Navier Stokes K orteweg system (dynamical case):

@t½+r ¢(½v) = 0@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

A ssumptions For a sequence" ! 0weassume that(1) ½" and v" are smooth solutions of theNSK in ­ T := ­ £ ]0;T[.(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ­ ; and inL1(­ ); j½"(x;t)j ·

c for all (x;t) 2 ­ T and c does not depend on ".(3) ½0 has only two values: ¯1;¯2. De neE0(t) := fx 2 ­ j½0(x;t) = ¯1g:(4) v" ! v0 a.e. in ­ T and v0 is su±ciently smooth in E0 and ­ nE0.(5) lim" 1

"R­¡"2jr ½" j2+W(½")

¢dx exists uniformly in t.

Navier Stokes K orteweg system (dynamical case):

@t½+r ¢(½v) = 0@t(½v) +r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½

Navier Stokes K orteweg system:

@t½+r ¢(½v) = 0@t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

Multiply by a smooth testfunction ψ

Integration by partsZ T

0

Z

­(½"@tà +(½"v" )r Ã) dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p(½")I )¢r Ãdxdt:

=Z T

0

Z

­

¡¹ v"¢ à +°"2½"r ¢½"âdxdt:

Z T

0

Z

­(@t½+r ¢(½v))Ãdxdt = 0

Z T

0

Z

­

¡@t(½v) +r ¢(½vvt +p(½)I )¢Ãdxdt =Z T

0

Z

­

¡¹ ¢ v+°"2½r ¢½¢Ãdxdt:

Z T

0

Z

­(½"@tÃ+(½"v" )r Ã)dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p(½")I )¢r Ãdxdt:

=Z T

0

Z

­

¡¹ v"¢Ã +°"2½"r ¢½"âdxdt:

Z T

0

Z

­(½"@tÃ+(½"v" )r Ã)dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p(½")I )¢r Ãdxdt:

=Z T

0

Z

­

¡¹ v"¢Ã +°"2½"r ¢½"âdxdt:

RT0R­ (½0@tà +(½0v0)r Ã) dxdt = 0;

Z T

0

Z

­(½"@tÃ+(½"v" )r Ã)dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p(½")I )¢r Ãdxdt:

=Z T

0

Z

­

¡¹ v"¢Ã +°"2½"r ¢½"âdxdt:

¡ RT0R­ ½0v0@tà +(½0v0vt0+p(½0)I )) r Ãdxdt

RT0R­ (½0@tà +(½0v0)r Ã) dxdt = 0;

Z T

0

Z

­(½"@tÃ+(½"v" )r Ã)dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p(½")I )¢r Ãdxdt:

=Z T

0

Z

­

¡¹ v"¢Ã +°"2½"r ¢½"âdxdt:

¡ RT0R­ ½0v0@tà +(½0v0vt0+p(½0)I )) r Ãdxdt

RT0R­ (½0@tà +(½0v0)r Ã) dxdt = 0;

RT0R­ ¹ v0¢ Ãdxdt

Z T

0

Z

­(½"@tÃ+(½"v" )r Ã)dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p(½")I )¢r Ãdxdt:

=Z T

0

Z

­

¡¹ v"¢Ã +°"2½"r ¢½"âdxdt:

¡ RT0R­ ½0v0@tà +(½0v0vt0+p(½0)I )) r Ãdxdt

RT0R­ (½0@tà +(½0v0)r Ã) dxdt = 0;

?

RT0R­ ¹ v0¢ Ãdxdt

Z T

0

Z

­(½"@tÃ+(½"v" )r Ã)dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p(½")I )¢r Ãdxdt:

=Z T

0

Z

­

¡¹ v"¢Ã +°"2½"r ¢½"âdxdt:

¡ RT0R­ ½0v0@tà +(½0v0vt0+p(½0)I )) r Ãdxdt

RT0R­ (½0@tà +(½0v0)r Ã) dxdt = 0;

=:R

RT0R­ ¹ v0¢ Ãdxdt

R : = "2Z T

0

Z

­(½"Ã)r ¢½"dxdt = ¡ "2

Z T

0

Z

­r ¢(½"Ã)¢½"dxdt

= ¡ "2Z T

0

Z

­(@j½"Ãj +½"@j Ãj )¢½"dxdt

= ¡ "2X

k

Z T

0

Z

­(@j½"Ãj +½"@j Ãj )@2k½"dxdt

= "2X

k

Z T

0

Z

­(@k@j½"Ãj +@j½"@kÃj +@k½"@j Ãj +½"@j@kÃj )@k½"dxdt

= "2X

k

Z T

0

Z

­(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj +½"@k½"@j@kÃj )dxdt

R : = "2X

k

Z T

0

Z

­(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj +½"@k½"@j@kÃj )dxdt

R : = "2X

k

Z T

0

Z

­(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj +½"@k½"@j@kÃj )dxdt

= "2X

k

Z T

0

Z

­(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj + 1

2@k½2"@j@kÃj )dxdt

= "2X

k

Z T

0

Z

­(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj ¡ 1

2½2"@j@2kÃj )dxdt

= "2X

k

Z T

0

Z

­(¡ 12(@k½")

2@j Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj ¡ 12½

2"@j@2kÃj )dxdt

= "2X

k

Z T

0

Z

­(@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2(@k½")2@j Ãj ¡ 1

2½2"@j@2kÃj )dxdt

= "2Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2+ 1

"2W(½")¶@j Ãj

¡ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt

= "2Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2+ 1

"2W(½")¶@j Ãj

¡ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt

= "2Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2+ 1

"2W(½")¶@j Ãj

¡ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt

Remember the assumptions:

A ssumptions For a sequence " ! 0weassume that(1) ½" and v" aresmooth solutions of theNSK in ­ T := ­ £ ]0;T[.

(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ­ ; and inL1(­ ); j½"(x;t)j ·c for all (x;t) 2 ­ T and c does not depend on ".

(3) De neE0(t) := fx 2 ­ j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ­ ¡ E0(t).

(4) v" ! v0 a.e. in ­ T and v0 is su±ciently smooth in E0 and ­ nE0.

(5) lim" 1"R­¡"2jr ½" j2+W(½")

¢dx exists uniformly in t.

= "2Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2+ 1

"2W(½")¶@j Ãj

¡ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt

Remember the assumptions:

A ssumptions For a sequence " ! 0weassume that(1) ½" and v" aresmooth solutions of theNSK in ­ T := ­ £ ]0;T[.

(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ­ ; and inL1(­ ); j½"(x;t)j ·c for all (x;t) 2 ­ T and c does not depend on ".

(3) De neE0(t) := fx 2 ­ j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ­ ¡ E0(t).

(4) v" ! v0 a.e. in ­ T and v0 is su±ciently smooth in E0 and ­ nE0.

(5) lim" 1"R­¡"2jr ½" j2+W(½")

¢dx exists uniformly in t.

= "2Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2+ 1

"2W(½")¶@j Ãj

¡ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt

Remember the assumptions:

A ssumptions For a sequence " ! 0weassume that(1) ½" and v" aresmooth solutions of theNSK in ­ T := ­ £ ]0;T[.

(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ­ ; and inL1(­ ); j½"(x;t)j ·c for all (x;t) 2 ­ T and c does not depend on ".

(3) De neE0(t) := fx 2 ­ j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ­ ¡ E0(t).

(4) v" ! v0 a.e. in ­ T and v0 is su±ciently smooth in E0 and ­ nE0.

(5) lim" 1"R­¡"2jr ½" j2+W(½")

¢dx exists uniformly in t.

= "2Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2+ 1

"2W(½")¶@j Ãj

¡ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt

Remember the assumptions:

A ssumptions For a sequence " ! 0weassume that(1) ½" and v" aresmooth solutions of theNSK in ­ T := ­ £ ]0;T[.

(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ­ ; and inL1(­ ); j½"(x;t)j ·c for all (x;t) 2 ­ T and c does not depend on ".

(3) De neE0(t) := fx 2 ­ j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ­ ¡ E0(t).

(4) v" ! v0 a.e. in ­ T and v0 is su±ciently smooth in E0 and ­ nE0.

(5) lim" 1"R­¡"2jr ½" j2+W(½")

¢dx exists uniformly in t.

= "2Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2+ 1

"2W(½")¶@j Ãj

¡ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt

Remember the assumptions:

A ssumptions For a sequence " ! 0weassume that(1) ½" and v" aresmooth solutions of theNSK in ­ T := ­ £ ]0;T[.

(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ­ ; and inL1(­ ); j½"(x;t)j ·c for all (x;t) 2 ­ T and c does not depend on ".

(3) De neE0(t) := fx 2 ­ j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ­ ¡ E0(t).

(4) v" ! v0 a.e. in ­ T and v0 is su±ciently smooth in E0 and ­ nE0.

(5) lim" 1"R­¡"2jr ½" j2+W(½")

¢dx exists uniformly in t.

Lemma (Luckhaus, M odica):

WehaveR ! 0 if " ! 0 and

"2Z T

0

Z

­@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt = "

Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+o("):

(1)

R = "2Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2+ 1

"2W(½")¶@j Ãj

¡ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt

Lemma (Luckhaus, M odica):

WehaveR ! 0 if " ! 0 and

"2Z T

0

Z

­@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt = "

Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+o("):

(1)

R = "2Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2+ 1

"2W(½")¶@j Ãj

¡ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt

Lemma (Luckhaus, M odica):

For " ! 0wehave

"2Z T

0

Z

­@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt

= "Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")

and thereforeR ! 0 for " ! 0:

Lemma (Luckhaus, M odica):

For " ! 0 wehave

"2Z T

0

Z

­@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt

= "Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")

and thereforeR ! 0 for " ! 0:

Z T

0

Z

­(½"@tÃ+(½"v" )r Ã)dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p(½")I )¢r Ãdxdt:

=Z T

0

Z

­

¡¹ v"¢Ã +°"2½"r ¢½"âdxdt:

" ! 0Z T

0

Z

­(½0@tÃ+(½0v0)r Ã)dxdt = 0:

¡Z T

0

Z

­½0v0@tÃ+

¡½v0vt0+p(½0)I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v0¢Ã)dxdt:

Z T

0

Z

­(½0@tÃ+(½0v0)r Ã) dxdt = 0:

¡Z T

0

Z

­½0v0@tÃ+

¡½v0vt0+p(½0)I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v0¢Ã) dxdt:

liquidvapor

Z T

0

Z

­(½0@tÃ+(½0v0)r Ã) dxdt = 0:

¡Z T

0

Z

­½0v0@tÃ+

¡½v0vt0+p(½0)I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v0¢Ã) dxdt:

liquidvapor

@t½0+r ¢(½0v0) = 0 in E0(t) and in ­ nE0(t)

localize

½0@tv0+r ¢¡½0v0vt0+p(½0)I )¢= ¹ ¢ v0 in E0(t) and in ­ nE0(t):

Z T

0

Z

­(½0@tÃ+(½0v0)r Ã) dxdt = 0:

¡Z T

0

Z

­½0v0@tÃ+

¡½v0vt0+p(½0)I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v0¢Ã) dxdt:

liquidvapor

@t½0+r ¢(½0v0) = 0 in E0(t) and in ­ nE0(t)

½0@tv0+r ¢¡½0v0vt0+p(½0)I )¢= ¹ ¢ v0 in E0(t) and in ­ nE0(t):

(1)

localize

Jump conditions ????

liquidvapor

Z T

0

Z

­(½0@tÃ+(½0v0)r Ã)dxdt = 0:

¡Z T

0

Z

­½0v0@tÃ+

¡½v0vt0+p(½0)I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v0¢Ã)dxdt:

[½0]nt +[½0v0]nx = 0 on ¡ :

liquidvapor

Z T

0

Z

­(½0@tÃ+(½0v0)r Ã)dxdt = 0:

¡Z T

0

Z

­½0v0@tÃ+

¡½v0vt0+p(½0)I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v0¢Ã)dxdt:

[½0]nt +[½0v0]nx = 0 on ¡ :

liquidvapor

Z T

0

Z

­(½0@tÃ+(½0v0)r Ã)dxdt = 0:

¡Z T

0

Z

­½0v0@tÃ+

¡½v0vt0+p(½0)I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v0¢Ã)dxdt:

Z

¡

¡[½0v0]ntÃ+[½0v0vt0+p(½0)]nxâd¾

=Z

¡(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾:

[½0]nt +[½0v0]nx = 0 on ¡ :Z

¡

¡[½0v0]ntÃ+[½0v0vt0+p(½0)]nxâd¾

=Z

¡(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã)d¾:

[½0]nt +[½0v0]nx = 0 on ¡ :Z

¡

¡[½0v0]ntÃ+[½0v0vt0+p(½0)]nxâd¾

=Z

¡(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã)d¾:

[½0v0]nt +[½0v0vt0+p(½0)]nx = ¡ ¹ nx[r v0] on ¡

Use any testfunction Á on ¡ and extend it to a function à on ­ such thatnxr à = 0 in a small layer around ¡ :

[v0]= 0 on ¡ :

[½0]nt +[½0v0]nx = 0 on ¡ :Z

¡

¡[½0v0]ntÃ+[½0v0vt0+p(½0)]nxâd¾

=Z

¡(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã)d¾:

[½0v0]nt +[½0v0vt0+p(½0)]nx = ¡ ¹ nx[r v0] on ¡

Use any testfunction Á on ¡ and extend it to a function à on ­ such thatnxr à = 0 in a small layer around ¡ :

Summary

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

Summary

Jump conditions: liquid

vapor

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

[v0]= 0 on ¡ :

[½0]nt +[½0v0]nx =0 on ¡ :

[½0v0]nt +[½0v0vt0+p(½0)]nx = ¡ ¹ nx[r v0] on ¡

Summary

Jump conditions: liquid

vapor

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

[v0]= 0 on ¡ :

[½0]nt +[½0v0]nx =0 on ¡ :

[½0v0]nt +[½0v0vt0+p(½0)]nx = ¡ ¹ nx[r v0] on ¡

Summary

Jump conditions: liquid

vapor

No curvature term !

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

Different scaling to get the curvature term

Phase field like scaling

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

In thestatic caseNSK reduces to

r ¢(p(½)I ) = °"2½r ¢½:

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

In thestatic caseNSK reduces to

r ¢(p(½)I ) = °"2½r ¢½:

This is equivalent to~W0(½) = °"2¢½+¸" :

Herewehaveused ~W(½) =½Ã(½) and p(½) =½2Ã0(½) .

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

In thestatic caseNSK reduces to

r ¢(p(½)I ) = °"2½r ¢½:

This is equivalent to~W0(½) = °"2¢½+¸" :

Here we have used ~W(½) = ½Ã(½) and p(½) = ½2Ã0(½) . This is just the EulerLagrangeequationwith theLagrangemultiplier ¸ " of thefollowingminimizationproblem: Minimize

~J "(½) :=Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

under the constraint R­ ½dx =M0 (conservation of mass):

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

In thestatic caseNSK reduces to

r ¢(p(½)I ) = °"2½r ¢½:

This is equivalent to~W0(½) = °"2¢½+¸" :

Here we have used ~W(½) = ½Ã(½) and p(½) = ½2Ã0(½) . This is just the EulerLagrangeequationwith theLagrangemultiplier ¸ " of thefollowingminimizationproblem: Minimize

~J "(½) :=Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

under the constraint R­ ½dx =M0 (conservation of mass):

d‘Alambert variation principle

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

Minimize

~J "(½) :=Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

under the constraint R­ ½dx =M0 (conservation of mass):

New scaling: Instead of

Minimize

~J "(½) :=Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

under the constraint R­ ½dx =M0 (conservation of mass):

New scaling: Instead of

consider

Minimize

J "(½) := 1"Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

under the constraint R­ ½dx =M0 (conservation of mass):

~J "(½) :=Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

d‘Alambert variation principle

~J "(½) :=Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

J "(½) := 1"Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

d‘Alambert variation principle

d‘Alambert variation principle

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

NSK : @t(½v) +r ¢(½vvt +p"(½)I ) = ¹ ¢ v+°"½r ¢½

~J "(½) :=Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

J "(½) := 1"Z

­

µ~W(½) +°"2 jr ½j

2

2¶dx (total energy)

d‘Alambert variation principle

d‘Alambert variation principle

p"(½) := 1"p(½) + 1¡ "

" d1

NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½

NSK : @t(½v) +r ¢(½vvt +p"(½)I ) = ¹ ¢ v+°"½r ¢½

@t½+r ¢(½v) = 0@t(½v) + r ¢(½vvt +p"(½)I ) = ¹ ¢ v+°"½r ¢½:

Variational formulation:Z T

0

Z

­(@t½+r ¢(½v))Ãdxdt = 0

Z T

0

Z

­

¡@t(½v) + r ¢(½vvt +p"(½)I )¢Ãdxdt =

Z T

0

Z

­(¹ ¢ v+°"½r ¢½)Ãdxdt:

Now consider for " ! 0 :

Z T

0

Z

­(½"@tà +(½"v")r Ã) dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p"(½")I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v"¢Ã +°"½"r ¢½"Ã)dxdt:

Integration by parts:

Z T

0

Z

­(½"@tà +(½"v")r Ã) dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p"(½")I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v"¢Ã +°"½"r ¢½"Ã)dxdt:

RT0R­ (½0@tà +(½0v0)r Ã) dxdt = 0;

Z T

0

Z

­(½"@tà +(½"v")r Ã) dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p"(½")I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v"¢Ã +°"½"r ¢½"Ã)dxdt:

RT0R­ (½0@tà +(½0v0)r Ã) dxdt = 0;

Z T

0

Z

­(½"@tà +(½"v")r Ã) dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p"(½")I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v"¢Ã +°"½"r ¢½"Ã)dxdt:

RT0R­ (½0@tà +(½0v0)r Ã) dxdt = 0;

?

?

Z T

0

Z

­(½"@tà +(½"v")r Ã) dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p"(½")I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v"¢Ã +°"½"r ¢½"Ã)dxdt:

1"R : =

Z T

0

Z

­°"½"r ¢½"Ã dxdt

= "Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2 ¡ 1

"2W(½")¶@j Ãj

+ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt:

As before:

1"R : =

Z T

0

Z

­°"½"r ¢½"Ã dxdt

= "Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2 ¡ 1

"2W(½")¶@j Ãj

+ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt:

Lemma (Luckhaus, M odica):

For " ! 0 wehave

"2Z T

0

Z

­@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt

= "Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")

and thereforeR ! 0:

1"R : =

Z T

0

Z

­°"½"r ¢½"Ã dxdt

= "Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2 ¡ 1

"2W(½")¶@j Ãj

+ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt:

Lemma (Luckhaus, M odica):

For " ! 0 wehave

"2Z T

0

Z

­@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt

= "Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")

and thereforeR ! 0:

1"R : =

Z T

0

Z

­°"½"r ¢½"Ã dxdt

= "Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2 ¡ 1

"2W(½")¶@j Ãj

+ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt:

o(1)

Lemma Luckhaus Modica

Lemma (Luckhaus, M odica):

For " ! 0 wehave

"2Z T

0

Z

­@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt

= "Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")

and thereforeR ! 0:

1"R : =

Z T

0

Z

­°"½"r ¢½"Ã dxdt

= "Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2 ¡ 1

"2W(½")¶@j Ãj

+ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt:

o(1)

Lemma Luckhaus Modica

o(1)

Lemma (Luckhaus, M odica):

For " ! 0 wehave

"2Z T

0

Z

­@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt

= "Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")

(and thereforeR ! 0:)

1"R : =

Z T

0

Z

­°"½"r ¢½"Ã dxdt

= "Z T

0

Z

­@k½"@j½"@kÃj ¡ (@k½")2@j Ãj + 3

2µ(@k½")2 ¡ 1

"2W(½")¶@j Ãj

+ 32"2W(½")@j Ãj ¡ 1

2½2"@j@2kÃj dxdt:

o(1)

Lemma Luckhaus Modica

O(ε)

?

Consider:Z T

0

Z

­

32"W(½")@j Ãj dxdt

We need several steps:

² Á(s) = Rs0

qW (r )2 dr, w" = Á±½" and w0 =Á±½0:

² lim"! 0R­

1"W(½"(x;t))Ãdx = lim"! 0

R­

"2jr ½"(x;t)j2Ãdx

= lim"! 0R­ jr w" jÃdx (HT)

² lim"! 0R­ Ãr w"dx = ¡ lim"! 0

R­ r Ãw"dx = ¡ R­ w0r Ãdx

= c0R¡ Ã(x)º0dHn¡ 1

² lim"! 0R­ Ã(x)jr w"(x)jdx = lim"! 0

R­ F (x;r w"(x))dx =

R­ F (x;w0(x))djDw0j

= R­ Ã(x)jw0(x)jdjDw0j = c0

R¡ t Ã(x)jnjdHn¡ 1 = c0

R¡ t Ã(x)dHn¡ 1: (LM)

² lim"! 0R­

32"W(½")@j Ãj dxdt = 3

2c0R¡ t @j Ãj (x)dHn¡ 1:

lim"! 0

Z T

0

Z

­°"½"r ¢½"Ã dxdt =

Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+32c0

Z T

0

Z

¡ t@j Ãj (x)dHn¡ 1:

SummaryZ T

0

Z

­(½"@tà +(½"v")r Ã) dxdt = 0:

¡Z T

0

Z

­½"v"@tÃ+

¡½v"vt" +p"(½")I )¢r Ãdxdt:

=Z T

0

Z

­(¹ v"¢Ã +°"½"r ¢½"Ã)dxdt:

By an additional assumption wehaveZ T

0

Z

­p"(½")r Ãdxdt !

Z T

0

Z

­p(x;t)r Ãdxdt

for somep which is smooth in each phase.

Thereforewecan go to the limit " ! 0:

Z T

0

Z

­(½0@tà +(½0v0)r Ã) dxdt = 0

¡Z T

0

Z

­½0v0@tà +

¡½v0vt0+p(x;t)I )¢r Ãdxdt =Z T

0

Z

­¹ v0¢ Ãdxdt+

=Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+ 3

2c0Z T

0

Z

¡ t@j Ãj (x)dHn¡ 1:

liquidvapor

@t½0+r ¢(½0v0) = 0 in E0(t) and in ­ nE0(t)

localize

½0@tv0+r ¢¡½0v0vt0+p(½0)I )¢= ¹ ¢ v0 in E0(t) and in ­ nE0(t):

[½0]nt +[½0v0]nx = 0 on ¡ :

Z T

0

Z

­(½0@tà +(½0v0)r Ã) dxdt = 0

¡Z T

0

Z

­½0v0@tà +

¡½v0vt0+p(x;t)I )¢r Ãdxdt =Z T

0

Z

­¹ v0¢ Ãdxdt+

=Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt+ 3

2c0Z T

0

Z

¡ t@j Ãj (x)dHn¡ 1:

liquidvaporlocalize

Z T

0

Z

¡ t

¡[½0v0]ntà +[½0v0vt0+p(x;t)]nxâd¾

=Z T

0

Z

¡ t(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾+

Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt

+ 32c0

Z T

0

Z

¡ t@j Ãj (x)dHn¡ 1:

Z T

0

Z

¡ t

¡[½0v0]ntà +[½0v0vt0+p(x;t)]nxâd¾

=Z T

0

Z

¡ t(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾+

Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt

+ 32c0

Z T

0

Z

¡ t@j Ãj (x)dHn¡ 1:

Sincer à = (nxr Ã)nx +(¿xr Ã)¿x

wealso have@iÃk = nxj@j Ãknxi +¿xj@j Ãk¿xi

and therefore

r ¢Ã = @iÃi =nxj@j Ãinxi +¿xj@j Ãi¿xi = nxr Ãinxi +(¿r Ãi )¿xi= nxr Ãinxi +r ¡ t ¢Ã

wherer ¡ t ¢Ã := (¿r Ãi )¿xi denotes thetangential divergenceto thesurface ¡ t.

32c0

Z T

0

Z

¡ t@j Ãj (x)dHn¡ 1:

r ¢Ã = @iÃi =nxj@j Ãinxi +¿xj@j Ãi¿xi = nxr Ãinxi +(¿r Ãi )¿xi= nxr Ãinxi +r ¡ t ¢Ã

wherer ¡ t ¢Ã := (¿r Ãi )¿xi denotes thetangential divergenceto thesurface ¡ t.Due to thespecial choiceof the testfunction à wehavenxr Ãi = 0and

@iÃi = r ¡ t ¢Ã:

Using theGauss theoremon surfaces weobtainZ

¡ t@j Ãj (x)dHn¡ 1 =

Z

¡ tr ¡ t ¢ÃdHn¡ 1 =

Z

¡ tr ¡ t ¢nxÃnxdHn¡ 1

=Z

¡ t· tÃnxdHn¡ 1:

Z T

0

Z

¡ t

¡[½0v0]ntà +[½0v0vt0+p(x;t)]nxâd¾

=Z T

0

Z

¡ t(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾+

Z T

0coZ

¡ t· tÃ(¢;t)nxdHn¡ 1dt

+ 32c0

Z T

0

Z

¡ t@j Ãj (x)dHn¡ 1:

Z

¡ t@j Ãj (x)dHn¡ 1 =

Z

¡ tr ¡ t ¢ÃdHn¡ 1 =

Z

¡ tr ¡ t ¢nxÃnxdHn¡ 1

=Z

¡ t· tÃnxdHn¡ 1:

Z T

0

Z

¡ t

¡[½0v0]ntÃ+[½0v0vt0+p(x;t)]nxâd¾

= ¡Z T

0

Z

¡ t¹ nx[r v0]Ãd¾+ 5

2c0Z T

0

Z

¡ t· tÃnxdHn¡ 1

Z T

0

Z

¡ t

¡[½0v0]ntÃ+[½0v0vt0+p(x;t)]nxâd¾

= ¡Z T

0

Z

¡ t¹ nx[r v0]Ãd¾+ 5

2c0Z T

0

Z

¡ t· tÃnxdHn¡ 1

[½0v0]nt +[½0v0vt0+p(x;t)]nx = ¡ ¹ nx[r v0]+ 52co· tnx

Conclusion• Introduction, numerical experiments for the NSK

system• Jump conditions across the interface for NSK,

static case: no curvature term• The pressure jump for the incompressible

Navier-Stokes equations: curvature term• Low Mach number limit for the NSK system:

curvature term• NSK system dynamical case: no curvature term• Phase field like scaling: curvature term