Global Regularity of 2-D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: High Regularity Case
|
摘要
This paper is a continuation work of [26] and studies the propagation of the high-order boundary regularities of the two-dimensional density patch for viscous inhomogeneous incompressible flow.We assume the initial density ρ_0=η_11?_0+η_21?_0~c,where(η_1,η_2)is any pair of positive constants and ?_0 is a bounded,simply connected domain with W~(k+2,p)(R~2)boundary regularity.We prove that for any positive time t,the density function ρ(t)=η_11(_?(t))+η_21_(?(t)c),and the domain ?(t) preserves the W~(k+2,p)-boundary regularity.
This paper is a continuation work of [26] and studies the propagation of the high-order boundary regularities of the two-dimensional density patch for viscous inhomogeneous incompressible flow.We assume the initial density ρ_0=η_11?_0+η_21?_0~c,where(η_1,η_2)is any pair of positive constants and ?_0 is a bounded,simply connected domain with W~(k+2,p)(R~2)boundary regularity.We prove that for any positive time t,the density function ρ(t)=η_11(_?(t))+η_21_(?(t)c),and the domain ?(t) preserves the W~(k+2,p)-boundary regularity.
This paper is a continuation work of [26] and studies the propagation of the high-order boundary regularities of the two-dimensional density patch for viscous inhomogeneous incompressible flow.We assume the initial density ρ_0=η_11?_0+η_21?_0~c,where(η_1,η_2)is any pair of positive constants and ?_0 is a bounded,simply connected domain with W~(k+2,p)(R~2)boundary regularity.We prove that for any positive time t,the density function ρ(t)=η_11(_?(t))+η_21_(?(t)c),and the domain ?(t) preserves the W~(k+2,p)-boundary regularity.
引文
[1]H.Abidi,Equations de Navier-Stokes avec densité et viscosité variables dans l’espace critique,Rev.Mat.Iberoam.,23(2007),537-586.
[2]H.Abidi and M.Paicu,Existence globale pour un fluide inhomogene,Ann.Inst.Fourier,57(2007),883-917.
[3]H.Abidi,G.Gui and P.Zhang,On the wellposedness of 3-D inhomogeneous Navier-Stokes equations in the critical spaces,Arch.Ration.Mech.Anal.,204(2012),189-230.
[4]H.Abidi,G.Gui and P.Zhang,Wellposedness of 3-D inhomogeneous Navier-Stokes equations with highly oscillating initial velocity field,J.Math.Pures Appl.,100(2013),166-203.
[5]H.Bahouri,J.-Y.Chemin and R.Danchin,Fourier Analysis and Nonlinear Partial Differential Equations,Grundlehren der mathematischen Wissenschaften,Springer,2010.
[6]A.-L.Bertozzi and P.Constantin,Global regularity for vortex patches,Commun.Math.Phys.,152(1993),19-28.
[7]J.-Y.Chemin,Calcul paradifférentiel précisé et applicationsa deséquations aux dérivées partielles non semilinéaires,Duke Math.J.,56(1988),431-469.
[8]J.-Y.Chemin,Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel,Invent.Math.,103(1991),599-629.
[9]J.-Y.Chemin,Persistance de structures géométriques dans les fluides incompressibles bidimensionnels,Ann.Sci.Ecole Norm.Sup.,26(1993),517-542.
[10]J.-Y.Chemin,Perfect incompressible fluids,Oxford Lecture Series in Mathematics and its Applications,14.The Clarendon Press,Oxford University Press,New York,1998.
[11]R.Danchin,Poches de tourbillon visqueuses,J.Math.Pures Appl.,76(1997),609-647.
[12]R.Danchin,Density-dependent incompressible viscous fluids in critical spaces,Proc.Roy.Soc.Edinburgh Sect.A,133(2003),1311-1334.
[13]R.Danchin and P.-B.Mucha,A Lagrangian approach for the incompressible Navier-Stokes equations with variable density,Commun.Pure Appl.Math.,65(2012),1458-1480.
[14]R.Danchin and P.-B.Mucha,Incompressible flows with piecewise constant density,Arch.Ration.Mech.Anal.,207(2013),991-1023.
[15]R.Danchin and P.-B.Mucha,The incompressible Navier-Stokes equations in vacuum,arXiv:1705.06061.
[16]R.Danchin and P.Zhang,Inhomogeneous Navier-Stokes equations in the half-space,with only bounded density,J.Funct.Anal.,267(2014),2371-2436.
[17]R.Danchin and X.Zhang,On the persistence of H?lder regular patches of density for the inhomogeneous Navier-Stokes equations,J.Ec.Polytech.Math.,4(2017),781-811.
[18]P.Gamblin and X.Saint-Raymond,On three-dimensional vortex patches,Bull.Soc.Math.France,123(1995),375-424.
[19]F.Gancedo and E.Garcia-Juarez,Global regularity of 2D density patches for inhomogeneous Navier-Stokes,Arch.Ration.Mech.Anal.,229(2018),339-360.
[20]T.Hmidi,Régularité H?ldérienne des poches de tourbillon visqueuses,J.Math.Pures Appl.,84(2005),1455-1495.
[21]T.Hmidi,Poches de tourbillon singulieres dans un fluide faiblement visqueux,Rev.Mat.Iberoam.,22(2006),489-543.
[22]J.Huang,M.Paicu and P.Zhang,Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity,Arch.Ration.Mech.Anal.,209(2013),631-682.
[23]D.Hoff,Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data,J.Differential Equations,120(1995),215-254.
[24]O.-A.Ladyzhenskaja and V.-A.Solonnikov,The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids.(Russian)Boundary value problems of mathematical physics,and related questions of the theory of functions,8,Zap.Naucn.Sem.Leningrad.Otdel.Mat.Inst.Steklov.(LOMI),52(1975),218-219.
[25]X.Liao and P.Zhang,On the global regularity of two-dimensional density patch for inhomogeneous incompressible viscous flow,Arch.Ration.Mech.Anal.,220(2016),937-981.
[26]X.Liao and P.Zhang,Global regularity of 2-D density patches for viscous inhomogeneous incompressible flow with general density:low regularity case,Accepted by Commun.Pure Appl.Math.,72(2019),835-884.https://onlinelibrary.wiley.com/toc/10970312/2019/72/4.
[27]X.Liao and P.Zhang,Global regularities of two-dimensional density patch for inhomogeneous incompressible viscous flow with general density,arXiv:1604.07922.
[28]P.-L.Lions,Mathematical topics in fluid mechanics.Vol.1.Incompressible models,Oxford Lecture Series in Mathematics and Its Applications,3.Oxford Science Publications,The Clarendon Press,Oxford University Press,New York,1996.
[29]M.Paicu and P.Zhang,Global solutions to the 3-D incompressible inhomogeneous NavierStokes system,J.Funct.Anal.,262(2012),3556-3584.
[30]M.Paicu,P.Zhang and Z.Zhang,Global unique solvability of inhomogeneous NavierStokes equations with bounded density,Commun.Partial Differential Equations,38(2013),1208-1234.
[31]P.Zhang and Q.Qiu,Propagation of higher-order regularities of the boundaries of 3-D vortex patches,Chinese Ann.Math.Ser.A,18(1997),381-390.
[32]J.Simon,Nonhomogeneous viscous incompressible fluids:existence of velocity,density,and pressure,SIAM J.Math.Anal.,21(1990),1093-1117.
?We calculate directly ■and then recursively for any l≥1,■
?We consider Dtvl instead of ?tvl since we would like to benefit from the density equation Dtρ=0 while ?tρ=-v·▽ρ has singularities since the densityρis discontinuous in space variable. The idea was used by D. Hoff in [23] for the study of the compressible Navier-Stokes system.
§We derive from H1-energy estimate for Dtvl and Sobolev embedding that Dtvl∈Lloc1(Lp)which controls ▽2vl∈Lloc1(Lp) up to lower order terms, see Eq.(1.29)below.
?The formulae for a,b come from the facts that div X=divv=0 and[Dt;?X]=0.
[2]H.Abidi and M.Paicu,Existence globale pour un fluide inhomogene,Ann.Inst.Fourier,57(2007),883-917.
[3]H.Abidi,G.Gui and P.Zhang,On the wellposedness of 3-D inhomogeneous Navier-Stokes equations in the critical spaces,Arch.Ration.Mech.Anal.,204(2012),189-230.
[4]H.Abidi,G.Gui and P.Zhang,Wellposedness of 3-D inhomogeneous Navier-Stokes equations with highly oscillating initial velocity field,J.Math.Pures Appl.,100(2013),166-203.
[5]H.Bahouri,J.-Y.Chemin and R.Danchin,Fourier Analysis and Nonlinear Partial Differential Equations,Grundlehren der mathematischen Wissenschaften,Springer,2010.
[6]A.-L.Bertozzi and P.Constantin,Global regularity for vortex patches,Commun.Math.Phys.,152(1993),19-28.
[7]J.-Y.Chemin,Calcul paradifférentiel précisé et applicationsa deséquations aux dérivées partielles non semilinéaires,Duke Math.J.,56(1988),431-469.
[8]J.-Y.Chemin,Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel,Invent.Math.,103(1991),599-629.
[9]J.-Y.Chemin,Persistance de structures géométriques dans les fluides incompressibles bidimensionnels,Ann.Sci.Ecole Norm.Sup.,26(1993),517-542.
[10]J.-Y.Chemin,Perfect incompressible fluids,Oxford Lecture Series in Mathematics and its Applications,14.The Clarendon Press,Oxford University Press,New York,1998.
[11]R.Danchin,Poches de tourbillon visqueuses,J.Math.Pures Appl.,76(1997),609-647.
[12]R.Danchin,Density-dependent incompressible viscous fluids in critical spaces,Proc.Roy.Soc.Edinburgh Sect.A,133(2003),1311-1334.
[13]R.Danchin and P.-B.Mucha,A Lagrangian approach for the incompressible Navier-Stokes equations with variable density,Commun.Pure Appl.Math.,65(2012),1458-1480.
[14]R.Danchin and P.-B.Mucha,Incompressible flows with piecewise constant density,Arch.Ration.Mech.Anal.,207(2013),991-1023.
[15]R.Danchin and P.-B.Mucha,The incompressible Navier-Stokes equations in vacuum,arXiv:1705.06061.
[16]R.Danchin and P.Zhang,Inhomogeneous Navier-Stokes equations in the half-space,with only bounded density,J.Funct.Anal.,267(2014),2371-2436.
[17]R.Danchin and X.Zhang,On the persistence of H?lder regular patches of density for the inhomogeneous Navier-Stokes equations,J.Ec.Polytech.Math.,4(2017),781-811.
[18]P.Gamblin and X.Saint-Raymond,On three-dimensional vortex patches,Bull.Soc.Math.France,123(1995),375-424.
[19]F.Gancedo and E.Garcia-Juarez,Global regularity of 2D density patches for inhomogeneous Navier-Stokes,Arch.Ration.Mech.Anal.,229(2018),339-360.
[20]T.Hmidi,Régularité H?ldérienne des poches de tourbillon visqueuses,J.Math.Pures Appl.,84(2005),1455-1495.
[21]T.Hmidi,Poches de tourbillon singulieres dans un fluide faiblement visqueux,Rev.Mat.Iberoam.,22(2006),489-543.
[22]J.Huang,M.Paicu and P.Zhang,Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity,Arch.Ration.Mech.Anal.,209(2013),631-682.
[23]D.Hoff,Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data,J.Differential Equations,120(1995),215-254.
[24]O.-A.Ladyzhenskaja and V.-A.Solonnikov,The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids.(Russian)Boundary value problems of mathematical physics,and related questions of the theory of functions,8,Zap.Naucn.Sem.Leningrad.Otdel.Mat.Inst.Steklov.(LOMI),52(1975),218-219.
[25]X.Liao and P.Zhang,On the global regularity of two-dimensional density patch for inhomogeneous incompressible viscous flow,Arch.Ration.Mech.Anal.,220(2016),937-981.
[26]X.Liao and P.Zhang,Global regularity of 2-D density patches for viscous inhomogeneous incompressible flow with general density:low regularity case,Accepted by Commun.Pure Appl.Math.,72(2019),835-884.https://onlinelibrary.wiley.com/toc/10970312/2019/72/4.
[27]X.Liao and P.Zhang,Global regularities of two-dimensional density patch for inhomogeneous incompressible viscous flow with general density,arXiv:1604.07922.
[28]P.-L.Lions,Mathematical topics in fluid mechanics.Vol.1.Incompressible models,Oxford Lecture Series in Mathematics and Its Applications,3.Oxford Science Publications,The Clarendon Press,Oxford University Press,New York,1996.
[29]M.Paicu and P.Zhang,Global solutions to the 3-D incompressible inhomogeneous NavierStokes system,J.Funct.Anal.,262(2012),3556-3584.
[30]M.Paicu,P.Zhang and Z.Zhang,Global unique solvability of inhomogeneous NavierStokes equations with bounded density,Commun.Partial Differential Equations,38(2013),1208-1234.
[31]P.Zhang and Q.Qiu,Propagation of higher-order regularities of the boundaries of 3-D vortex patches,Chinese Ann.Math.Ser.A,18(1997),381-390.
[32]J.Simon,Nonhomogeneous viscous incompressible fluids:existence of velocity,density,and pressure,SIAM J.Math.Anal.,21(1990),1093-1117.
?We calculate directly ■and then recursively for any l≥1,■
?We consider Dtvl instead of ?tvl since we would like to benefit from the density equation Dtρ=0 while ?tρ=-v·▽ρ has singularities since the densityρis discontinuous in space variable. The idea was used by D. Hoff in [23] for the study of the compressible Navier-Stokes system.
§We derive from H1-energy estimate for Dtvl and Sobolev embedding that Dtvl∈Lloc1(Lp)which controls ▽2vl∈Lloc1(Lp) up to lower order terms, see Eq.(1.29)below.
?The formulae for a,b come from the facts that div X=divv=0 and[Dt;?X]=0.