GREEN’S FUNCTION OF 2-D COMPRESSIBLE NAVIER-STOKES EQUATIONS ON THE HALF SPACE ZHANG WEI NATIONAL UNIVERSITY OF SINGAPORE 2013 GREEN’S FUNCTION OF 2-D COMPRESSIBLE NAVIER-STOKES EQUATIONS ON THE HALF SPACE ZHANG WEI (B.Sc., Fudan University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2013 DECLARATION I hereby declare that this thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Zhang Wei 20 November, 2013 i Acknowledgements First and foremost, I would like to express my sincere gratitude to my supervisor Professor Yu Shih-Hsien for his guidance and instructions during my Ph.D studies and research work In the past five years, with his kindness and patience, he has taught me immense knowledge not only on academic fields, but also on attitudes towards work and life, which will benefit me in my lifetime He is always ready to share his own ideas and experience in his research work without reservation, and encourages me to take good challenges At the end of my scholarship, he offered me a research assistant position to continue my work Without his support, it is impossible to complete this thesis Besides, I would like to thank my friends Huang Xiaofeng, Tu Linglong, Zhang Xiongtao and Wang Haitao, who have provided me insightful suggestions and shard their opinions on my research work from time to time Special thanks given to Dr Deng Shijing, who presided a seminar on shock waves when she worked at the Department of Mathematics The seminar enriched my academic awareness and equipped me necessary techniques to finish this thesis Last but not the least, I appreciate opportunities offered by Department of Mathematics of National University of Singapore I will cherish memories in the last five and half years forever ii Contents Introduction 1.1 Literature Review 1.2 Results Preliminaries 2.1 Basic Identities for Laplace Transformation and Fourier Transform 2.2 Notations 10 Fundamental Solution 3.1 Long-Wave Component Inside Finite Mach Number Region 3.2 Short-Wave Component Inside Finite Mach Number Region 3.3 Fundamental Solution Outside Finite Mach Number Region 15 17 35 39 Master Relation and Wave Propagators 43 4.1 Well-posedness Assumption and Master Relation 47 4.2 Structures of Wave Propagators 48 Green’s Function 59 Conclusion 69 6.1 Further work 69 Bibliography 71 iii Summary This thesis discusses solutions of 2-D compressible Navier-Stokes equations, especially solutions of their linearized form around a constant state with presence of zero Dirichlet boundary It is divided into three parts Firstly, the fundamental solution, or Green’s identity of linearized 2-D compressible Navier-Stokes are studied Results in this part are mainly through analysis on Fourier variables For structures inside the finite Mach number region {x||x|  Mt}, by Long-wave and Short-wave decomposition, it shows the fundamental solution consists of two leading parts: one pure diffusion, the other diffusion waves Outside the finite Mach number region, the fundamental solution decays exponentially in space Secondly, Master Relation and wave propagators are introduced, and structures of these wave propagators are shown Master Relation reveals essential connections between boundary data, which is key factors to construct below Green’s function It can also be observed that Master Relation is composed of two types of wave propagators: one called the interior wave propagator and the other the surface wave propagator Therefore, it is inevitable to investigate their structures At last, Green’s function of linearized 2-D compressible Navier-Stokes equations with presence of zero Dirichlet boundary is represented With integration of the fundamental solution and Master Relation, Green’s function can be represented explicitly Structures of wave propagators help us to achieve its whole picture iv Chapter Introduction Compressible Navier-Stokes equations are used to describe the motion of compressible flow In the science and engineering discipline, the pointwise structure of their solutions is a fundamental subject and has vast applications See [24], [3] The pointwise structure of the fundamental solution or Green’s functions is the subject of great interest Depending on it, the wave propagation structure of compressible Navier-Stokes equations near given constant states can be obtained Currently, the main tools to study this subject are the mathematical modeling and scientific computations as in [3] However, understanding of the precise pointwise structure of Green’s functions is still limited, especially with the presence of boundary conditions In this thesis, we will consider 2-D compressible Navier-Stokes equations under zero Dirichlet boundary conditions around a given constant state, and give the structure of their Green’s function Considering n dimensional space Rn , the well-known compressible NaiverStokes equations can be written as ( @t ⇢ + r · m = ⌘ ⇣ N ⇣ ⌘ ⇣ ⇣ ⌘⌘ m m m @t m + r · + rP (⇢) = µ1 ⇢ + (µ1 + µ2 )r r · m ⇢ ⇢ (1.1) n Here, ⇢(x, t) R and m(x, t) R represent the unknown density and momentum at time t and x Rn P (⇢) represents the pressure; µ1 and µ2 are the viscosity coefficients satisfying µ1 > and n µ1 + µ2 r · f represents the divergence in x of the vector function f , and rg representsN ⌘gradient in x of the scalar⇣function g The i th component ⇣ the ⌘ i of r · m ⇢ m can be expressed by r · m⇢m In this thesis, we are interested in the compressible Navier-Stokes Equations in 2-D half space under the zero Dirichlet boundary condition around a constant state And without loss of generality, we assume µ1 = and Chapter Introduction µ1 + µ2 = Thus equations studied here can be written as ( @t ⇢ + r · m = ⌘ ⇣ N ⇣ ⌘ m m @t m + r · + rP (⇢) = m ⇢ ⇢ (1.2) for x R2 with x1 and x2 R In addition, we have the following + initial and boundary conditions: < ⇢(x1 , x2 , 0) = ⇢0 (x1 , x2 ), m (x , x , 0) = m10 (x1 , x2 ), : 1 m2 (x1 , x2 , 0) = m20 (x1 , x2 ); m1 (0, x2 , t) = 0, m2 (0, x2 , t) = (1.3) (1.4) Now considering the constant equilibrium solution (⇢, m1 , m2 ) = (1, 0, 0), the linearized equations of (1.2) around this solution can be expressed as < @ t ⇢ + @ x1 m + @ x2 m = (1.5) @ m + @ x1 ⇢ = 4m1 : t @ t m2 + @ x2 ⇢ = 4m2 with the initial condition < ⇢(x1 , x2 , 0) = ⇢0 (x1 , x2 ) 1, m (x , x , 0) = m10 (x1 , x2 ), : 1 m2 (x1 , x2 , 0) = m20 (x1 , x2 ), (1.6) under boundary conditions (1.4) Our goal of this thesis is to find the precise pointwise structure of the Green’s Function of the above equations 1.1 Literature Review This section presents a survey of literature pertinent to studies on the structure of the solutions to the compressible Navier-Stokes equations Previous studies that considered the existence and uniqueness of the equations with or without the presence of boundary conditions are reviewed In addition, progresses in the research of the pointwise structure of the fundamental solution in one dimensional Navier-Stokes equations are shown, which lead to results on nonlinear stability of the solution in the whole space around constant states Furthermore, studies that have attempted to find large-time asymptotic behavior of the high dimensional compressible Navier-Stokes Chapter Green’s Function In this chapter, our goal is to achieve an explicit representation for Green’s function of (1.5) and (1.4) Master Relation plays an important role in this place since it provides complete boundary data from limited Dirichlet data Unless we can know all information on the boundary, it is impossible to find pointwise structure of Green’s function Proof of Theorem 1.2 Proof Recall G(x1 , x2 , t) is the fundamental solution or Green’s identity of (1.5) Denote u⇤ (x, t) = (⇢⇤ , m⇤ , m⇤ )T Therefore, with the same proce1 dures in proof of (2.7) and (2.11), u⇤ (x, t) can be expressed as u⇤ (x, t) = Since 0 1 B C s) B 0 C u⇤ (0, z2 , s)dsdz2 @ A 0 0 0 B C RR B C @z1 u⇤ (0, z2 , s)dsdz2 G(x1 , x2 z2 , t) @ R2 A 0 1 0 B C RR + R2 @z1 G(x1 , x2 z2 , t) B C u⇤ (0, z2 , s)dsdz2 @ A 0 (5.1) RR R2 G(x1 , x2 z2 , t 0 B C u⇤ (0, z2 , s) = B m⇤ (0, z2 , s) C @ A 59 Chapter Green’s Function and 0 B ⇤ B ⇤ @z1 u⇤ (0, z2 , s) = B O1 (z2 ,s) m1 (0, z2 , s) @ O2 ⇤ m⇤ (0, z2 , s) (z2 ,s) C C C A from Master Relation, besides m⇤ (0, z2 , s) = m1 (0, z2 , s) RR = R+ GS (0, z2 , y1 , y2 , s) 21 1T ⇢(y1 , y2 , 0) B C B C B GS (0, z2 , y1 , y2 , s) C · B m1 (y1 , y2 , 0) C dy1 dy2 @ 22 A @ A S G23 (0, z2 , y1 , y2 , s) m2 (y1 , y2 , 0) where (GS , GS , GS ) represents the second row of GS From (4.7), we can 21 22 23 write out the Green’s function Gb (x1 , x2 , y1 , y2 , t) explicitly as G S ZZ M1 dz2 ds + R2 ZZ M2 dz2 ds R2 ZZ (5.2) M3 dz2 ds R2 where M1 = G(x1 , x2 z2 , t s) GS (0, z2 , y1 , y2 , s) GS (0, z2 , y1 , y2 , s) GS (0, z2 , y1 , y2 , s) 22 23 B 21 C B C ·@ 0 A 0 (5.3) M2 = G(x1 , x2 ·@ z2 , t s)· GS (0, z2 , y1 , y2 , s) 21 GS (0, z2 , y1 , y2 , s) 21 ⇤ (z2 ,s) ⇤ (z2 ,s) O1 GS (0, z2 , y1 , y2 , s) 22 O2 GS (0, z2 , y1 , y2 , s) 22 60 ⇤ (z2 ,s) ⇤ (z2 ,s) O1 GS (0, z2 , y1 , y2 , s) 23 O2 GS (0, z2 , y1 , y2 , s) 23 (z2 ,s) ⇤ O1 ⇤ O2 (z2 ,s) A (5.4) Chapter Green’s Function M3 = @x1 G(x1 , x2 z2 , t s) 0 B C · B GS (0, z2 , y1 , y2 , s) GS (0, z2 , y1 , y2 , s) GS (0, z2 , y1 , y2 , s) C 22 23 @ 21 A 0 (5.5) This prove the theorem Because G and GS have explicit representations, we will only discuss Oi (i = 1, 2) here Their structures can be derived from structures of wave propagators in the last chapter Notice A2 B2 + C(t s) Cs (A + B)2 Ct (5.6) for any real number A and B This inequality will be used frequently in proofs of the following theorem Theorem 5.1 There exists a constant C such that p > e (x2 +t) 2Ct > > x2  t + t > t+1 > > > p p > > t + t  x2  t > t+1 > > > p p < e C|x2 | or t  x2  t t |O1 (x2 , t)|  C +C > t+1 > x2 > e 2Ct p p > > t  x2  t > t+1 > > > (x t)2 > > p > e 2Ct > : x2 t t t+1 |O2 (x2 , t)|  C e C|x2 | (t + 1) +C > > > > > > > > > > > > > < > > > > > > > > > > > > > : e (x2 +t)2 2Ct (t+1) (t+1) x2 2Ct (t+1) p e t)2 e (x2 2Ct (t+1) 61 p x2  t + t p p t + t  x2  t p p or t  x2  t t t  x2  x2 t p p t t Chapter Green’s Function Proof These two estimates can be proved similarly So it is sufficient to prove the structure for O2 (x1 , x2 ) From the definition of O2 , O2 = (@t @x2 @ x2 @t @x2 )S @t @x2 (S ⇤(x2 ,t) = K1 + K2 ⇤(x2 ,t) 2) ) + Ce C|x2 | Ct I Estimates for K1 : = = 3 (@t @x2 @x2 @t @x2 )S > > C(@ @ 3 > @x2 @t @x2 )(H1 ⇤x2 > t x2 > < > > > > > Ce : > e < C > Ce : x2 Ct ep t + H0 ⇤x2 x2 Ct ep t t C|x2 | (x2 +t)2 Ct t C (x2 t)2 Ct t e t1 C|x2 | Ct + Ce t C|x2 | t1 (5.7) II Estimates for K2 From Lemma 4.4- 4.6, = C ⇤ R1R +C +C +C +C +C R R1R Rt R 1 R R Rt R t Rt R t U (y2 , s)dy2 ds (x2 y2 )2 C(t s) (t s)2 e Rt (x2 y2 )2 C(t s) (t s)2 e ⇣ e R (x2 y2 )2 C(t s) (t s)2 e R (x2 y2 )2 C(t s) (t s)2 L 1F (y2 )e s ⇣ s @ x2 ) e (@t @ x2 ) e ( 2) (x2 y2 )2 4(t s) (x2 y2 )2 4(t s) = J1 + J2 + J3 + J4 + J5 + J6 ⌘ U (y2 , s) dy2 ds dy2 ds L 1F (@ R t R (5.8) (y2 )e s ( 2) (y2 )e s ⇣ s dy2 ds dy2 ds L F 62 ⌘ s ( 2) (y2 )e s s ⌘ dy2 ds Chapter Green’s Function where n 2 U (x2 , t) = (@t + @x2 )2 E(x2 , t) + (@t x2 Ct @x2 @t )(e t ⇤t e ) + @t (te t ⇤t e x2 Ct ) o and E(x2 , t) defined in the Lemma 4.5 By the same techniques in the previous proof of estimates for K1 , |J1 |, |J2 |  C x2 Ct e x2 (t + 1) e Ct |J3 |, |J5 |  C ; (t + 1)2 , (5.9) For J4 , |J4 |  C Rt 1 +C Rt R y2 ) (x2 C(t s) p e (t s)2 |y2 |s s R (x2 s2 y2 ) C(t s) p e (t s)2 |y2 | s s = J41 + J42 + J43 ; pC2 n s Ce y2 dy2 ds (|y2 | s)2 Cs s + Ce (|y2 |+s) C o dy2 ds (5.10) Considering J41 , |J41 |  C RtR +C (x2 Rt t R (x2 if x2 t Z t |y2 |s p p s e (t (x2 y2 )2 C(t s) t or x2  e p C s)2 s t+ s2 pC2 p p s2 dy2 ds s y2 pC2 y2 ) C(t s) p e (t s)2 |y2 |s s For the first term of J41 , if |x2 |  Z y2 ) C(t s) p e (t s)2 |y2 |s s s2 s dy2 ds t, x2 C e Ct dy2 ds  C ; 2 (t + 1) (t + 1)2 y2 t, since in this situation, (x2 y2 )2 C(t s) y2 e 63 (|x2 | t)2 C(t s) , Chapter Green’s Function we have RtR if t+ p y2 ) (x2 C(t s) p e (t s)2 |y2 |s s p s y2 dy2 ds  C R  Ce t  x2  Z s2 C t Z p t or p |y2 |s p e (t s t  x2  t p t e C p s)2 s s2 ds (|x2 | t)2 Ct (t+1)2 ; s2 ; t, (x2 y2 )2 C(t s) (|x2 | t)2 Ct (t+1)2 y2 dy2 ds  C (t + 1)2 For the second term of J41 , since p thus, if |x2 |  Z if x2 t+ t p t Z if t p t, Z |y2 |s t  x2  Z t t p |y2 |s e (t s t or x2  t t Z 1 p  3, s2 y s4 Z p t+ s p |y2 |s (x2 y2 )2 C(t s) e (t t or p s s)2 s p t, (x2 y2 )2 C(t s) e (t s2 p s2 t  x2  t (x2 y2 )2 C(t s) s)2 y2 C s)2 s p p C C p s s2 64 y2 p dy2 ds  C dy2 ds  C e x2 Ct (t + 1) e (|x2 | t)2 Ct (t + 1) t, y2 dy2 ds  C (t + 1) ; Chapter Green’s Function Therefore, |J41 |  > > > > < > C > > > : x2 p |x2 |  t; p p x2 t t or x2  t + t; p p p p t + t  x2  t or t  x2  t t (5.11) e Ct C (t+1)2 , e (|x2 | t)2 Ct (t+1)2 , C , (t+1)2 Considering J42 , |J42 |  C| RtR +C|  C| Rt t RtR R (x2 +C| +C| +C| Rt y2 s t RtR Rt t R p (x2 s)2 2Ct e s y2 s p p y2 ) e p y2  s+ s e e dy2 ds| (|y2 | s)2 Cs s4 (x2 y2 )2 2C(t s) (t s)2 e dy2 ds| (|y2 | s)2 2Cs s4 dy2 ds| (x2 s)2 2Ct e (x2 y2 )2 2C(t s) (t s)2 e (|y2 | s)2 2Cs s4 dy2 ds| (x2 +s)2 2Ct e (x2 y2 )2 2C(t s) (t s)2 e (|y2 | s)2 2Cs s4 dy2 ds| e s y2  s+ s R (|y2 | s)2 Cs s4 e C(t s) p e (t s)2 |y2 | s s y2 ) (x2 C(t s) p e (t s)2 |y2 | s s e (x2 +s)2 2Ct e (x2 y2 )2 2C(t s) (t s)2 e (|y2 | s)2 2Cs s4 dy2 ds|, we have J42 x2 > e (x2 t) 2Ct > > C (t+1)2 + C e 2Ct > (t+1)2 > > (x2 t)2 > > e 2Ct > C > > (t+1)2 > > > > C > (1+t) > < x2 e 2Ct  > C (t+1)2 > > > C > (1+t)2 > > > > > e (x2 +t) 2Ct > C > > (t+1)2 > > x2 > > e (x2 +t) 2Ct e 2Ct : C + C (t+1)2 (t+1) p x2 t + t p p t t  x2  t + t p p t  x2  t t p |x2 |  t p p t + t  x2  t p p t t  x2  t + t p x2  t t 65 (5.12) Chapter Green’s Function Considering J43 , |J43 | = C  C  C Rt 1 nR ⇣ R t 1 x2 (x2 y2 ) C(t s) p e (t s)2 |y2 | s s e Ct (1+t)2 R |y2 | +e |x| + Rt 1 C|x2 | Ct ⌘ R e (|y2 |+s) C |x| |y2 | o e dy2 ds (x2 y2 )2 C(t s) (t s)2 e (|y2 |+s) C dy2 ds (5.13) Therefore, from (5.10), (5.11),(5.12) and (5.13), we have |J4 |  Ce C|x2 | Ct + x2 > e (x2 t) 2Ct > C > + C e 2Ct , > (t+1) (t+1)2 > > > > > > C > (1+t)2 > > > < > > x2 > > e 2Ct > C > > (t+1)2 > > > > > x2 > > e (x2 +t) 2Ct : C e 2Ct + C (t+1)2 (t+1)2 x2 p p t t; p t  x2  t t p p t  x2  p |x2 |  t; or t+ x2  t+ p t; t (5.14) With the same procedures, J6 has the same estimates Thus from (5.8), (5.9) and (5.14), | 1⇤ 2|  Ce C|x2 | Ct + x2 > e (x2 t) 2Ct > C e 2Ct > + C (t+1)2 , > (t+1) > > > > > > C > (1+t)2 > > > < > > x2 > > e 2Ct > C > > (t+1)2 > > > > > x2 > > e (x2 +t) 2Ct : C + C e 2Ct (t+1)2 (t+1)2 66 x2 p or t p t; p t  x2  t p t  x2  p |x2 |  t; t+ x2  t+ p t p t (5.15) t; Chapter Green’s Function Next similar to prove estimates for K1 , @ t @ x2 S = < C@t @x2 (H1 ⇤x2 : Ce < e C = : Ce x2 Ct ep t C|x2 | (x2 +t)2 Ct t C|x2 | +Ce + H ⇤x (x2 t)2 Ct t + Ce x2 Ct ep t ) + Ce C|x2 | Ct t t1 C|x2 | Ct , t 1; t  , (5.16) Now using techniques in proving structures of J4 , from (5.15) and (5.16), |K2 |  C e C|x2 | (t + 1) +C > > > > > > > > > > > > < > > > > > > > > > > > > : e (x2 +t)2 2Ct (t+1) (t+1) x2 e 2Ct (t+1) e (x2 t)2 2Ct (t+1) p x2  t + t p p t + t  x2  t p p or t  x2  t t p p t  x2  t x2 t p t Finally, (5.7) and (5.17) lead to the conclusion in the lemma 67 (5.17) Chapter Conclusion This thesis includes two results on linearized 2-D compressible NavierStokes equations The first one is the construction of fundamental solution of (1.5) It shows that the fundamental solution consists two leading parts: one is pure diffusion part and the other is diffusion waves In construction of the fundamental solution, we mainly discuss its structures inside finite Mach number region |x|  Mt The Long-wave and Short-wave decomposition is the key tool in this discovery The second result is the construction of Green’s function of linearized compressible Navier-Stokes equations with presence of zero Dirichlet data To understand its structure, it is necessary to make clear relations between different boundary data, where Master Relation plays a key role Furthermore, since Maser Relation can be considered as composition of wave propagators, through investigation of structures of wave propagators, the whole picture of Green’s function will be completed 6.1 Further work Based on depicts of Green’s function, it is possible to investigate nonlinear stability and large-time asymptotic behavior of solutions of compressible Navier-Stokes equations around constant solutions It is interesting to combine Green’s function with weighted energy estimates to solve this problem, like work done by Zeng in [30] for the one dimensional case, and Kobyashi and Kagei in 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[31] Zeng, Y.: Lp asymptotic behavior of solutions to hyperbolic–parabolic systems of conservation laws Arch Math 66,310-319 (1996) (Not cited.) 73 ... transform of the function f (x) 2. 2 Notations We begin with the introduction of the fundamental solution and Green’s function Definition 2. 4 The fundamental solution of (1.5) we discussed in this thesis... consider 2- D compressible Navier- Stokes equations under zero Dirichlet boundary conditions around a given constant state, and give the structure of their Green’s function Considering n dimensional... methods to deal with the question with the presence of boundary, it is necessary to find Green’s function of the corresponding linearized equations under specified boundary conditions Due to lack of