Greens function for viscous system

GREEN’S FUNCTION FOR VISCOUS SYSTEM WANG HAITAO (B.Sc., Shanghai Jiao Tong University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2014 To my parents Wang Hu and Li Yanping and my wife Zhang Rong Acknowledgements First and foremost, I owe my deepest gratitude to my dedicated supervisor Prof. Yu Shih-Hsien for his generous encouragements and invaluable discussions. He made me aware that the meaningful research should be the thorough understanding of the problem, of the phenomena, rather than the one only loading with a lot of fancy tools and theorems. His insightful observations, his passion for discovery and jovial character will surely benefit me in my lifetime. It is also my great pleasure to thank Prof. Han Fei from our department for so many discussions about mathematics and advices on research career. I would also take this opportunity to express my appreciation to Prof. Liu Chunlei, Prof. Wang Weike, Dr. Yin Hao and Dr. Li Liangpan from Shanghai Jiao Tong University. It is their guidance and encouragement led me entering the mathematical world. I would also like to thank my friend and collaborate Zhang Xiongtao for his inspiring supports and discussions. Thanks also go to my other fellow postgraduate friends for their friendship and help, including Du Linglong, Huang Xiaofeng, Zhang Wei, too numerous to list here. Last but not least, I am forever indebted to my parents and my wife, for their love and supports. v vi Acknowledgements Wang Haitao Oct 2014 Contents Acknowledgements Summary v ix Introduction 1.1 General Description of Initial-Boundaray value problems . . . . . . . 1.2 Algebraic-Complex Scheme . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Fundamental solution, Green’s funciton and Dirichlet-Neumann map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Laplace-Fourier transforms. . . . . . . . . . . . . . . . . . . . 1.2.3 The Dirichlet-Neumann map in the transformed variable, the Master Relationship . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Fundamental solution in transformed variables and symbol identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Purpose and outline of the thesis . . . . . . . . . . . . . . . . . . . . 10 Algebraic-Complex Scheme and two toy models 13 vii viii Contents 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 A simple Toolbox for the Laplace Transformation . . . . . . . . . . . 17 2.3 Convection heat Equation with robin boundary condition . . . . . . . 18 2.4 Linearized Compressible Navier-Stokes Equation in 1-D with Robin condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Multidimensional Compressible Navier-Stokes equation 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Fundamental solution for Cauchy Problem . . . . . . . . . . . . . . . 38 3.3 3.2.1 Fundamental solution in Fourier variables . . . . . . . . . . . 39 3.2.2 Long-short wave decomposition . . . . . . . . . . . . . . . . . 42 3.2.3 Long wave component inside finite Mach region . . . . . . . . 43 3.2.4 Short wave component inside finite Mach number region . . . 68 3.2.5 Fundamental solution outside finite Mach number region . . . 80 Green’s function for half space problem . . . . . . . . . . . . . . . . . 85 3.3.1 Fundamental solution in (x1 , ξ , s) variables . . . . . . . . . . . 85 3.3.2 Green’s function in (ζ, ξ , s) variables . . . . . . . . . . . . . . 89 3.3.3 Green’s function in (x1 , ξ , s) variables . . . . . . . . . . . . . 93 3.3.4 Symbol identification and inversion of Green’s function . . . . 97 Conclusions and Discussions 105 Bibliography 107 Summary Initial-boundary value problems are the fundamental problems in partial differential equations. Among them, the half space problem is the essential one. The classical approaches for it emphasize more on the well-posed aspect while the point-wise behavior is not fully understood. To gain more insights into the point-wise behavior, Liu and Yu [16] initiated a new program—Algebraic Complex Scheme recently . In this thesis, we aim at developing the Algebraic Complex Scheme and applying it to study the Green’s function for viscous system. To be more specific, firstly toy models, the 1-D convection heat equation and 1D compressible Navier-Stokes equation with Robin condition were investigated. The explicit full boundary data and the detailed quantitative description were obtained up to exponentially decaying terms. Next, we considered the multi-dimensional compressible Navier-Stokes equation in half space. We first obtained the point-wise estimates of fundamental solution for Cauchy problem by long-short wave decomposition and weighted energy estimate. Then we applied Liu-Yu algorithm to find the Green’s function in transformed variable. By comparing representation of fundamental solution and Green’s function in transformed variables, we obtained the Green’s function exactly in terms of composition of fundamental solution, heat kernel and their derivatives. ix Chapter Introduction Our understanding of the fundamental processes of the natural world is based on a large extent on partial differential equations (PDEs). The study of PDEs can be traced back to 18th century, as a mathematical language to describe the mechanics of continuous medium and as the principal tool to analytical study of physical science. Examples include the vibration of solids, the spread of heat, the diffusion of chemicals and flow of fluids. More recently, PDE also appears naturally in modern physics such as quantum mechanics and general relativity, for which the Schr¨odinger’s equation and Einstein’s equation are the fundamental and central equations respectively. The PDEs also arise in finance, for example, the Black–Scholes equation gave a quantitative description of financial market. Among these equations, the evolutionary one, i.e., the PDEs depending on time are usually more interesting, since the solution of them can give the dynamical description of the model. 1.1 General Description of Initial-Boundaray value problems Typically, a PDE admits many solutions. To single out the unique solution which models the real world, one needs to impose additional conditions. These conditions 3.3 Green’s function for half space problem 97 form of λ1 and λ2 , λ1 λ2 + |ξ |2 = λ1 λ2 − |ξ |2 λ21 λ22 − |ξ |4 ε ν+ = c2 s (λ1 λ2 + |ξ |2 ) (3.3.48) s2 + (ν + ε)|ξ |2 s + c2 |ξ |2 ε ν+ ≡ c2 s (λ1 λ2 + |ξ |2 ) D(ξ , s) . Hence we can rewrite the Green’s function as Gb (x1 , ξ , s)  sλ1 +ε|ξ |2 (λ1 +λ2 ) −λ1 x1 e D √ − ε(λ1 λ2 +|ξD| ) −1ξ  2  ε sλ2 + ν+ cs ε ν+ cs (λ1 λ2 +|ξ |2 )|ξ |2 (e−λ1 x1 −e−λ2 x1 ) −λ1 x1 = e − +  D  ε ν+ c2 s λ +|ξ |2 (λ +λ ) e−λ1 x1 −e−λ2 x1 √−1ξ ε ) )( s (ε −λ2 x1 − e I − n−1 D |ξ |2 (λ ( +λ2 ) T e e−λ1 x1 −e−λ2 x1 (λ1 λ2 +|ξ |2 )ξ ξ T D where D(ξ , s) = s2 + (ν + ε)|ξ |2 s + c2 |ξ |2 . Symbol identification and inversion of Green’s function In order to find the Green’s function in (x1 , x , t) variables, we have to invert the representation in (x1 , ξ , s) variables. From Proposition 3.33, it suffices to obtain Fourier-Laplace inverse of three functions: e−λi x1 , i = 1, and 1/D. It turns out that immediate Fourier-Laplace inverse would be highly nontrivial, for example, one can refer to [13] and [18]. Here we adopt a different approach. In previous sections, we have obtained the detailed point-wise estimates of fundamental solution in Theorem 3.22, Theorem 3.27, and its representation in (x1 , ξ , s) variables (3.3.16). In this section, we will relate the symbols e−λi x1 , i = 1, in Green’s function to those in fundamental solution and invert 1/D explicitly to get Green’s function. Throughout this section, we assume x1 > 0. √ ) −1ξ T D ν+ cs (3.3.49) 3.3.4  −λ1 x1 (e−λ1 x1 −e−λ2 x1 )      98 Chapter 3. Multidimensional Compressible Navier-Stokes equation Lemma 3.34. Assume x1 > 0. e−λ1 x1 = νs + c2 G12 (x1 , ξ , s) e−λ2 x1 = −2ε∂x1 (3.3.50) e−λ2 x1 ε 2λ2 (3.3.51) Proof. They are from (3.3.16). Lemma 3.35. For i = 1, 2, 3, ∂t G1i = −∂x1 G2i − ∇ · G3i (3.3.52) ∂t G2i = −c2 ∂x1 G1i + ε (3.3.53) + η∂x21 G2i + η∂x1 ∇ · G3i ∂t G3i = −c2 ∇ G1i + η∂x1 ∇ G2i + ε ∂t2 G1i = + η∇ ∇ T G3i c2 G1i − ν (∂x1 G2i + ∇ · G3i ) where ∇ = (∂x2 , · · · , ∂xn )T , = n j=1 (3.3.54) (3.3.55) ∂x2j . Proof. We rewrite (3.1.5) as,      −∂x1 −∇ T ρ ρ           ∂t m1  = −c2 ∂x1 ε + η∂x21 η∂x1 ∇ T  m1       m −c2 ∇ η∂x1 ∇ m ε + η∇ ∇ T (3.3.56) The first three equalities are just rephrasing of the matrix equation. As a consequence, we have the last equality. |x|2 Lemma 3.36. Assume x1 > 0. Let E(x, t) = e− 4εt , (4πεt)n/2 O(x, t) = c2 G12 −ν (∂x1 G22 + ∇ · G32 ). (i). −λ1 x1 = 2O(x, t); Ψ1 (x, t) ≡ Fξ−1→x ◦ L−1 s→t e (3.3.57) −λ2 x1 = −2ε∂x1 E(x, t); Ψ2 (x, t) ≡ Fξ−1→x ◦ L−1 s→t e (3.3.58) (ii). (iii). Ai (x , t) ≡ Fξ−1→x ◦ L−1 s→t [λi ] = −∂x1 Ψi (x, t) (3.3.59) x1 =0 3.3 Green’s function for half space problem Proof. For part (i), from (3.3.16), we have G12 = 99 e−λ1 x1 , νs+c2 thus e−λ1 x1 = 2(νs + c2 )G12 (x1 , ξ , s). By the property of Laplace transform L[ d f ] = sL[f ] − f (t), dt and G12 (x, 0) = 0, we have L−1 s→t [sG12 (x1 , ξ, s)] = ∂t G12 (x1 , ξ , t). Taking inverse Fourier transform, Fξ−1→x ◦ L−1 s→t [sG12 ] = 2(c + ν∂t )G12 (x1 , x , t). By Lemma 3.35, ∂t G12 = −∂x1 G22 − ∇ · G32 , we conclude that −λ1 x1 Fξ−1→x ◦ L−1 = 2(c2 + ν∂t )G12 = 2O(x, t). s→t e Part (ii) is consequence of Lemma 3.34 and Fξ−1→x ◦ L−1 s→t e−λ2 x1 = E(x, t). ε 2λ2 Lemma 3.37. Let S(x , t) ≡ Fξ−1→x ◦ L−1 s→t . D(ξ , s) There exists C1 > such that,for l ∈ N, |β| ≤ l, when t ≥ 1,  Dxβ S(x , t) − e c2 − 2α t [(n+l)/2]  Aj (x , t) ≤ χ3 (D ) j=1 O(1)Wn−1 (x , t) ∗x |x |2 −C t e t n−1+|β| − + O(1)Wn−1 t (x , t) ∗x e t |x |2 C1 t n−1+|β| + O(1)e−(|x |+t)/C1 ; 100 Chapter 3. Multidimensional Compressible Navier-Stokes equation when t ≤ 1,  Dxβ S(x , t) − e c2 − 2α t [(n+l)/2] Aj (x , t) − χ3 (D ) χ3 (D )  [(n+l)/2] j=1 Bj (x , t) j=1 ≤ O(1)e−|x |/C1 . where Aj s, Bj s are defined in (3.3.60). Proof. L−1 s→t 1 = L−1 s→t D(ξ , s) s + (ν + ε)|ξ |2 s + c2 |ξ |2 = e− (ν+ε)2 |ξ |4 −c2 |ξ |2 ν+ε |ξ |2 t+t ≡ e−α|ξ | t+λ(|ξ |)t − e− (ν+ε)2 |ξ |4 − e−α|ξ | 2λ(|ξ |) t−λ(|ξ ν+ε |ξ |2 t−t (ν+ε)2 |ξ |4 −c2 |ξ |2 − c2 |ξ |2 |)t . We will use long wave short wave decomposition technique to find Fourier inverse of above function. We introduce cut-off functions χ1 (ξ ) ≡ 1{|ξ |R} , χ2 (ξ ) = − χ1 (ξ ) − χ3 (ξ ), where κ and R are positive number, κ is sufficiently small while R is sufficiently large. Fξ−1→x =Fξ−1→x e−α|ξ | t+λ(|ξ |)t − e−α|ξ | 2λ(|ξ |) (χ1 + χ2 + χ3 ) e−α|ξ | t−λ(|ξ t+λ(|ξ |)t |)t − e−α|ξ | 2λ(|ξ |) t−λ(|ξ ≡S1 + S2 + S3 . Case I: Finitie Mach Region {|x | ≤ Mct}. We consider the case t ≥ firstly. When |ξ | is very small, √ α2 −1c|ξ | − |ξ |2 c √ = −1c|ξ | + A |ξ |2 , λ(|ξ |) = |)t 3.3 Green’s function for half space problem 101 where A(|ξ |2 ) is an analytic function in |ξ |2 and A = O(|ξ |2 ). In addition we have 1+A = + B(|ξ |2 ), B is also an analytic function in |ξ |2 and of order |ξ |2 . In this case, e−α|ξ | t+λ(|ξ |)t − e−α|ξ | 2λ(|ξ |) t−λ(|ξ |)t sin (c|ξ |t(1 + A)) c|ξ |(1 + A) sin (c|ξ |t) −α|ξ |2 t e cos (c|ξ |tA) (1 + B) = c|ξ | =e−α|ξ | 2t A1 + cos (c|ξ |t) e−α|ξ | sin (c|ξ |tA) (1 + B) . c|ξ | 2t A2 Here Ai , i = 1, can be extended to analytic functions of ξ in a neighborhood of 0, and can be bounded by eO(1)|ξ | t . Similar as Proposition 3.12, taking κ small enough, we can show Dxβ S1 (x , t) ≤ − O(1)Wn−1 (x , t) ∗x e t |x |2 C1 t n−1+|β| − + O(1)Wn−1 t (x , t) ∗x where Wn−1 is the inverse Fourier transform of sin(c|ξ |t) , c|ξ | e t |x |2 C1 t + O(1)e−t/C1 , n−1+|β| that is the fundamental solution of wave equation in n − dimension. When |ξ | is very large, λ(|ξ |) = α|ξ |2 1− c2 α2 |ξ |2 c2 = α|ξ | − + 2α ∞ j=1 aj , |ξ |2j then e−α|ξ | t+λ(|ξ |)t − e−α|ξ | 2λ(|ξ |) t−λ(|ξ |)t =e ∞ c − 2α t c2 −2j aj (t)|ξ | j=1 ∞ ≡ e− 2α t bj (t)|ξ |−2j +e j=1 ∞ Aj (ξ , t) + j=1 ∞ c −2α|ξ |2 t+ 2α t Bj (ξ , t) j=1 (3.3.60) 102 Chapter 3. Multidimensional Compressible Navier-Stokes equation where aj (t), bj (t) are polynomials in t of degree no larger than j. Similar as Proposition 3.19 and 3.20, we can prove  c2 − 2α t Dxβ S3 (x , t) − e  [(n+|β|)/2] Aj (x , t) ≤ O(1)e−t/C1 . χ3 (D ) j=1 When κ < |ξ | < R, e−α|ξ | t+λ(|ξ |)t − e−α|ξ | 2λ(|ξ |) t−λ(|ξ |)t ≤ O(1)e−t/C1 , since the range of ξ is compact, it is easy to see |Dxβ S2 (x , t)| ≤ O(1)e−t/C1 . When < t < 1, applying similar argument, we obtain  Dxβ S3 (x , t) − e c2 − 2α t [(n+|β|)/2] [(n+|β|)/2] Bj (x , t) ≤ O(1). Aj (x , t) − χ3 (D ) χ3 (D ) j=1  j=1 Case II: Outside Finite Mach Region {|x | > Mct}. Observe that D is the transformation of solution of the following equation   ∂t2 − (ν + ε)∂t − c2 u=0  u(x , 0) = 0, u (x , 0) = δ(x ) t Then following the similar weighted energy argument as for Theorem 3.27, we can prove that |x | −C Dxβ S(x , t) ≤ Ce . Summarizing above results complete the proof. Theorem 3.38 (Green’s Function). Let Gb (x1 , x , t) be the Green’s function for boundary value problem (3.3.20), i.e. , the solution is given by     ρ(x1 , x , t)   t mb (x∗ , s)    dx∗ ds. Gb (x1 , x − x∗ , t − s)  m (x1 , x , t) =   n−1 R mb (x∗ , s) m (x1 , x , t) 3.3 Green’s function for half space problem 103 Let G(x, t) be the fundamental solution for Cauchy problem. Let Ψi , Ai , i = 1, 2, S be same in Lemma 3.36 and 3.37. We have the following representation of Green’s function Gb ,  (Gb )11 (Gb )12      Gb = (Gb )21 (Gb )22    (Gb )31 (Gb )32 (Gb )11 = (∂t A1 − ε (A1 + A2 )) ∗ S Ψ1 , (3.3.61) S) ∇ T Ψ1 , (3.3.62) (Gb )12 = −ε (A1 ∗ A2 ∗ S − (Gb )21 = Ψ1 + ε ν + c2 I (−A1 ∗ A2 ∗ S + (Gb )22 = −ε ∂t A2 − ν + c2 I (Gb )31 = − (Ψ1 − Ψ2 ) , S) (A1 + A2 ) ∗ S ∇ T (Ψ1 − Ψ2 ) , ν∂t + c2 A1 − ε ν + c2 I (3.3.63) (3.3.64) (A1 + A2 ) ∗ S ∇ (Ψ1 − Ψ2 ) , (3.3.65) (Gb )32 = Ψ2 In−1 + ε ν + c2 I (A1 ∗ A2 ∗ S − S) ∇ ∇ T (Ψ1 − Ψ2 ) . (3.3.66) where t I [f ] (t) = f (τ )dτ, t (f ∗ g) (x , t) = Rn−1 f (x − x∗ , t − τ )g(x∗ , τ )dx∗ dτ, and t (f x1 f (x∗1 , x∗ , τ )g(x1 − x∗1 , x − x∗ , t − τ )dx∗ dx∗1 dτ. g) (x1 , x , t) = Rn−1 should be understood in the sense of distribution. Proof. This theorem is consequence of Proposition 3.33, Lemma 3.36 and Lemma 3.37. Chapter Conclusions and Discussions The main objective of this dissertation was to study the Green’s function for viscous system and the point-wise behavior of the solution of initial-boundary value problem. In order to achieve this, the Algebraic-Complex Scheme was employed and further extended. We have applied the new method to three classical equations, convection heat equation with Robin condition, 1-D compressible Navier-Stokes equation with Robin condition with background velocity, and multi-dimensional compressible Navier-Stokes equation in half space. For the first two equations, the full boundary data were constructed. For the last equation, the point-wise structure of fundamental solution and the representation of Green’s function have been obtained. In view of our results, it appears that the algebraic-complex scheme provides an efficient method to handle problem with various boundary condition. Currently, the common methods to tackle nonlinear problem include energy estimates, functional analysis and pseudo-differential operator and their combinations. Admittedly, these methods are powerful for proving global well-posedness and stability result. However, they provide insufficient information on local behavior of the solution. Compared to these methods, the algebraic-complex scheme exploits the delicate analyticity properties, thus describing the local point-wise behavior explicitly. 105 106 Chapter 4. Conclusions and Discussions Based on the representation of Green’s function and point-wise estimate of fundamental solution, it is possible to investigate nonlinear stability and large-time asymptotic behavior of solutions of compressible Navier-Stokes equation in half space, like what have been done in [12]. This kind of result would provide more precise point-wise information than [7]. However, there are several limitations in the algebraic-complex scheme. Firstly, it can only deal with the constant coefficient problems or piecewise constant coefficient problem, since the algebraic-complex scheme depends on the analyticity of solution in transformed variables while the variable coefficient would break this analyticity. Therefore, it would not be possible to generalize algebraic-complex scheme to most general variable coefficient problem. A possible extension is to investigate the analytic coefficient problem. This direction may rely on how to incorporate hyper-function theory [8] into algebraic-complex scheme. The second limitation is that currently we only know how to apply algebraic-complex method to half-space boundary. For more general boundary, although we can partition them into small segments and stretch them to straight one, the analyticity will not be preserved under these operations. Hence algebraic-complex scheme lose its power in this case. However, we still can consider some special curved boundary such that certain deformation will preserve the analyticity. Therefore, the interesting future directions include extending algebraic-complex scheme to problems with special coefficients or problems with special curved boundary. Another interesting future area is inverse problem of multi-layer problem. One example is to gain the inside information such as depth of different medium layer only by measuring the boundary properties. 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Zygmund, Trigonometric series, vol. 1, Cambridge university press, 2002. GREEN’S FUNCTION FOR VISCOUS SYSTEM WANG HAITAO NATIONAL UNIVERSITY OF SINGAPORE 2014 GREEN’S FUNCTION FOR VISCOUS SYSTEM Wang Haitao 2014 [...]... Section 2, we will prepare some simple toolbox for the Laplace transformation and its inverse transformation In Section 3, 4 we will follow the ABCD steps to develop the full boundary data for (2.1.2) and (2.1.3) 2.2 A simple Toolbox for the Laplace Transformation Let f (t) be a function defined in t ≥ 0, its Laplace transformation F (s) and the inverse transformation (the Bromwich integral) are given... for the Cauchy problem for system (1.2.1), is defined as      n n Aj ∂ x j − ∂t + j=1 Bij ∂xi ∂xj G(x, t) = 0, (1.2.3) i,j=1     G(x, 0) = δ(x)Im , where In is the n × n identity matrix There are many highly developed tools for finding the fundamental solution, for example harmonic analysis cf [25] Complex analytic technique is also useful sometimes, cf [31] The Green’s function G(x, y, t) for. .. ))m×m are rational functions in λj1 , · · · , λjl The main effort is to invert the transforms for the entries Kij 1.2.4 Fundamental solution in transformed variables and symbol identification In principle, once we have inverted the Master relationship, the full boundary data will be constructed Then by fundamental solution for Cauchy problem and (1.2.6), one can construct the Green’s function However,... Green’s function By algebraic-complex scheme, we obtain the Green’s function in transformed variables To invert back to physical variable, we compute the transformed representation of fundamental solution and compare the symbol between them From the symbol identification, we get the Green’s function Chapter 2 Algebraic-Complex Scheme and two toy models In this chapter, we used the Laplace-Laplace transformation... these roots represent poles for the rational functions (1.2.11) and the inverse Laplace transform in x1 is given as the sum of residue at the poles by the Bromwich integral (1.2.13) for inverting the Laplace transform in x1 : eλj x1 Res (soln(ζ, ξ , s; ub , un ); ζ = λj ) u(x1 , ξ , s) = (1.2.14) p(λj ,ξ ,s)=0 The standard well-posedness condition that lim u(x, t) = 0 yields a system of x1 →∞ algebraic... fundamental function for corresponding whole space problem, they were able to shift the initial condition to boundary and obtain a homogenous initial condition and nonhomogeneous boundary condition problem Then by combining the Laplace-Fourier transform, the problem was converted to a purely algebraic problem in transform variables The delicate complex analysis and decomposition of transform domain,... Purpose and outline of the thesis The main goal of this dissertation is to study the Green’s function for viscous system by the framework of algebraic-complex scheme In Chapter 2, we demonstrate the efficiency of Algebraic complex scheme by investigating two toy models From algebraic-complex scheme, to find the Green’s function, the key ingredients are the full boundary data, thus our main concerns are construction... ∂xj u = Bij ∂xi ∂xj u x1 > 0,  j=1 i,j=1 (1.2.7)    u(x, 0) = 0 Taking Fourier transform with respect to tangential variable x = (x2 , · · · , xn ) and Laplace transform to x1 and t variable, denoting the transformed variables by ξ , ζ, s respectively, then one obtains an algebraic polynomial system in transformed variables 6 Chapter 1 Introduction ∞   u(x , ξ , s) = ˆ 1    u(ζ, ξ , s) =... (t) and w(t) are the given controlled boundary data We use the above two systems to demonstrate the simplicity and efficiency of the A-C scheme, since there were no such exponentially sharp estimates obtained for those 2.1 Introduction 15 problems before Even for the whole space problem, the construction of the fundamental solution for (2.1.3) is very non-trivial; and it was obtained in 1990’s, see [31]... new pointwise understandings of some classical PDEs, for example, heat equation and Navier-Stokes equation [27], Lamb’s problem [28] In the rest of this chapter, the main idea of the Algebraic-Complex Scheme will be introduced For a general viscous hyperbolic system in half space, n ∂t u + n Aj ∂ x j u = j=1 Bij ∂xi ∂xj u (1.2.1) i,j=1 The unknown functions are u = u(x, t) ∈ Rm The space variables . GREEN’S FUNCTION FOR VISCOUS SYSTEM WANG HAITAO (B.Sc., Shanghai Jiao Tong University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT. we aim at developing the Algebraic Complex Scheme and applying it to study the Green’s function for viscous system. To be more specific, firstly 2 toy models, the 1-D convection heat equation and. some boundedness and asymptotic results. However, Fourier transformation is only applicable for function defined on whole space. Therefore, in the presence of boundary, Fourier analysis method faces