Asymptotics and Borel Summability: Applications to MHD, Boussinesq equations and Rigorous Stokes Constant Calculations

Transcription

1 Asymptotics and Borel Summability: Applications to MHD, Boussinesq equations and Rigorous Stokes Constant Calculations DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Heather Rosenblatt, B.S. and M.S. Graduate Program in Mathematics The Ohio State University 23 Dissertation Committee: Saleh Tanveer, Advisor Ovidiu Costin Barbara Keyfitz

2 c Copyright by Heather Rosenblatt 23

3 Abstract We consider two problems where Borel summation based methods can be used to obtain information about solutions to di erential equations. In the first problem, we analyze the initial value problem for the Boussinesq equation for fluid motion and temperature field as well as the magnetic Bénard equation which models electro-magnetic e ects on fluid flow under some simplifying assumptions. This method has previously been applied to the Navier-Stokes equation in [4], [6], and [7], which is a limiting case for each of these equations. We show that this approach can be used to prove local existence for the Boussinesq and the magnetic Bénard equation, in two or three dimensions. We prove that an equivalent system of integral equations in each case has a unique solution, which is exponentially bounded for p 2 R +, p being the Laplace dual variable of /t. This implies local existence of a classical solution to Boussinesq and magnetic Bénard in a complex t-region that includes a real positive time (t)-axis segment. Further, it is shown that within this real time interval, for analytic initial data and forcing f, the solution remains analytic and has the same analyticity strip width. Further, under these conditions, the solution is Borel summable, implying that the formal series in time is Gevrey- asymptotic for small t. We also determine conditions on the computed solution to the integral equation in each case over a finite interval [,p ] that results in a better estimate for existence time for the corresponding solution to the partial di erential equation. The second problem is to give rigorous bounds on the Stokes constant values for a nonlinear ordinary di erential equation (ODE) that arises in the context of selection of limiting finger width in viscous fingering -the so called Sa man-taylor problem. Specifically, it was proved [52] that the selected finger width asymptotically corresponds to values of a ii

4 parameter C in a nonlinear ODE such that the Stokes constant on the real positive line vanishes. The full asymptotic expansion for the solution, G(y), includes not only inverse powers of the independent variable y, but also exponentially small corrections. In particular, for y on the positive real axis, Im(G) S(C)y 3/4 e 23/2 3 y 3/2 for arbitrary C, see [52]. We prove rigorous estimates on S(C) and find intervals in C for which S(C) =, in agreement with earlier numerical computations, see [9]. iii

5 To my father, who willingly spent many hours discussing mathematics with me, igniting my interest and curiosity. iv

6 Acknowledgments I want to thank my family: Dad, Mom, Audrey, Ben, Ivana, Dave, and Becky. Without your love and support this would probably never have been written. Special thanks also go to my advisor, Saleh Tanveer, whose advice was invaluable. Thank you for all your time and support. Lastly, I would like to thank Jon Hood. Your support and company have made life fun. v

7 Vita 22-present Teaching Associate at The Ohio State University Master of Science in Mathematics from The Ohio State University Departmental Fellowship at The Ohio State University Presidential Fellowship at The Ohio State University Bachelor of Science in Mathematics from University of Missouri Summer Research Fellow at University of Missouri Publications ) Mitrea, Dorina and Heather Rosenblatt. A general converse theorem for mean-value theorems in linear elasticity, Mathematical Methods in the Applied Sciences Volume 29, Issue 2, pages , August 26 2) Rosenblatt, Heather and Saleh Tanveer. Existence, Uniqueness, Analyticity, and Borel Summability of Boussinesq and Magnetic Benard Equations, arxiv under identifier.468 Fields of Study Maor Field: Mathematics Studies in application of asymptotic and Borel summability methods to di erential equations: Saleh Tanveer, PhD Thesis Advisor. vi

8 Table of Contents Page Abstract ii Dedication iv Acknowledgments v Vita vi Chapters Introduction. Overview of Thesis Motivation for Studying These Problems Thesis Content Background to the Mathematical Methods 7 2. Asymptotic Tools and Definitions Classical Asymptotics Inadequacy of Classical Asymptotics Borel Summability Borel Summability Applied to a Nonlinear Problem Stokes Phenomena Stokes Phenomena: An Example with Laplace transform Definition of Stokes Constant Stokes Phenomena: Airy Function Classical Results for the Navier-Stokes Equation Existence of Weak Solutions Uniqueness Smooth Solutions Classical Results for MHD and Boussinesq Equation Borel Summability of Boussinesq and MHD Equations Introduction Main Results Local Existence and Uniqueness of Solution Formulation of Integral Equation: Borel Transform Norms in p Existence of a Solution in Dual Variable vii

9 3.3.4 Proof of Local Existence Borel-Summability Estimates on the Solution in the Borel Plane Estimates p(ĥ,ŝ)(k, l p) p(ŵ, ˆQ)(k, p) Extension of Existence Time Improved Radius of Convergence Improved Growth Estimates Based on Knowledge of the Solution in [, p] Stokes Constant Calculation A Simple Example to Outline Idea Applying the Idea to the Sa man-taylor Problem Asymptotics of Coe cients of Formal Series Solution Relationship between the Stokes Constant and the Asymptotic Coefficients Proof of Lemma Bounds on the Scaling Factor q Proof of Lemma Di culty with the Approach Improvement on the Stokes Constant Bounds Summary and Conclusion 2 5. The Main Points Future Research Bibliography 5 Appendices A Derivation of Navier-Stokes Equation Continuum Model 9 A. Conserved Quantities A.. Conservation of Mass A..2 Conservation of Momentum A.2 Derivation of the Navier-Stokes Equation A.3 Derivation of the Boussinesq and MHD Equations B Additional Background Material for Boussinesq and MHD Theorems 25 B. Fourier Inequalities in Two Dimensions viii

10 Chapter Introduction. Overview of Thesis In this thesis, we apply asymptotic tools and Borel summability methods to two modern problems of physical importance. The first problem concerns two nonlinear partial di erential equations (PDE) governing fluid flow in di erent settings. Under most conditions, the equation governing constant density fluid flow is the Navier-Stokes equation, which is derived from basic physical laws such as conservation of mass and momentum. However, if in a constant density fluid, local temperature di erences produce a buoyancy force or a magnetic field provides a body force on the fluid, the flow is modeled by Boussinesq and magnetic Bénard (MHD) equations respectively (see Appendix A for derivation). The Boussinesq equation is u t u = P [u ru ae 2 ] + f, u(x, ) = u (x) (.) t µ = u r, (x, ) = (x), where d = 2 or 3 is the dimension, u : R d R +! R d, and : R d R +! R. Also, P = I r (r ) is the Hodge proection operator to the space of divergence free vector fields, is the nondimensional fluid viscosity, µ is the thermal di usion coe cient, e 2 is the unit vector aligned opposite to gravity, and the parameter a is proportional to gravity. Here (u, ) corresponds to the fluid velocity and temperature field. The magnetic Bénard

11 equation is v t v = P [v rv B t µ µ B rb]+f, v(x, ) = v (x) (.2) B = P [v rb B rv], B(x, ) = B (x), where v, B : R d R +! R d. Here v is the fluid velocity, B is the magnetic field, while,, µ and are constants related to fluid viscosity, density, magnetic permeability and electric conductivity respectively. Study of these two variations of the Navier-Stokes equation constitutes the first part of this thesis. We prove existence of local solutions to each of these two equations using a technique based on Borel summation as opposed to the usual Sobolev energy methods. The method of proof automatically gives time analyticity for < t > from the representation of the solution. The results also include Borel summability for analytic initial conditions, which gives a Gevrey asymptotic result for small t. The second problem is to rigorously bound the Stokes constant for a nonlinear ordinary di erential equation (ODE) with a parameter that arises in applications. Specifically, we bound the Stokes constant on the positive real line for y d apple y d dy dy G + apple G 2 = y2 + C, satisfying G(y) y as y!, arg(y) 2, 2 3, (.3) and obtain an interval in the parameter C in which the Stokes constant S = S(C) must vanish..2 Motivation for Studying These Problems Fluid flow is ubiquitous in everyday experience and displays a wide range of phenomena depending on the geometry of flow conditions. More than a hundred years ago, a continuum model ignoring individual particle interaction for fluid motion was derived that is applicable to a wide range of conditions. This is called the Navier-Stokes equation (see for example [3]) and is derived from simple consideration of conservation of mass, momentum, energy, equation of state and thermodynamic relations. For flow conditions when the fluid velocity is far smaller than the speed of sound in the medium, one obtains the incompressible 2

12 Navier-Stokes approximation where the conservation of mass and momentum are decoupled from the rest of the physical quantities (see Appendix A for derivation). Despite this simplification, the incompressible limit is believed to accurately model physical flows in most ordinary everyday experiences including turbulent flows for small viscosity and/or large flow velocity and length scales. The nondimensional Reynolds number (the inverse of the nondimensional viscosity, ) characterizes the flow; flow of honey or flow through microchannels correspond to low Reynolds number flow, whereas ets, pipe flows, and geophysical fluid flows correspond to high Reynolds number, i.e. small viscosity flows. Thus, the incompressible limit is believed to accurately model most high Reynolds number flows. In situations where thermal e ects are significant enough to cause a buoyant force but not significant enough to alter momentum in the flow, one arrives at the Boussinesq equation. In other situations where one considers flow of a magnetic conducting fluid, one must include the coupling of Maxwell s equation with fluid flow and arrives at the magnetohydrodynamic (MHD) equation under some simplifying assumptions (see Appendix A for derivation). A fundamental question is whether or not the incompressible Navier-Stokes equation, or its variations, Boussinesq and MHD equations, are always physically relevant for high Reynolds number turbulent flows. This depends on whether or not the Navier-Stokes initial value problem has smooth solution globally in time. If instead the solution blows up, then the continuum model is called into question and the derivation, which neglected compressibility and thermodynamic e ects, may not be accurate. Further, since in nature we see smooth solutions to fluid flow problems over long time intervals, an accurate model of fluid flow should have smooth global solutions. On the mathematical side, the resolution to this question is likely to be applicable to a number of di erent equations and give insight into the fundamental nature of these equations. The question of global existence of a smooth solution to Navier-Stokes equation in 3- D is a celebrated Clay problem and remains open since Leray raised the question in the 93s. Classical Sobolev space and energy methods have thus far proved futile in proving the existence of a global solution except when Reynolds number is small, i.e. viscosity e ects are su ciently large. This is because the known energy norms which are controlled independent 3

13 of T are only those of C ([,T],L 2 (R 3 ) and L 2 ([,T],H (R 3 )). This provides the basis of Leray s (93) proof of global existence of weak solutions. However, these solutions live in a space where the velocity and its gradient can blow up pointwise, which violates the continuum assumption under which the incompressible Navier-Stokes continuum model is derived. Therefore, to resolve the question of global existence, we are motivated to use an alternate mathematical formulation, provided by the Borel summability approach, that does not rely at all on energy bounds. Our method provides an alternate existence and uniqueness theory for a class of Navier- Stokes-like nonlinear PDEs for which the question of global existence of solutions to the PDE becomes one of asymptotics for a known solution to the associated nonlinear integral equation. While the global asymptotics problem is di cult and yet to be resolved, we show in Theorem how information about solutions on a finite interval in the dual variable, for specific initial condition and forcing, may be used for obtaining better exponential bounds in the Borel plane and therefore extending existence time for classical solutions to the PDEs. An important problem in Laplacian growth and pattern formation is the understanding of patterns due to displacement of a more viscous fluid by a less viscous fluid in a Hele-Shaw cell or a porous medium, see [42], [38], [3], and [48]. (A Hele-Shaw cell is a pair of parallel plates separated by a small gap.) In studying this displacement, Sa man and Taylor (958) [42], noticed that a steadily traveling finger emerges with relative finger width,, close to a half, where theoretical calculation with neglect of surface tension showed no restriction on. The selection of remained an important open mathematical problem for many years though a considerable body of work using numerical and formal asymptotic calculation in the 98s suggested that the selection of in the small surface tension limit results from exponentially small terms in surface tension, see [36], [4], [3], [47], and [48]. The Sa man- Taylor problem was also linked to dendritic crystal growth. There were similar discoveries for the velocity and tip radius of a needle crystal growing from a pure supercooled melt, see Kessler et al s articles [3], [32], [33]. Then Xie and Tanveer, [52], [49], and [5], proved that the selection problem was equivalent to proving that the Stokes constant on the real positive y axis for the nonlinear ODE 4

14 (.3) was zero for certain values of C, related to the finger width. Earlier numerical computation gave the values of C for which S(C) =, see [9]. However, there is no rigorous proof of this statement other than the result by Kohut et al, [2], where a rigorous bound on the Stokes constant was found, showing it was nonzero for a related nonlinear ODE without any parameter. That computation was relevant to completing the proof in [52] that there are no fingers selected with < 2. The results in this thesis address the problem in [52] for 2 ( 2, m) for some 2 < m <. As mentioned above, exponential asymptotics play a critical role in many physical problems ranging from quantum mechanics to crystal growth, see [43]. For instance, in the context of (.3), it is known that the asymptotic series expansion as y!for arg(y) 2, 2 3 is in the form G(y) y + X =2 where a s are all real, [52]. However, this is insu a y G(y), (.4) cient information to determine Im(G) along the positive real axis as y!, which is critical to the selection problem in viscous fingering. For that purpose, we must account for exponentially small corrections. Indeed accounting for the exponentially small correction, it is known, see [52], that Im(G) Sy 3/4 e 23/2 3 y 3/2, (.5) as y!,wheres = S(C) is referred to as the Stokes constant. The condition on C that makes S vanish is of critical importance and is related to the selection of finger width in viscous fingering. While there are well known numerical procedures that have been used to compute S(C), see for example [9], none of these are rigorous except for [2]. Here we show how a rigorous theory similar to that in [2] can be constructed involving a parameter C and more precise bounds determined for the calculation of S(C). The same methodology can be used in many other Stokes constant problems arising from physical considerations. 5

15 .3 Thesis Content In Chapter 2, we provide a brief background for the mathematical techniques used in the thesis. This includes an introduction to some of the tools used in Borel summability and exponential asymptotics. This area of research is quite large; we give only a minimal description to be helpful to the reader. However, we present some of the key ideas and give references which provide an introduction and background for the techniques used directly in the thesis. Specifically, we define Borel summability and Stokes constants. We also give a short review of the classical Leray results on weak solutions of Navier-Stokes-related nonlinear equations. In Chapter 3, we apply Borel summability methods to the Boussinesq and MHD equations, two equations related to the Navier-Stokes equation. We show in Theorem 3.2. that Borel summability methods give existence and uniqueness of local solutions. Importantly, this provides an alternate existence and uniqueness theory for a class of nonlinear PDEs for which the question of global existence of a solution to the PDE becomes one of asymptotics for a known solution to the associated nonlinear integral equations. Further, time analyticity for < t > follows readily from the solution representation. We also prove that the classical H 2 (R d ) solution, which is unique, has the Laplace transform representation given here, provided initial data and forcing are in L \ L in Fourier space. In Chapter 4, we switch gears slightly and consider a problem associated with selection of finger width for viscous fingering in a Hele-Shaw cell. We give rigorous bounds for the Stokes constant of a nonlinear problem which depends on a parameter. This completes the work done in [52]. In that paper, it was rigorously proved that the selection problem for finger width 2 2, m, for some m > 2, asymptotically corresponds to values of parameter C such that the leading order Stokes constant S(C) vanishes in the asymptotics of solution G to the leading order inner-problem: y d apple y d dy dy G + G 2 = y2 + C, satisfying G(y) y as y!. Here we find an interval in C where S(C) =. 6

16 Chapter 2 Background to the Mathematical Methods 2. Asymptotic Tools and Definitions Over the past thirty odd years, many maor developments and advancements have been made in asymptotic analysis. In many problems arising in physical applications, classical asymptotics in the sense of Poincaré has proved inadequate. Classical asymptotics does not distinguish between functions that di er by terms that are exponentially small terms relative to other terms in the asymptotic expansion. More refined notions such as Borel sums are needed to understand the behavior of such functions, which arise naturally as solutions to ODEs, discrete maps, and in some cases PDEs. Under some conditions, Borel summability gives a one-to-one correspondence between an actual function and the full asymptotic expansion that includes all possible exponential corrections. The development of Borel summability for formal power series and the more general development of Ecalle- Borel summability or Generalized Borel summability of transseries give a widely applicable technique for recovering actual solutions from formal solutions of di erential systems. This will be the basis of the mathematical methods used in this thesis. 2.. Classical Asymptotics Asymptotics concerns limiting behavior of functions as certain points are approached in a particular direction. The Taylor series of an analytic function is a special case of an asymptotic series, but more generally asymptotics concerns the behavior near singular points. 7

17 Definition 2... An asymptotic expansion, f, as a point is approached in some direction in the complex plane, is a formal series f = X n f n (x) (2.) in which each term is smaller then its predecessor. Explicitly, f n+ (x) =o( f n (x)) as x! x along a specified direction. Definition A formal expansion f is asymptotic (in the sense of Poincaré) to a function, f, atapointx, if for every N f(x) NX k= f k = o( f N (x)) as x! x along the specified direction. (2.2) In the above, it is assumed that f k 6=. In many applications, fn (x) =a n x n,inwhich case a less restrictive definition that allows some of the coe cients to be zero is desirable. Definition An asymptotic power series, P n c nx n is asymptotic to f as x! if for every N, NX f(x) c k x k = o(x N ) as x! along a specified direction. (2.3) k= There is no loss of generality in taking x! (or x!) since the change of variables z = x x (or z =/x) can accommodate other cases. While asymptotic expansions are often not convergent, they can be used to accurately approximate a function. They also prove very useful in developing formal solutions to di erential equations, which when Borel summable, result in true solutions. For analytic functions near regular points, the asymptotic series is ust the Taylor series. For example, if f(x) =ln( x), then f = P n= xn n and f n = xn. We get progressively n better approximations of f(x) for small x as we include more and more terms in the sum, and this provides a reasonable, accurate way to approximate our function. However, convergent series need not be asymptotic. For example, consider g(z) =e /z for small z. We can use the Laurent expansion to see that e /z = P n= ( ) n n!z, and the series n converges for each value of z 6=. However, if we consider a small value of z, sayz = 6, 8

18 the number of terms that we have to take from the sum to get a good approximate solution is very large. Notice that for this value of z the terms in the sum are growing in absolute value until n is quite large. Thus, while the series gives us a convergent representation of the function, it is basically unusable to approximate the function. This series is an example of a convergent series which is not an asymptotic series. Indeed f n (z) = ( )n n!z n qualify as an asymptotic sequence since as z! f n+ 6= o( f n ). does not On the other hand, a divergent series can be asymptotic. Consider the integral equation f(x) =e x Z x e t dt (2.4) t where x!along R +. We can integrate (2.4) by parts successively n times and get f(x) = x + x (n) x n nx k= (k)e x+ + (n + )e x Z x e t dt. tn+ For f n = (n) x n and any N, x N (f NX f n )=x N n= N X k= (k)e x+ + (N + )e x Z x e t dt N+ t!, which tends to as x!on R + by L Hospital. More generally, the same is true along any ray in the right half complex plane: e i R + for 2 ( /2, /2). Recalling (2.3), this means is asymptotic to f as x!on R +. f(x) = X n= (n) x n, Thus, we see that a divergent expansion can be asymptotic. It is also to be noted that di erent functions can have the same asymptotic expansion in the sense of Poincaré since, for example, e x R x 2 expansion as (2.4). e t t dt has the same asymptotic Further, we also see that integration by parts can be used, in some cases, to derive asymptotic series from integral functions. There is a wealth of literature detailing how to derive the asymptotics of functions and integral functions in a variety of forms. These techniques including Laplace method and Watson s Lemma, Steepest descent method, and 9

19 WKB method and are explained in more detail in many introductory books, for example [2] and []. The three examples given here illustrate the fundamental concept that not all convergent series are asymptotic nor are all asymptotic series convergent. In fact, as we will see in the following sections, most asymptotic series arising in physical problems are not convergent Inadequacy of Classical Asymptotics Consider the asymptotic behavior of f satisfying f f = for x approaching. (2.5) x2 This problem can be solved explicitly in terms of an integral, and the asymptotics of the integral gives the required behavior. Nonetheless, we follow a more indirect argument that is readily generalizable for much more complicated problems and will motivate Borel summation. We look for a balance and see that f x 2 is a consistent balance where f provides a lower order correction. Including this correction in f, wehavef x 2 2 x 3 and so on. In a systematic manner, this series can be generated through the recurrence f n = x 2 + f n for n 3with f 2 = x 2 or through plugging in the formal series P k= and equating powers. Thus, the expected behavior of all small solutions as x!is f f X n!( ) n+ n= a k x k+ x n+. (2.6) However, since x = is an irregular singular point of (2.5), this formal series is not convergent for any value of x 6=. While the series is not convergent, the asymptotic relationship f f can be proved for any small solution f of (2.5) as x!along any complex ray. The problem with representation (2.6), and more generally with classical Poincaré asymptotics as a whole, is that it does not incorporate the full freedom present in the ODE. Notice that to any solution which is small in the left half plane we can add a homogeneous solution Ce x and obtain a second solution that is also small in the left half plane. Solutions that decay in the left half plane as x!are not unique. Thus, loosely speaking,

20 f f + Ce x. (2.7) This expression cannot be made rigorous with classical asymptotics. Consider the problem where one has f f in a part of the right half plane. This solution is unique since no multiple of the exponential maybe added. Upon analytic continuation into the left half plane it is known that the constant C which was zero in the right half plane becomes nonzero as the negative real axis is crossed. One seeks to determine this specific C. The exponential is of smaller order than all terms in the asymptotic power series, f. Hence, f does not incorporate the true behavior of f such as Im(f) on the negative real axis. The fact that classical asymptotics cannot see exponentially small corrections cannot simply be fixed by adding a homogeneous solution to the expansion nor need there be only one exponential correction as is the case for (2.5). Consider the nonlinear problem y y = x 2 y 2. (2.8) We can characterize small solutions as before and plug in the formal series P a k k=.we x k+ equate powers of /x to obtain y ỹ x 2 + X k=2 a k x k+, the full asymptotic expansion in the sense of Poincaré. The exponentially small corrections in the left half plane are no longer ust Ce x. However, one can systematically plug-in y = X (Ce x ) k f k (x) (2.9) k= and equate coe cients of the exponential e kx to obtain the recurrence relation It can be shown that f k f k = P a,k = x (k )f k + fk =2f Xk f k + f f k. f = and therefore formally X (Ce x ) k k= X = a,k x,

21 which is the full transseries corresponding to all small solutions to (2.8) in the left half complex plane. These exponentially small corrections cannot be approached through classical asymptotics, but can be approached through Borel summation. It turns out that there exists a unique function f k such that f k f k and (2.9) is convergent. We will use the two examples here in more detail to motivate the definitions and illustrate how Borel techniques are actually used Borel Summability To begin with, we define the Laplace and inverse Laplace transform as usual. Definition If e c p F (p) 2 L (, ), thelaplacetransform L(F )(x) := Z e xp F (p)dp exists and defines a function which is analytic in the half plane H c = {x : Re(x) >c}. Further, the function is continuous on the closure of H c and tends to zero as x!in H c. The inverse Laplace transform is well-defined on the space of functions which are analytic in H c, continuous on the closure of H c, and tend to zero suitably quickly as x!in H c. Moreover, the inverse Laplace transform is given explicitly by integration in the complex plane, L (f)(p) = 2 i Z c+i c i e px f(x)dx. Recall (2.5) from the last section, which has a small solution f as x!with f f X n= n!( ) n+ x n+. If we take a formal inverse Laplace transform of the series f and use the fact that L(p k )= (k + ) x k+ = k!, (2.) xk+ we get a new series P n= ( p)n, which does converge for small p to p +p. Now this function 2

22 is well defined for all p 2 R +. Taking its Laplace transform, we get f(x) = Z e xp p dp. (2.) +p We can now check that f given by (2.) is actually a solution to (2.5) for <x >. Thus, we have recovered a function that solves the equation from a formal series solution. This example demonstrates the idea behind Borel summability and its beauty. We take a formal series solution without worrying about convergence and apply a Borel transform to it which acts as a formal inverse Laplace transform. This will make any series, which was originally factorially divergent, convergent in the new variable p in the Borel plane. Finally, if the analytical continuation on [, ) isl loc and exponentially bounded, a Laplace transform recovers an actual function to which the formal series is asymptotic. We formalize this process in the definition below. Remark The di erence between an inverse Laplace transform and a Borel transform is that for k, (k + ) L x k+ (p) =p k only for <p >. For <p <, contour deformation to the right half plane gives. For a Borel transform, B( (k+) x k+ )(p) =p k for all p. Remark If this method seems a little unmotivated at first, we can come to the same representation by using a integral representation of and formally switching the sum and integration. Alternatively we have, using the change of variables t = xp and dt = xdp, f X n= (n + )( ) n+ x n+ = X n= R t n e t dt( ) n+ x n+ = Z e xp ( ) X ( p) n dp. n= This gives us the same expression for f as (2.). Definition The Borel transform, B, is a map from formal power series in /x to power series in p given by applying to each term (s + ) B (p) =p s. (2.2) x s+ 3

23 Thus, the Borel transform of a formal series is defined to be another formal series:! X c k X c k p k B x k (p) = (k). (2.3) k= The Borel transform of the product of two series f and g in /x is given by k= B( f g) :=B( f) B( g), where is the Laplace convolution defined by [F G](p) := Z p F (p s)g(s)ds. (2.4) Definition The Borel sum along the ray R + is defined by three steps (assuming they are possible): Borel transform f!b( f). Convergently sum the series for B( f) for small p and analytically continue it along R + calling this function F (p). Take the Laplace transform of F and call it the Borel sum LB( f). In other words, LB( f) := R F (p)e xp dp. When we can do the operations in the second and third steps, we say the formal series is Borel summable. The operator LB applied to f is called the Borel sum. This requires that the function, which we get for small p, is analytically continuable on the real axis and that its analytic continuation is in L (e ap dp) for some value of a. Remark This definition requires that the new series for small p be convergent. The Borel transform of a formal power series results in a series with smaller coe cients (see (2.3)). Thus, a Gevrey- asymptotic series becomes convergent on applying a Borel transform. More generally, if the coe cients are Gevrey-m for m 6=, then after a change of variables the Borel transform of the series will be convergent for small p. Definition 2... A formal series P c k k= is said to be Gevrey-m if there are C x k and C 2 4

24 such that for all k c k applec C k 2 (k!) m. (2.5) The definition of Borel summability was drawn from O. Costin s book, [], which also provides other good examples and generalizes the idea of Borel summability given here to other rays besides R + and to transseries Borel Summability Applied to a Nonlinear Problem Our first example was chosen specifically to be straightforward and to motivate the definition of Borel summability. While it motivates the definition well, it fails to give a real feel for the process that is normally used in Borel summation of ODEs. we had a closed form of the function for small p, namely This occurred because p +p, which was already analytic along R +. This allowed us to easily say that the analytic continuation, F (p), is p +p and is exponentially bounded. To get a more accurate idea of how the method is normally applied, let s consider the nonlinear problem (2.8). This problem also serves as a good introduction to Borel summability of PDEs in Chapter 3. Consider y y = x 2 y 2. (2.6) If we try to develop a power series ỹ and then take the Borel transform leading to a convergent series for small p, it will be very di cult to tell what the analytic continuation would be along R +, let alone if it is exponentially bounded. So, instead we apply a formal Borel transform directly to the di erential equation (2.6). If y is a series in inverse powers of x starting with x 2 and if Y (p) :=B(y) converges for small p then by direct calculation B(y ):= py (p). Thus, formally, (p + )Y (p) = p Y Y. This can be rewritten as Y (p) = p (Y Y )(p) + =: N [Y ](p). (2.7) p + +p 5

25 We now want to show that N is a contractive operator on a space of functions that has our desired properties. Let S := {p : p < or arg(p) 2 ( +, )} for some > and small and define a norm Y =sup( + p 2 )e p Y (p). p2s Then the space A of functions which are analytic in S and continuous on its closure equipped with norm is a Banach space. Lemma 2... For some constant K independent of p p + apple K. Proof. Split the domain into two pieces. If p 2S and p < then ( + p 2 )e p p +p apple 2 e p p apple 2 sup e k k. On the other hand, since S does not contain the singularity at p = k2r, if p 2Sand p then ( + p 2 )e p p +p is exponentially small for large and bounded by K for some choice of K. Combine these two case and the Lemma follows. Lemma If Y and Y 2 are in A then so is the convolution and Y Y 2 apple M Y Y 2, where M =4. Proof. Analyticity and continuity are inherited from the analyticity and continuity of Y and Y 2. For p 2S,lets lie on the straight line from to p. Then p = p s + s and p s = p s, so [Y Y 2 ](p) apple Y Y 2 Z p apple Y Y 2 Z p apple M Y Y 2 + p 2. 6 e p [ + ( p s ) 2 ][ + s 2 ] ds e p [ + ( p s ) 2 ][ + s 2 ] ds e p

26 To see the last bound notice that by a change of variables and by using the fact that tan (s ) is bounded by 2,wehave Z p Z p /2 [ + ( p s ) 2 ][ + s 2 ] ds =2 apple 8 4+ p 2 [ + ( p s ) 2 ][ + s 2 ] ds Z p /2 +s 2 ds apple 4 + p 2. Lemma For large enough, N is a contractive mapping of the ball B = Y : Y 2A, Y < 2K onto itself, and (2.7) has a unique solution in B. Proof. Notice +p is bounded in S by. So, for C = M, Lemma 2..2 gives Hence, for Y 2 B and large enough so Y Y +p apple C Y 2. 4CK > >, N [Y ] apple K + C Y 2 apple K + 4CK2 2 < 2K and N maps B into itself. Moreover, Y Y Y 2 Y 2 = Y (Y Y 2 )+Y 2 (Y Y 2 ). Thus, N [Y ] N[Y 2 ] apple C( Y + Y 2 ) Y Y 2 apple 4CK Y Y 2. So, for large enough so 4CK > mapping theorem (2.7) has a unique solution in B. >, N is a contractive map. Thus, by the contraction Lemma For x 2 R + with x>, there exists a solution to (2.6) in the form y (x) :=L(Y )= 7 Z e xp Y (p)dp, (2.8)

27 where Y is a solution to (2.7). Moreover, for large x it has the asymptotic expansion y (x) x 2 + X Proof. The expression y = L(Y ) is convergent for x> su ciently large since e p Y (p) 2 k=3 a k x k. L (, ). Moreover, L( py )=y and L(Y Y )=y2 from an application of Fubini s theorem. Hence, y satisfies (2.6) simply because Y solves (2.7). Since Lemma 2..3 guarantees Y and Y Y are analytic at the origin, Y (P )= P k= A kp k. Then noticing that p p k = p k++ R s ( s) k ds, wehave Since the Taylor expansion of [Y Y ](p) =p X B k p k. k= p +p also begins with p, (2.7) implies A = and Y (p) = X A k p k. k= Iterating the convolution argument above shows that B = and the convolution actually starts with p 2. Thus, A = since the Taylor series of p +p begins with p. From Watson s Lemma, we have the large x expansion actually starts with p 2. Thus, A =sincethe Taylor series of p +p begins with p. From Watson s Lemma, we have the large x expansion y (x) X k= A k k! x k+ = x 2 + X a k x k. k=3 Remark The space A was chosen so that our solution Y is analytic in a ball around p =and is analytic and exponentially bounded on R +. Lemma 2..3 corresponds to the second step in our definition of Borel summation and Lemma 2..4 corresponds to the third step which recovers an actual solution to the equation. This method of applying a Borel transform directly to the di erential equation is often easier to work with than applying a Borel transform to the series itself since that latter process is unwieldy. 8

28 2..5 Stokes Phenomena It is well known that the asymptotics of solutions to linear and nonlinear ODE exhibit Stokes phenomena near irregular singular points. Definition The Stokes phenomenon is a sudden change in the asymptotic behavior of a function in the complex plane across particular rays called Stokes lines Stokes Phenomena: An Example with Laplace transform To give a simple example of this phenomenon, consider as before f(x) = Z e xp p dp (2.9) +p and the large x asymptotic behavior in di erent complex sectors. This function is analytic for arg(x) 2 ( 2, 2 ) and has an asymptotic series in /x. Further, for x 2 R+, we can deform the contour of integration in p so that f(x) = This new integral is analytic for arg(x) 2 ( Z e i /4 e xp p +p dp. 3 4, 4 ) and agrees with f(x) on the real axis, so is an analytic continuation of (2.9). We can again rotate the p contour up to get analytic continuation until arg(x) = + and f(x) = Z e i i e xp p +p dp. This is analytic for arg(x) 2 ( 3 2 +, 2 + ). However, to continue further we need to rotate past the singular ray e i R + where the singularity of the integrand occurs at p =. We notice that using complex analysis and residues Z e i i e xp p +p dp Z e i +i e xp p +p dp =2 iex. So, f(x) = Z e i +i e xp p dp +2 iex +p 9

29 which is analytic for arg(x) 2 ( continuation in x as before if desired. 3 2, 2 ), and we can continue with an analytic Thus, the asymptotic series of f changes across arg(x) = since an expansion become the integral expansion plus the exponential, 2 ie x. Further, while 2 ie x does not change the asymptotic behavior of f when arg(x) = since the exponential is small, it will become the leading term when arg(x) = the asymptotic expansion goes from a decaying power series in x 3 2. Thus, to an exponentially large leading order as the complex sector changes. 2 i is referred to as the Stokes constant at the Stokes line, arg(x) =. Remark Recall that classical Poincaré asymptotics was incomplete because it could not give any information about the exponentially small corrections. Borel summability and analysis in the Borel plane do give information about the exponential corrections and Stokes phenomena. As seen in this example, the exponentially small corrections arise from singularities in the Borel plane. Analysis of singularities in the Borel plane gives information about the full asymptotic expansion, which includes exponentially small terms. Remark The actual asymptotic expansion of f can be completely calculated using Watson s Lemma. However, we don t really need the specific expansion to understand the Stokes phenomenon, and we include it only for completeness and to remind the reader that this is the same example used to motivate the definition of Borel summability above. The asymptotic behavior of f for large x with arg(x) 2 (, ) is f X k= ( ) k+ k! x k+. However, for arg(x) 2 ( 3, ) f 2 ie x + X k= ( ) k+ k! x k+. The nonlinear solution y (x) also exhibits Stokes phenomenon. Recall from Lemma 2..4 that y (x) = Z e xp Y (p)dp, 2

30 where Y solves (2.7). In [3], Costin and Tanveer computed the behavior near p = and proved Lemma For p 2 D, := {p : p + <,arg( + p) 2 (, )}, Y (p) has the representation Y (p) = A ( + p) +p + log( + p)a ( + p)+a 2( + p) for some A and A 2 analytic near zero. So, we can follow the same steps as in the linear example. We rotate x down and p up until at arg(x) =, wepickup2 ia ()e x to the leading order as x!. Hence, S =2 ia (). Notice that unlike the linear problem the singularity at p = is a branch singularity with cut along the negative real axis from to. However, this does not change our Stokes constant calculation to the leading order, though the value of S is di erent from the linear problem Definition of Stokes Constant More generally, it is known (see []) that after a suitable change of variables a generic system of ODEs has small solutions with a convergent representation in some complex sector where all are small in the form ~y = X k,...k m k... km m ~y ~k (x), (2.2) where ~ k =(k,...,k m ), m apple n (where n is the dimension of ~y.), = C e x x for some, and ~y ~k (x) X = ~a,k x =: ỹ ~k (2.2) with ~a, =, ~a, = and ~a, = ~ (,,, ). The ỹ ~k are generally divergent and the constant C umps by S as Stokes lines arg(x) = arg( ) are crossed. If C is initially zero on one side of the Stokes line, then the constant multiplying e x x is born as one The actually problem treated in [3] is not (2.6) but is the same after taking the change of variables p! p and y! y. 2

31 crosses this Stokes line; this is the Stokes constant S. For example, the Stokes constant for f solving (2.5) at arg(x) = is 2 i. Definition A Stokes constant, S, is the ump of C across a Stokes line. Thus, each Stokes constant is associated to a ray called the Stokes line. Remark If the Stokes constant happens to be zero, that particular exponential if absent on one side of the Stokes line is not resurrected on crossing the Stokes line, i.e. the Stokes phenomenon does not in fact occur. Remark Stokes constants are related to the behavior of functions in the Borel plane. If a function is Borel Summable along R + and in the Borel plane has a single pole along a ray arg(p) = at p = p s 6=then the Stokes constant at arg(x) =, which is a Stokes line, is 2 i times the residue at the pole. The example given above with f solving (2.5) provides an illustration of this fact. For other types of singularities besides a pole, the Stokes constant at arg(x) = is still related to the singularity Stokes Phenomena: Airy Function As a second example to illustrate how we can compute Stokes constants for functions which are in integral form but not as a Laplace transform, consider the Airy equation y xy =, which has one solution A i (x), the Airy function. For x 2 R +, the Airy function is given by A i (x) = Z e t3 3 2 i C where C is a contour starting at e i 3 and ending at e i 3. We then compute the asymptotic behavior of A i for large x in di erent complex sectors using steepest descent method and get A i (x) 2 p x /4 e 2 3 x3/2 ( +...) for arg(x) 2 A i (x) 2 p x /4 e 2 i 3 x3/2 + 2 p x /4 e 2 3 x3/2 22 xt dt, 2 3, 2 3 ( +...) for arg(x) 2 2 3, 4 3.

32 Thus, the Airy function exhibits a Stokes phenomenon at with arg(x) = 2 3 being a Stokes line. We first balance the terms appearing in the exponential by using the change of variables, t = p xq. Then A i (x) = p Z x e x3/2 (q 2 i C q 3 /3) dq. Let f(q) = q q 3 /3. Near the end points of the contour f s leading behavior is real. We want to find the line of steepest descent for the exponential where all the change is happening in the real part. Or in other words, we seek the line where Im(f) =constant. Matching behavior at infinity, we want Im(f) =. Notice that this is also true all along the real axis, so our contour can pass through anywhere. You then look at the critical points for f. Notice that f (q) = q 2 so the critical points are at ±. We want our contour to pass through the critical point on R +. A Taylor expansion at q =tellsus f = 2 3 (q ) 2 3 (q )3 + O((q ) 4 ). Thus, a contour passing through this point with imaginary part must stay on the real axis or be perpendicular to the real axis. This is true at both critical points. Our end behavior rules out staying on the real axis, and we now know the steepest descent contour comes down and passes perpendicularly through q =. Because f is now monotone along our contour choice, the asymptotic behavior is given by A i (x) p Z x + e x3/2 (q q3 /3) dq, 2 i + for some complex with arg( ) 6= so that + is on the steepest descent contour, = Im(q q 3 /q) and is small. We again use the Taylor series and let = (q ) 2 3 (q )3. Then q ( ) 2 p. Splitting the integral into two similar pieces gives A i (x) Applying Watson s lemma, we have p Z x e 2/3x3/2 e x3/2 2 p d. (2.22) A i (x) p x e 2/3x3/2 2 (/2)(x3/2 ) /2 = 2 p x /4 e 2 3 x3/2. 23

33 for arg(x) 2 3, 3. Now that we have the contour of steepest descent, we play the same rotating game as in the first example to produce the Stokes phenomenon. Here the match is to rotate the contour for di erent x to keep the leading order real, arg(q 3 x 3/2 ) =. When arg(x) = 2 3,theq contour has swung /3 in the negative direction. Now, our contour starts at e 2 i/3 and connects to on R +. The only way to do this and keep f real is to pass through both critical points, picking up exponential corrections from both of them. As the contour at q = comes and leaves at right angles, we will pick up only half of the contribution from Watson s lemma. Rotate a little more and we get the full contribution from Watson s lemma at both critical points. So, when arg(x) 2 2 3, 4 3 our contour passes fully through each critical point and gives A i (x) 2 p x /4 e 2 i 3 x3/2 + 2 p x /4 e 2 3 x3/2 for arg(x) 2 2 3, 4 3 Remark Notice that after an appropriate change of variables this example follows. the first one. Thus, (2.22) shows us that the Airy function is integral which can be treated as in our first example. p x e 2/3x3/2 times a Laplace 2.2 Classical Results for the Navier-Stokes Equation The first part of this thesis is concerned with two nonlinear PDEs governing fluid flow in certain settings. These equations are the Boussinesq equation which governs constant density fluid flow where local temperature causes a buoyancy force and the MHD equation where a magnetic field provides a body force on the constant density fluid. A limiting case as a! (or µ!) of the Boussinesq equation (or the MHD equation) is the u +(u r)u = rp + u + f, with r u = and u(x, ) = u (x), where u : R d R +! R d, p : R d R +! R d. The incompressible Navier-Stokes equation governs the motion of fluids in a wide range of situations. A physical derivation of the equation is given in Appendix A. Before describing specific results for the Boussinesq and 24

34 MHD equations, it is appropriate to briefly summarize results for the incompressible Navier- Stokes equation since the mathematical methods are similar. The global existence of smooth solution in 3-D to the Navier-Stokes equation is a formidable open problem despite having been extensively studied in the past 85 years. There are many monographs on the Navier-Stokes equation which focus on di erent aspects of the known results and approaches, see for example [5], [25], [8]. In short, for the 3-D Navier-Stokes equation, global weak solutions have been known to exist since Leray s work in the 93 s (see [5]). Smooth solutions are known to exist on a time interval [,T] where T scales inversely to a Sobolev norm of the initial condition and forcing, [3], and global solutions exist for su ciently large Reynolds number (small viscosity ) or su ciently small initial condition and forcing. However, without these assumptions, the question of global existence in 3-D remains open. In the following, we present some of the known fundamental existence theorems. We also highlight the di erence between the 2-D and 3-D case and the di culty in proving global existence in 3-D Existence of Weak Solutions One of the classical methods of proving existence of solutions to PDEs for a finite domain is Galerkin approximation, which is standard material for a graduate level introduction to PDE course. For many PDEs, Galerkin approximation can be combined with energy estimates to prove the existence of solutions. In general, the energy methods require a set of apriori estimates for energy norms of solutions to the regularized system of equations given by Galerkin approximation, for which ODE theory gives a solution. For infinite domains, mollification is often used or combined with arguments on a sequence of finite open subsets, O n, of the infinite domain. Then compactness arguments allow us to take the regularization limit to prove existence of solutions. While the compactness results for finite domains often do not hold on infinite domains, by restricting to a finite subdomain O we get local results which are su cient to pass to the regularization limit. Classically, this method can be applied to the Navier-Stokes equation and more general equations such as ours, where the highest order derivative is a 25