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BELTRAMI FLOWS a Thesis Submitted to the Kent State University Honors BELTRAMI FLOWS Athesissubmittedtothe Kent State University Honors College in partial fulfillment of the requirements for Departmental Honors by Alexander Margetis May, 2018 Thesis written by Alexander Margetis Approved by , Advisor , Chair, Department of Mathematics Accepted by , Dean, Honors College ii Table of Contents Acknowledgments.................................. iv 1 Introduction .................................. 1 2 Sobolev-Spaces & the Lax-Milgram theorem ............. 2 2.0.1 HilbertspaceandtheLax-Milgramtheorem . 3 2.0.2 Dirichlet Problem . 5 2.0.3 NeumannProblem ........................ 8 3 Constructing a solution of (1.1) ...................... 11 3.1 The bilinear form . 12 3.2 A solution of (3.3) results in a solution of (3.1) . 16 References . 19 iii Acknowledgments I would like to thank my research advisor, Dr. Benjamin Jaye for everything he has done. His support and advice in this thesis and everything else has been invaluable. I would also like to thank everyone on my defense committee. I know you did not have to take that time out of your busy schedule. iv Chapter 1 Introduction ABeltramiflowinthree-dimensionalspaceisanincompressible(divergencefree) vector field that is everywhere parallel to its curl. That is, B = curl(B)= r⇥ λB for some function λ.Theseflowsarisenaturallyinmanyphysicalproblems. In astrophysics and in plasma fusion Beltrami fields are known as force-free fields. They describe the equilibrium of perfectly conducting pressure-less plasma in the presence of a strong magnetic field. In fluid mechanics, Beltrami flows arise as steady states of the 3D Euler equations. Numerical evidence suggests that in certain regimes turbulent flows organize into a coherent hierarchy of weakly interacting superimposed approximate Beltrami flows [PYOSL]. Given the importance of Beltrami fields, there are several approaches to proving existence of solutions, for instance use the calculus of variations [LA], and use fixed point arguments [BA]. In this thesis we instead use a Hilbert space approach through the Lax-Milgram lemma. Our goal will be to find a weak solution to the Beltrami flow, for a constant λ in ⌦andaboundaryfunctiong on @⌦satisfying @⌦ gdS =0, R B = λB in ⌦ 8r⇥ > (1.1) > B =0in⌦ >r· <> B n = g on @⌦ > · > > :> 1 Chapter 2 Sobolev-Spaces & the Lax-Milgram theorem Sobolev Spaces give rise to what are called weak solutions to di↵erential equations - and do so because they are based on the notion of the weak derivative. We recall the classical integration by parts formula for an open set ⌦with a smooth boundary: If f and g are continuously di↵erentiable functions in ⌦(we shall write C1(⌦) for the collection of such functions), then @f @g (2.1) gdx = f dx + fgn dS, @x − @x i Z⌦ j Z⌦ j Z@⌦ where n =(n1,n2,n3)istheoutwardunitnormalto⌦. Wesaythatanintegrable function f has a weak derivative Djf if Djf is an integrable function, and for any C1(⌦) function g with g =0on@⌦, @g (D f)gdx = f dx. j − @x Z⌦ Z⌦ j We will see that by moving away from the classical, or strong, derivative we will be able to prove some quite powerful theorems about the existence of solutions to uniformly elliptic equations. The Lax–Milgram theorem, gives conditions under which a bilinear function can be ”inverted” to show the existence and uniqueness of a weak solution to a given boundary value problem. 2 3 2.0.1 Hilbert space and the Lax-Milgram theorem Definition 2.0.1. An inner product space is a vector space V over R equipped with a function , : V V R satisfying: h· ·i ⇥ ! 1. (Bilinearity) af + bg, h = a f,h + b g, h , and f, ag + bh = a f,g + b f,h , h i h i h i h i h i h i for any f,g,h V and a, b R 2 2 2. (Symmetry) f,g = g, f for any x, y V . h i h i 2 3. (Non-degeneracy) For f V , f,f > 0 for f =0. 2 h i 6 Given an inner product space (V, , ), we can associate a norm to V through the h i formula f = f,f . k k h i p The function satisfies the following three properties k·k 1. (Non-degeneracy) f =0ifandonlyiff =0. k k 2. (Homoegeneity) if f V and λ F,then λf = λ f . 2 2 k k | |k k 3. (The triangle inequality) if f,h V ,then f + g f + g . 2 k kk k k k Therefore, we may define the distance between f and g as f g .Thisgivesus k − k notions of convergence and continuity, etc. We say that V is a Hilbert space if it is a complete inner product space, meaning that every sequence f that is Cauchy, i.e. { n} for every ">0thereexistsN N (2.2) 2 such that f f <"for every n, m N, k n − mk ≥ 4 has a limit in V ,meaningthatthereisf V with 2 for every ">0thereexistsN N such that fn f <"for every n N. 2 k − k ≥ The most common example of a Hilbert space that we shall see is a Sobolev space. We first define L2(⌦) as the collection of (Lebesgue) function f :⌦ R with ! f 2dx < .WeputaninnerproductonL2(⌦) with the formula ⌦ | | 1 R f,g = fgdx. h i2 Z⌦ The Cauchy-Schwarz inequality 1/2 1/2 (2.3) f g dx f 2dx g 2dx for any f,g L2(⌦) ⌦ | || | ⌦ | | ⌦ | | 2 Z ⇣Z ⌘ ⇣Z ⌘ ensures that f,g is finite if f,g L2(⌦). We next define h i2 2 W 1,2(⌦) = f such that f L2(⌦) and, for every j =1, 2, 3 (2.4) 2 n the weak derivative D f L2(⌦) . j 2 o We can endow this space with an inner product via 3 f,g 1,2 = f,g + D f,D g , h iW h i2 h j j i2 j=1 X where each term makes sense because of (2.3). With an abuse of notation, we define the weak gradient f =(D ,f,D f,D f), r 1 2 3 5 and set 3 f, g = D f,D g . hr r i2 h j j i2 j=1 X We next state the Lax-Migram lemma, which may be found in Chapter 6 of the book by Evans [E]. Theorem 2.0.2 (Lax–Milgram Lema). Suppose that V is a Hilbert space, and : A V V R satisfies the properties: ⇥ ! 1. is continuous: there is γ>0 such that (v, φ) γ v φ for every v, φ A |A | || |||| | 2 V . 2. is elliptic: there is ↵>0 such that (u, u) ↵ u 2 for every u V . A A ≥ || || 2 Then, for every L : V R that is continuous in the sense that there is λ>0 such ! that L(φ) λ φ , there is a unique solution u V of (u, φ)=L(φ) for every | | || ||v 2 A φ V . 2 We first show how the Lax-Milgram lemma can be used to solve some classical problems in PDE. 2.0.2 Dirichlet Problem Define the Laplacian ∆u =div( u). For a function f L2(⌦), our goal is to find r 2 asolutionu of the following problem ∆u = f in ⌦ (2.5) 8− > <>u =0 on@⌦ > :> 6 For the Dirichlet Problem we pick the function space V = W 1,2(⌦) = u W 1,2,u=0 in @⌦ , 0 2 1 which is formally the closure of the space C (⌦) with u =0on@! in the norm 1,2 k·kW associated to the Sobolev inner product , 1,2 introduced in the last chapter. We h· ·iW choose and L as A (v, φ)= u φdxand L(φ)= fφdx. A r ·r Z Z Assuming for the moment that we can verify conditions (1) and (2) of the Lax- Milgram theorem, we would arrive at a function v such that u φdx = fφdx for all φ W 1,2(⌦). r ·r 2 0 Z⌦ Z⌦ This is considered a weak solution of the Dirichlet problem (2.5) because if we were permitted to integrate by parts on the left hand side, we would arrive at (due to the zero boundary condition) ( ∆u)φdx = fφdx for all φ W 1,2(⌦), − 2 0 Z⌦ Z⌦ that is, ∆u = f in ⌦. − 7 We now return to check (1) and (2). Applying the Cauchy-Schwarz inequality (v, φ) = u φdx |A | | r ·r Z 1/2 1/2 (2.6) v 2 dx φ 2 dx |r | |r | ⇣Z ⌘ ⇣Z ⌘ v 1,2 φ 1,2 , || ||W || ||W we get that a is continuous so (1) holds. For property (2), we observe that (u, u)= u udx= u 2 dx. A r ·r |r | Z⌦ Z⌦ Therefore, in order to establish (2), we need to compare u 2 dx with the ⌦ |r | 2 2 2 Sobolev norm u 1,2 = u dx + u dx. Noticing thatR the Sobolev norm k kW (⌦) ⌦ ⌦ |r | has an extra term in it, we needR an inequalityR of the following form: Theorem 2.0.3 (Sobolev). There exists C0 > 0 (depending on ⌦) such that (2.7) u2 dx C u 2 dx for u W 1,2(⌦). 0 |r | 2 0 Z⌦ Z⌦ Inequalities such as (2.7) are called either Friedrich, Sobolev, or Poincar´einequal- ities – we shall state a number of them in this thesis and refer to Chapter 1 of [Maz] for the proofs. Notice that the obstacle to achieving a statement such as (2.7) are constant functions, since if u is constant then u = 0. However, the only constant r 1,2 function in W0 (⌦) is the 0 function, so we do not face this obstacle. Sobolev and Poincar´einequalities essentially state that the constant functions are the only obsta- cle to obtaining an inequalities which says that the integral of a function is dominated 8 by the integral of its (weak) gradient.
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