i ii Abstract Motivated by the Navier-Stokes equations, which are a set of unsolved equations related to fluid motion in R, we explored the incompressibility condition and the Neumann boundary problem. After exploring, we noticed that using iterated Riesz transforms of the boundary data could be used to get information about the velocity field directly from the boundary data. We then used this same logic to look at the same problem in a spherical space, and found no definition for the Riesz transforms. By using the incompressibility condition and solving the Neumann boundary problem on a sphere, using both separation of variables and potential theory, we can define the Riesz transforms of a function on the sphere. This allows us to use boundary data on a sphere to describe the divergence-free field, or in the case of the incompressible Navier-Stokes equation, allows us to solve for the velocity field given initial data. 1/29 1 Introduction 1.1 Motivation and Objective One of the great, unanswered questions in physics and mathematics is related to solving the incompressible Navier-Stokes equations in R3. The Clay Mathematics Institute has put up a one-million dollar prize for a correct, verified solution [5]. As a Millennium Prize Problem, it is an important problem that is currently being used to design a number of practical things by using "approximations" that idealize the systems they're modeling. This includes many sorts of fluid dynamics problems such as designing boat hulls, airplanes, aerodynamic cars, and other systems. The motivation to define the Riesz transforms on the sphere comes from the motion of fluids, which are described by the 3-dimensional incompressible Navier-Stokes equations. We will be using the Riesz transform, which we will define on both the interior and exterior of the sphere, so that we can use the boundary data we measure around a sphere to solve for the velocity field in the incompressible Navier-Stokes equations. By defining the Riesz transforms on the sphere, we can use the result of the transforms to explore systems using measurable physical data such as the velocity of water passing over a surface. So we will begin by discussing the Navier-Stokes equations, what they mean, and then how they relate to Riesz transforms. From there, we discuss the definition of the Riesz transform and how it is related to finding solutions to the equation. Then we will solve our Neumann boundary problem on the sphere, and we will find a result for the Riesz transform, in spherical coordinates, related to Neumann boundary data. Our goal was to find a definition for Riesz transforms on the sphere in both infinite series and integral forms using analysis and potential theory. Although Riesz transforms are well understood and have applications in mathematics and physics, we were unable to find any explicit representations for the Riesz transforms on the sphere. In the following sections, we show you the background we used to provide the basis of our definition, the steps we followed to get there, and then show the result and its potential uses. 1.2 Navier-Stokes Equations 3 The Navier-Stokes equations are used to describe fluid velocities u(x; t) = (u1(x; t); u2(x; t); u3(x; t)); x 2 R ; t ≥ 0; and will include some initial data u(x; 0) = u0(x). These equations are based on Newton's laws of motion and conservation of mass for fluids of constant density. The Navier-Stokes equation can be expressed as: 2/29 @uu ρ + (u · r)u = νr2u − rp + ff; r · u = 0; u(x; 0) = u (x): (1) @t 0 P3 @ The definitions of the operators are: r = (@=@x1; @=@x2; @=@x3), u · r = uj , and j=1 @xj 2 P3 2 2 r = r · r = j=1 @ =@xj , where the last two are evaluated component-wise for the vector functions. The term ρ is the density, p(x; t) is a scalar function representing pressure, and f (x; t) represents external force, and ν > 0 is the kinetic viscosity. The incompressibility condition is the requirement that r · u = 0. These equations make up a non-linear system of four partial differential equations that govern four unknown scalars, three coordinates of velocity (u1; u2; u3) and a scalar quantity for pressure. The equations are used to model the motion of a viscous, incompressible, homogeneous fluid in free space and are derived from conservation of mass and momentum of packets of fluid. We place these under the assumption that stress and strain rates have a localized, linear relationship among these packets of fluid. The left hand side of (1) represents the acceleration in a reference frame where one views the motion as a parade these fluid packets where ρ is the mass density and u(x; t) is the velocity of the fluid packet at that location and time. The right hand side represents the forces acting on the fluid packet. The first term νr2u represents the viscous friction forces, which is a result of the shear stresses from interactions between the molecules. The term f ≡ f (x; y; z; t) represents body forces, such as gravity or other external fields. The −∇p term defines the pressure force on a fluid packet, and is the gradient of the pressure, which is a force that maintains the homogeneous fluid density under our incompressibility condition [2]. 1.3 Riesz Transforms on R3 Riesz transforms are a multi-dimensional generalizations of the Hilbert transform, which means that Riesz transforms are singular integral operators that consist of a convolution of one function with another that has a singularity at the origin. These transforms are important in the studying of harmonic potentials in harmonic analysis and potential theory. This makes them useful for many different systems, and are connected to many areas of study, but it has not been characterized for many spaces and effectively has not been explored or investigated much beyond Marcel Riesz's original descriptions. The most famous form of the Riesz transform on Rn is the singular integral: [3] 3/29 Z yj Rj(f)(x) = lim cn n+1 f(x − y)dy; (j = 1; :::; n) (2) "!1 jy|≥" jyj n+1 Γ( 2 ) yj where the normalization constant is cn = π(n+1)=2 , the non-constant part of the Riesz kernel is yn+1 , and f(x) is the function to be transformed. However, there are other ways of describing the definition of the Riesz transform, and we will discuss those later. 4/29 2 Background and Methods 2.1 Incompressibility Following all the information so far, one may ask why we can justify using the incompressibility condition we stipulated, that r · u = 0. The discussion for this begins with the idea of mass conservation. In our system, we are not creating or destroying mass, we are simply looking at a fixed subspace of some larger region we care about. Let W be a fixed subspace of D, which is a region in R3 that is filled with some fluid. The rate of change of mass in W with respect to time is d Z @ρ m(W; t) = (x; t)dV; (3) dt W @t where dV is the volume element of W , ρ is the mass density of the fluid, and we can take the time derivative inside the integral because our space W is fixed and does not change with time. It's easy to see that the rate of change of mass in W is equal to the rate at which mass is crossing the boundaries of W or @W . Using equation (3) and the divergence theorem, we can show that Z @ρ Z Z (x; t)dV = − ρu · ηdA = − r · (ρu)dV (4) W @t @W W and finally Z @ρ + r · (ρu)dV = 0: (5) W @t Since this is to hold for all W, we can generate the continuity equation for this flux, resulting in @ρ + r · (ρu) = 0 (6) @t If we then decide that our fluid is homogeneous, which is to say that our density (ρ) is a constant, then it must follow that r · u = 0 is true for equation (6) to be true. Because of this, we can justify using r · u = 0 to solve this problem for incompressible fluids. 5/29 2.2 Fourier Transform on Rn It's useful to look at Fourier transforms now because we will use them in further definitions and because they are helpful for understanding what transforms can do and are used for. The Fourier transform is a transform that takes a "time-based" function and decomposes it into the frequencies that describe the function. This method of describing the same function using different variables and form means that you can more easily identify different parts of the equation or perform different operations that are easier. The Fourier transform is typically defined as Z 1 F(f)(ξ) = ce−iξxf(x)dx: (7) −∞ where c is a normalization constant. Using this transform allows us to take a function in "x space" and convert it to "ξ space." As a result, we can see that @f F = −iξjF(f) (8) @xj is true as well. We will use these equations and their properties for helping show how the Riesz transform is defined. 2.3 One Way to Define Riesz Transform For us to move forward with developing a definition for the Riesz transform on the sphere, it is important for us to show how we came up with the justification for our form for the transform.