Lectures on Wave Equation

This is a note for the lectures given on Oct. 21st and 23rd, 2014 in lieu of D. Tataru, for the course MAT222 at UC Berkeley. 1. Wave equation The purpose of these lectures is to give a basic introduction to the study of linear wave equation. Let d ≥ 1. The wave operator, or the d’Alembertian, is a second order partial differential operator on R defined as (1.1) := −∂ t + ∂ x1 + · · ·+ ∂ xd = −∂ 2 t +4, where t = x is interpreted as the time coordinate, and x, · · · , x are the coordinates for space. The corresponding PDE is given by (1.2) φ = F, where φ and F are, in general, real-valued distributions on an open subset of R. As usual, when the forcing term F is absent, we call (1.2) the homogenous wave equation. In general, (1.2) is referred to as the inhomogeneous wave equation. As suggested by our terminology, the wave equation (1.2) is a evolutionary PDE, and a natural problem to ask is whether one can solve the initial value (or Cauchy) problem: (1.3) { φ =F, (φ, ∂tφ) {t=0}=(φ0, φ1). We will use the notation Σt for the constant t-hypersurface in R; hence Σ0 = {t = 0}. We are being deliberately vague about the function spaces that φ, φ0 and φ1 live in; we will give a more concrete description as we go on. Remark 1.1. Note that we prescribe not only φ(0) but also its time derivative ∂tφ(0). This is necessary because (1.2) is second order in time. Observe that prescription of φ(0) and ∂tφ(0) is enough to determine all derivatives of φ at Σ0, and we can write down the formal power series of φ at each point on Σ. If φ0, φ1 and F are analytic, then these formal power series would converge and give a local solution to (1.3) by the Cauchy-Kowalevski theorem. The wave equation models a variety of different physical phenomena, including: • Vibrating string. It was for this example that (1.2) (with F = 0 and d = 1) was first derived by Jean-Baptiste le Rond d’Alembert. • Light in vacuum. From Maxwell’s equation in electromagnetism, it can be seen that each component of electric and magnetic fields satisfies (1.2) with F = 0 and d = 3. 1 • Propagation of sound. The wave equation (1.2) arises as the linear approximation of the compressible Euler equations, which describe the behavior of compressible fluids (e.g., air). • Gravitational wave. A suitable geometric generalization of the wave equation (1.2) turns out to be the linear approximation of the Einstein equations, which is the basic equation of the theory of general relativity for gravity. Needless to say, a good understanding of the linear operator (1.1) is fundamental for the study of any of the above topics in depth. Our goal is to present basics of analysis of the d’Alembertian . We will introduce three approaches: (1) Fourier analytic method, (2) Energy integral method, (3) Approach using fundamental solution. Each has its own strength and weakness, but nevertheless they all turn out to be useful in further studies. For a systematic introduction to wave equations, it will be natural to have a discussion of the symmetries of (1.1) at this point. However, as this is a lecture with time constraint, we will be in favor of a quicker introduction and simply jump right into the analysis, deriving the symmetries of (1.1) that we need as we go on. By taking this route, it is hoped that the central role of the symmetries in the study of (1.1) would appear naturally. 2. Fourier analytic method Note that (1.1) is a constant coefficient partial differential operator; therefore, translations in time and space commute with , i.e., (2.1) ( φ(t+ ∆t, x, . . . , x) ) = ( φ)(t+ ∆t, x, . . . , x), ( φ(t, x, . . . , x + ∆x, . . . , x) ) = ( φ)(t, x, . . . , x + ∆x, . . . , x), This property suggests that Fourier analysis will be effective for studying , since Fourier analysis exploits the global translation symmetries of R. Indeed, the Fourier analytic method turns out to be the quickest of the three for solving (1.3), and it will be the subject of our discussion below. Applying Fourier transform in x to (1.2), we obtain the equation (2.2) ∂ t φ̂(t, ξ) + |ξ|φ̂(t, ξ) = F̂ (t, ξ). Fix ξ ∈ R such that ξ 6= 0; then the preceding equation is a second order ODE in t. We easily checked that {eit|ξ|, e−it|ξ|} forms a fundamental system for this ODE. Using the variation of constants formula, we see that a solution to (2.2) for each ξ is given by (2.3) φ̂(t, ξ) = c+e it|ξ| + c−e −it|ξ| + ∫ t 0 ( eF̂+(s, ξ) + e F̂−(s, ξ) ) ds, 1We are using the convention f̂(ξ) = ∫ f(x)e−ix·ξ dx and f(x) = ∫ f̂(ξ)eix·ξ dξ (2π)d for the Fourier transform. 2 where c± are to be determined from the initial data (φ̂0, φ̂1), and F̂± can be computed from F̂ . Carrying out the algebra using Euler’s identity e±it|ξ| = cos(t|ξ|)± i sin(t|ξ|), we can rewrite the preceding formula as follows: (2.4) φ̂(t, ξ) = cos(t|ξ|)φ̂0(ξ) + sin(t|ξ|) |ξ| φ̂1(ξ) + ∫ t 0 sin((t− s)|ξ|) |ξ| F̂ (s, ξ) ds. The formula (2.4) describes the evolution of a single Fourier mode f̂(ξ) under the wave equation (1.2) for every ξ 6= 0. Combining this result for different ξ’s under the assumption that (φ0, φ1) ∈ H × Hk−1 and F ∈ Lt ([0, T ];Hk−1 x ) (which is natural in view of the Plancherel theorem), we obtain the following solvability result for the wave equation: Theorem 2.1 (Solvability of wave equation). Let k ∈ Z+ := {1, 2, . . .} and T > 0. The initial value problem (1.3) is solvable on [0, T ] × R for (φ0, φ1) ∈ H × Hk−1 and F ∈ Lt ([0, T ];H k−1 x ) with a unique solution φ(t, x) ∈ Ct([0, T ];H x)∩C t ([0, T ];Hk−1 x ). The spatial Fourier transform φ̂(t, ξ) of φ(t, x) is described by the formula (2.4). Proof. The existence of a solution follows from simply verifying that φ given by (2.4) solves the equation (1.2). The fact that the solution φ belongs to Ct([0, T ];H k x) ∩ C t ([0, T ];Hk−1 x ) is a consequence of the following energy inequality : (2.5) ‖(φ, ∂tφ)(t)‖Hk x×H x ≤ C‖(φ0, φ1)‖Hk x×H x + C ∫ t