3.2 Linear problems with variable coefficients

Next, we generalize the theory to linear evolution problems with variable coefficients. That is, we consider equations of the following form:
∑ ut = P (t,x,∂∕ ∂x)u ≡ Aν(t,x)D νu, x ∈ ℝn, t ≥ 0, (3.72 ) |ν|≤p
where now the complex m × m matrices A ν(t,x) may depend on t and x. For simplicity, we assume that each matrix coefficient of A ν belongs to the class C∞ ([0,∞ ) × ℝn) b of bounded, C∞-functions with bounded derivatives. Unlike the constant coefficient case, the different k-modes couple when performing a Fourier transformation, and there is no simple explicit representation of the solutions through the exponential of the symbol. Therefore, Definition 1 of well-posedness needs to be altered. Instead of giving an operator-based definition, let us define well-posedness by the basic requirements a Cauchy problem should satisfy:

Definition 3. The Cauchy problem

u(t,x) = P (t,x,∂∕∂x )u(t,x), x ∈ ℝn, t ≥ 0, (3.73 ) t u(0,x ) = f(x), x ∈ ℝn, (3.74 )
is well posed if any f ∈ C∞ (ℝn ) 0 gives rise to a unique C ∞-solution u(t,x ), and if there are constants K ≥ 1 and α ∈ ℝ such that
αt ∥u (t,⋅)∥ ≤ Ke ∥f ∥ (3.75 )
for all f ∈ C ∞ (ℝn) 0 and all t ≥ 0.

Before we proceed and analyze under which conditions on the operator P (t,x, ∂∕∂x ) the Cauchy problem (3.73View Equation, 3.74View Equation) is well posed, let us make the following observations:

3.2.1 The localization principle

Like in the constant coefficient case, we would like to have a criterion for well-posedness that is based on the coefficients Aν(t,x) of the differential operator alone. As we have seen in the constant coefficient case, well-posedness is essentially a statement about high frequencies. Therefore, we are led to consider solutions with very high frequency or, equivalently, with very short wavelength. In this regime we can consider small neighborhoods and since the coefficients A ν(t,x) are smooth, they are approximately constant in such neighborhoods. Therefore, intuitively, the question of well-posedness for the variable coefficient problem can be reduced to a frozen coefficient problem, where the values of the matrix coefficients Aν(t,x) are frozen to their values at a given point.

In order to analyze this more carefully, and for the sake of illustration, let us consider a first-order linear system with variable coefficients

∑n ut = P (t,x, ∂∕∂x )u ≡ Aj (t,x)-∂--u + B (t,x)u, x ∈ ℝn, t ≥ 0, (3.77 ) j=1 ∂xj
where A1,...,An, B are complex m × m matrices, whose coefficients belong to the class C ∞b ([0,∞ ) × ℝn ) of bounded, C ∞-functions with bounded derivatives. As mentioned above, the Fourier transform of this operator does not yield a simple, algebraic symbol like in the constant coefficient case.7 However, given a specific point p0 = (t0,x0) ∈ [0,∞ ) × ℝn, we may zoom into a very small neighborhood of p0. Since the coefficients Aj (t,x) and B (t,x ) are smooth, they will be approximately constant in this neighborhood and we may freeze the coefficients of j A (t,x) and B (t,x) to their values at the point p0. More precisely, let u (t,x ) be a smooth solution of Eq. (3.77View Equation). Then, we consider the formal expansion
u(t0 + 𝜀t,x0 + 𝜀x) = u (t0,x0) + 𝜀u(1)(t,x) + 𝜀2u(2)(t,x) + ..., 𝜀 > 0. (3.78 )
As a consequence of Eq. (3.77View Equation) one obtains
∑n [ (1) (2) ] u(1)(t,x) + 𝜀u(2)(t,x) + ...= Aj (t + 𝜀t,x + 𝜀x) ∂u---(t,x) + 𝜀 ∂u--(t,x ) + ... t t 0 0 ∂xj ∂xj j=1 [ ] + B(t0 + 𝜀t,x0 + 𝜀x ) u(t0,x0) + 𝜀u(1)(t,x) + ... .
Taking the pointwise limit 𝜀 → 0 on both sides of this equation we obtain
∑ n ∂u(1) u(t1)(t,x ) = Aj (t0,x0) ---j-(t,x) + F0 = P0 (t0,x0, ∂∕∂x )u (1)(t,x) + F0, (3.79 ) j=1 ∂x
where F0 := B (t0,x0)u(t0,x0). Therefore, if u is a solution of the variable coefficient equation ut = P (t,x,∂∕∂x )u, then, u(1) satisfies the linear constant coefficient problem u (1t)(t,x ) = P0(t0,x0, ∂∕∂x )u(1) + F0 obtained by freezing the coefficients in the principal part of P (t,x,∂∕∂x )u to their values at the point p 0 and by replacing the lower-order term B(t,x)u by the forcing term F0. By adjusting the scaling of t, a similar conclusion can be obtained when P (t,x,∂∕∂x ) is a higher-derivative operator.

This leads us to the following statement: a necessary condition for the linear, variable coefficient Cauchy problem for the equation u = P(t,x,∂ ∕∂x)u t to be well posed is that all the corresponding problems for the frozen coefficient equations vt = P0(t0,x0,∂∕ ∂x)v are well posed. For a rigorous proof of this statement for the case in which P(t,x,∂ ∕∂x) is time-independent; see [397Jump To The Next Citation Point]. We stress that it is important to replace P (t,x, ∂∕∂x ) by its principal part P0(t,x,∂ ∕∂x) when freezing the coefficients. The statement is false if lower-order terms are retained; see [259Jump To The Next Citation Point, 397Jump To The Next Citation Point] for counterexamples.

Now it is natural to ask whether or not the converse statement is true: suppose that the Cauchy problems for all frozen coefficient equations vt = P0(t0,x0,∂∕ ∂x)v are well posed; is the original, variable coefficient problem also well posed? It turns out this localization principle is valid in many cases under additional smoothness requirements. In order to formulate the latter, let us go back to the first-order equation (3.77View Equation). We define its principal symbol as

∑n P0(t,x,ik) := i Aj (t,x)kj. (3.80 ) j=1
In analogy to the constant coefficient case we define:

Definition 4. The first-order system (3.77View Equation) is called

  1. weakly hyperbolic if all the eigenvalues of its principal symbol P0(t,x,ik) are purely imaginary.
  2. strongly hyperbolic if there exist M > 0 and a family of positive definite, Hermitian m × m matrices H (t,x,k), (t,x, k) ∈ Ω × Sn −1, whose coefficients belong to the class Cb∞(Ω × Sn−1), such that
    − 1 ∗ M I ≤ H (t,x, k) ≤ M I, H (t,x,k)P0(t,x,ik) + P0(t,x,ik) H (t,x,k) = 0, (3.81)
    for all (t,x, k) ∈ Ω × Sn− 1, where Ω := [0,∞ ) × ℝn.
  3. symmetric hyperbolic if it is strongly hyperbolic and the symmetrizer H (t,x,k) can be chosen independent of k.

We see that these definitions are straight extrapolations of the corresponding definitions (see Definition 2) in the constant coefficient case, except for the smoothness requirements for the symmetrizer H (t,x,k).8 There are examples of ill-posed Cauchy problems for which a Hermitian, positive-definite symmetrizer H (t,x,k) exists but is not smooth [397] showing that these requirements are necessary in general.

The smooth symmetrizer is used in order to construct a pseudo-differential operator

∫ ∫ ---1--- ik⋅x n --1---- −ik⋅x n [H (t)v](x ) := (2π )n∕2 H (t,x,k ∕|k |)e ˆv(k)d k, ˆv(k) = (2 π)n∕2 e v(x)d x, (3.82 )
from which one defines a scalar product (⋅,⋅)H(t), which, for each t, is equivalent to the L2 product. This scalar product has the property that a solution u to the equation (3.77View Equation) satisfies an inequality of the form
d --(u,u)H (t) ≤ b(T )(u,u)H (t), 0 ≤ t ≤ T, (3.83 ) dt
see, for instance,  [411Jump To The Next Citation Point]. Upon integration this yields an estimate of the form of Eq. (3.75View Equation). In the symmetric hyperbolic case, we have simply [H (t)v] = H (t,x)v(x) and the scalar product is given by
∫ (u,v)H (t) := u(x)∗H (t,x)v(x)dnx, u,v ∈ L2 (ℝn). (3.84 )
We will return to the application of this scalar product for deriving energy estimates below. Let us state the important result:

Theorem 3. If the first-order system (3.77View Equation) is strongly or symmetric hyperbolic in the sense of Definition 4, then the Cauchy problem (3.73View Equation, 3.74View Equation) is well posed in the sense of Definition 3.

For a proof of this theorem, see, for instance, Proposition 7.1 and the comments following its formulation in Chapter 7 of [411]. Let us look at some examples:

Example 18. For a given, stationary fluid field, the non-relativistic, ideal magnetohydrodynamic equations reduce to the simple system [120Jump To The Next Citation Point]

Bt = ∇ ∧ (v ∧ B ) (3.85 )
for the magnetic field B, where v is the fluid velocity. The principal symbol for this equation is given by
P0 (x,ik)B = ik ∧ (v(x) ∧ B ) = (ik ⋅ B )v(x) − (ik ⋅ v(x))B. (3.86 )
In order to analyze it, it is convenient to introduce an orthonormal frame e1,e2,e3 such that e1 is parallel to k. With respect to this, the matrix corresponding to P (x, ik ) 0 is
( ) 0 0 0 i|k|( v2(x) − v1(x) 0 ) , (3.87 ) v3(x) 0 − v1(x)
with purely imaginary eigenvalues 0, − i|k |v1(x). However, the symbol is not diagonalizable when k is orthogonal to the fluid velocity, v (x ) = 0 1, and so the system is only weakly hyperbolic.

One can still show that the system is well posed, if one takes into account the constraint ∇ ⋅ B = 0, which is preserved by the evolution equation (3.85View Equation). In Fourier space, this constraint forces B1 = 0, which eliminates the first row and column in the principal symbol, and yields a strongly hyperbolic symbol. However, at the numerical level, this means that special care needs to be taken when discretizing the system (3.85View Equation) since any discretization, which does not preserve ∇ ⋅ B = 0, will push the solution away from the constraint manifold, in which case the system is weakly hyperbolic. For numerical schemes, which explicitly preserve (divergence-transport) or enforce (divergence-cleaning) the constraints, see [159] and [136], respectively. For alternative formulations, which are strongly hyperbolic without imposing the constraint; see [120].

Example 19. The localization principle can be generalized to a certain class of second-order systems [261][308Jump To The Next Citation Point]: For example, we may consider a second-order linear equation of the form

n 2 n n v = ∑ Ajk (t,x )--∂----v + ∑ 2Bj(t,x)-∂--v + ∑ Cj (t,x)-∂--v + D (t,x)v + E (t,x)v,(3.88 ) tt ∂xj∂xk ∂xj t ∂xj t j,k=1 j=1 j=1
x ∈ ℝn, t ≥ 0, where now the m × m matrices Ajk, Bj, Cj, D and E belong to the class C ∞ ([0,∞ ) × ℝn ) b of bounded, C ∞-functions with bounded derivatives. Zooming into a very small neighborhood of a given point p0 = (t0,x0) by applying the expansion in Eq. (3.78View Equation) to v, one obtains, in the limit 𝜀 → 0, the constant coefficient equation
∑n ∂2v (2) ∑n ∂v(2) v(2tt)(t,x) = Ajk (t0,x0) -------(t,x ) + 2Bj (t0,x0 )--t-(t,x) + F0, (3.89 ) j,k=1 ∂xj∂xk j=1 ∂xj
∑ n ∂v F0 := Cj (t0,x0 )--j(t0,x0) + D (t0,x0)vt(t0,x0) + E (t0,x0)v(t0,x0), (3.90 ) j=1 ∂x
where we have used the fact that (1) ∑n j ∂v v (t,x) = tvt(t0,x0 ) + j=1x ∂xj(t0,x0). Eq. (3.89View Equation) can be rewritten as a first-order system in Fourier space for the variable
( ) |k|ˆv n ˆU = ( ˆv − i∑ Bj (t ,x )k ˆv ) , (3.91 ) t j=1 0 0 j
see Section 3.1.5. Now Theorem 2 implies that the problem is well posed, if there exist constants M > 0 and δ > 0 and a family of positive definite m × m Hermitian matrices h(t,x,k), (t,x,k) ∈ Ω × Sn−1, which is C ∞-smooth in all its arguments, such that M − 1I ≤ h(t,x,k) ≤ M I and h(t,x,k)R (t,x, k) = R (t,x, k)∗h(t,x,k) ≥ δI for all (t,x,k) ∈ Ω × Sn −1, where ∑n R (t,x,k) := (Aij(t,x) + Bi(t,x)Bj (t,x))kikj i,j=1.

In particular, it follows that the Cauchy problem for the Klein–Gordon equation on a globally-hyperbolic spacetime M = [0,∞ ) × ℝn with α, βi,γij ∈ C ∞b ([0,∞ ) × ℝn ), is well posed provided that α2 γij is uniformly positive definite; see Example 17.

3.2.2 Characteristic speeds and fields

Consider a first-order linear system of the form (3.77View Equation), which is strongly hyperbolic. Then, for each t ≥ 0, n x ∈ ℝ and n−1 k ∈ S the principal symbol P0(t,x,ik) is diagonalizable and has purely complex eigenvalues. In the constant coefficient case with no lower-order terms (B = 0) an eigenvalue iμ(k) of P0(ik) with corresponding eigenvector a(k ) gives rise to the plane-wave solution

iμ(k)t+ik⋅x n u (t,x) = a (k )e , t ≥ 0,x ∈ ℝ . (3.92 )
If lower-order terms are present and the matrix coefficients Aj(t,x) are not constant, one can look for approximate plane-wave solutions, which have the form
−1 u(t,x) = a𝜀(t,x)ei𝜀 ψ(t,x), t ≥ 0,x ∈ ℝn, (3.93 )
where 𝜀 > 0 is a small parameter, ψ (t,x) a smooth-phase function and a (t,x) = a (t,x) + 𝜀a (t,x) + 𝜀2a (t,x ) + ... 𝜀 0 1 2 a slowly varying amplitude. Introducing the ansatz (3.93View Equation) into Eq. (3.77View Equation) and taking the limit 𝜀 → 0 yields the problem
n ∑ j ∂ψ-- iψta0 = P0(t,x,i∇ ψ)a0 = i A (t,x )∂xja0. (3.94 ) j=1
Setting ω(t,x) := ψt(t,x) and k(t,x) := ∇ ψ(t,x), a nontrivial solution exists if and only if the eikonal equation
det [iωI − P0(t,x,ik)] = 0 (3.95 )
is satisfied. Its solutions provide the phase function ψ(t,x) whose level sets have co-normal ωdt + k ⋅ dx. The phase function and a0 determine approximate plane-wave solutions of the form (3.93View Equation). For this reason we call ω(k) the characteristic speed in the direction k ∈ Sn −1, and a0 a corresponding characteristic mode. For a strongly hyperbolic system, the solution at each point (t,x) can be expanded in terms of the characteristic modes ej(t,x,k) with respect to a given direction n−1 k ∈ S,
m ∑ (j) u(t,x) = u (t,x, k)ej(t,x, k). (3.96 ) j=1
The corresponding coefficients u(j)(t,x,k) are called the characteristic fields.

Example 20. Consider the Klein–Gordon equation on a hyperbolic spacetime, as in Example 17. In this case the eikonal equation is

( ) i(ω − βjkj ) |k| j 2 2 ij 0 = det[iωI − Q0 (ik)] = det − α2γijkk ∕ |k| i(ω − βjk ) = − (ω − β kj) + α γ kikj, (3.97 ) i j j
which yields ∘ ------- ω± (k) = βjkj ± α γijkikj. The corresponding co-normals ω± (k )dt + kjdxj is null; hence the surfaces of constant phase are null surfaces. The characteristic modes and fields are
( ) ( i|k| ) 1 U1 U2 e±(k) = ∘ -ij---- , u(±)(k) = -- ----∓ -∘---ij----- , (3.98 ) ∓ α γ kikj 2 i|k | α γ kikj
where U = (U ,U ) = (|k|v, v − iβjk v) 1 2 t j and v is the Klein–Gordon field.

Example 21. In the formulation of Maxwell’s equations discussed in Example 15, the characteristic speeds are 0, √ --- ± α β and ±1, and the corresponding characteristic fields are the components of the vector on the right-hand side of Eq. (3.54View Equation).

3.2.3 Energy estimates and finite speed of propagation

Here we focus our attention on first-order linear systems, which are symmetric hyperbolic. In this case it is not difficult to derive a priori energy estimates based on integration by parts. Such estimates assume the existence of a sufficiently smooth solution and bound an appropriate norm of the solution at some time t > 0 in terms of the same norm of the solution at the initial time t = 0. As we will illustrate here, such estimates already yield quite a lot of information on the qualitative behavior of the solutions. In particular, they give uniqueness, continuous dependence on the initial data and finite speed of propagation.

The word “energy” stems from the fact that for many problems the squared norm satisfying the estimate is directly or indirectly related to the physical energy of the system, although for many other problems the squared norm does not have a physical interpretation of any kind.

For first-order symmetric hyperbolic linear systems, an a priori energy estimate can be constructed from the symmetrizer H (t,x) in the following way. For a given smooth solution u (t,x ) of Eq. (3.77View Equation), define the vector field J on Ω = [0, ∞ ) × ℝn by its components

μ ∗ μ J (t,x) := − u (t,x )H (t,x)A (t,x)u(t,x), μ = 0,1,2,...,n, (3.99 )
where A0(t,x) := − I. By virtue of the evolution equation, J satisfies
∂ ∑n ∂ ∂μJ μ(t,x ) ≡ --J 0(t,x) + ---kJk(t,x) = u(t,x)∗K (t,x)u(t,x), (3.100 ) ∂t k=1 ∂x
where the Hermitian m × m matrix K (t,x) is defined as
∑ n ∂ [ ] K (t,x) := H (t,x)B (t,x) + B (t,x)∗H (t,x ) + Ht (t,x ) − ---k H (t,x)Ak (t,x) . (3.101 ) k=1 ∂x
If K = 0, Eq. (3.100View Equation) formally looks like a conservation law for the current density J. If K ⁄= 0, we obtain, instead of a conserved quantity, an energy-like expression whose growth can be controlled by its initial value. For this, we first notice that our assumptions on the matrices H (t,x), B(t,x) and Ak (t,x) imply that K (t,x) is bounded on Ω. In particular, since H (t,x) is uniformly positive, there is a constant α > 0 such that
K (t,x ) ≤ 2αH (t,x), (t,x) ∈ Ω. (3.102 )
Let ⋃ ΩT = Σt 0≤t≤T be a tubular region obtained by piling up open subsets Σt of t = const hypersurfaces. This region is enclosed by the initial surface Σ 0, the final surface Σ T and the boundary surface ⋃ 𝒯 := 0≤t≤T ∂Σt, which is assumed to be smooth. Integrating Eq. (3.100View Equation) over ΩT and using Gauss’ theorem, one obtains
∫ ∫ ∫ ∫ J 0(t,x)dnx = J 0(t,x )dnx − eμJμ(t,x)dS + u(t,x)∗K (t,x)u(t,x)dtdnx, (3.103 ) ΣT Σ0 𝒯 ΩT
where eμ is the unit outward normal covector to 𝒯 and dS the volume element on that surface. Defining the “energy” contained in the surface Σt by
∫ ∫ 0 n ∗ n E (Σt) := J (t,x)d x = u(t,x ) H (t,x )u (t,x )d x (3.104 ) Σt Σt
and assuming for the moment that the “flux” integral over 𝒯 is positive or zero, one obtains the estimate
∫T ( ∫ ) ∗ n E (ΣT) ≤ E (Σ0 ) + ( u(t,x ) K (t,x )u (t,x )d x) dt 0 Σt T ∫ ≤ E (Σ0 ) + 2α E (Σt )dt, (3.105 ) 0
where we have used the inequality (3.102View Equation) and the definition of E(Σt ) in the last step. Defining the function ∫ h(T ) := 0TE (Σt)dt this inequality can be rewritten as
d-( −2αt) −2αt dt h(t)e ≤ E (Σ0)e , 0 ≤ t ≤ T, (3.106 )
which yields 2αT αh (T) ≤ E (Σ0)(e − 1) upon integration. This together with (3.105View Equation) gives
E (Σt ) ≤ e2αtE (Σ0), 0 ≤ t ≤ T, (3.107 )
which bounds the energy at any time t ∈ [0,T] in terms of the initial energy.

In order to analyze the conditions under which the flux integral is positive or zero, we examine the sign of the integrand eμJ μ(t,x). Decomposing eμdx μ = N [adt + s1dx1 + ...+ s2dxn] where s = (s1,...,sn) is a unit vector and N > 0 a positive normalization constant, we have

e Jμ(t,x) = N (t,x)u(t,x)∗[a(t,x)H (t,x ) − H (t,x)P (t,x, s)]u(t,x), (3.108 ) μ 0
where ∑n P0(t,x,s) = Aj (t,x )sj j=1 is the principal symbol in the direction of the unit vector s. This is guaranteed to be positive if the boundary surface 𝒯 is such that a(t,x) is greater than or equal to all the eigenvalues of the boundary matrix P0(t,x,s), for each (t,x) ∈ 𝒯. This is equivalent to the condition
u ∗H (t,x )P (t,x, s)u a(t,x) ≥ sup -----------0-------- for all (t,x) ∈ 𝒯 . (3.109 ) u∈ℂm,u⁄=0 u∗H (t,x)u
Since H (t,x)P0(t,x,s) is symmetric, the supremum is equal to the maximum eigenvalue of P0 (t,x, s). Therefore, condition (3.109View Equation) is equivalent to the requirement that a (t,x ) be greater than or equal to the maximum characteristic speed in the direction of the unit outward normal s.

With these arguments, we arrive at the following conclusions and remarks:

Example 22. We have seen that for the Klein–Gordon equation propagating on a globally-hyperbolic spacetime, the characteristic speeds are the speed of light. Therefore, in the case of a constant metric (i.e., Minkowksi space), the past cone C − (p0) defined in Eq. (3.111View Equation) coincides with the past light cone at the event p0. A slight refinement of the above argument shows that the statement remains true for a Klein–Gordon field propagating on any hyperbolic spacetime.

Example 23. In Example 21 we have seen that the characteristic speeds of the system given in Example 15 are 0, √ --- ± αβ and ±1, where αβ > 0 is assumed for strong hyperbolicity. Therefore, the past cone − C (p0) corresponds to the past light cone provided that 0 < α β ≤ 1. For αβ > 1, the formulation has superluminal constraint-violating modes, and an initial perturbation emanating from a region outside the past light cone at p0 could affect the solution at p0. In this case, the past light cone at p 0 is a proper subset of C− (p ) 0.

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