The boundary condition is called maximal dissipative if the associated boundary spaces are maximal nonpositive for all .
Maximal dissipative boundary conditions were proposed in [189, 275] in the context of symmetric positive operators, which include symmetric hyperbolic operators as a special case. With such boundary conditions, the IBVP is well posed in the following sense:
Definition 9. Consider the linearized version of the IBVP (5.1, 5.2, 5.3), where the matrix functions and and the vector function do not depend on . It is called well posed if there are constants and such that each compatible data and gives rise to a unique -solution satisfying the estimatefor all . If, in addition, the constant can be chosen strictly positive, the problem is called strongly well posed.
This definition strengthens the corresponding definition in the Laplace analysis, where trivial initial data was assumed and only a time-integral of the -norm of the solution could be estimated (see Definition 6). The main result of the theory of maximal dissipative boundary conditions is:
Theorem 7. Consider the linearized version of the IBVP (5.1, 5.2, 5.3), where the matrix functions and and the vector function do not depend on . Suppose the system is symmetric hyperbolic, and that the boundary conditions (5.3) are maximal dissipative. Suppose, furthermore, that the rank of the boundary matrix is constant in .
Then, the problem is well posed in the sense of Definition 9. Furthermore, it is strongly well posed if the boundary matrix is invertible.
This theorem was first proven in [189, 275, 344] for the case where the boundary surface is non-characteristic, that is, the boundary matrix is invertible for all . A difficulty with the characteristic case is the loss of derivatives of in the normal direction to the boundary (see ). This case was studied in [293, 343, 387], culminating with the regularity theorem in , which is based on special function spaces, which control the -norms of tangential derivatives and normal derivatives at the boundary (see also ). For generalizations of Theorem 7 to the quasilinear case; see [218, 388].
A more practical way of characterizing maximal dissipative boundary conditions is the following. Fix a boundary point , and define the scalar product by , . Since the boundary matrix is Hermitian with respect to this scalar product, there exists a basis of eigenvectors of , which are orthonormal with respect to . Let be the corresponding eigenvalues, where we might assume that the first of these eigenvalues are strictly positive, and the last are strictly negative. We can expand any vector as , the coefficients being the characteristic fields with associated speeds . Then, the condition (5.90) at the point can be written as
In conclusion, a maximal dissipative boundary condition must have the form of Eq. (5.94), which describes a linear coupling of the outgoing characteristic fields to the incoming ones, . In particular, there are exactly as many independent boundary conditions as there are incoming fields, in agreement with the Laplace analysis in Section 5.1.1. Furthermore, the boundary conditions must not involve the zero speed fields. The simplest choice for is the trivial one, , in which case data for the incoming fields is specified. A nonzero value of would be chosen if the boundary is to incorporate some reflecting properties, like the case of a perfect conducting surface in electromagnetism, for example.
Example 29. Consider the first-order reformulation of the Klein–Gordon equation for the variables ; see Example 13. Suppose the spatial domain is , with the boundary located at . Then, and the boundary matrix is
Example 30. For Maxwell’s equations on a domain with -boundary , the boundary matrix is given by
Recall that the constraints and propagate along the time evolution vector field , , , provided the continuity equation holds. Since is tangent to the boundary, no additional conditions controlling the constraints must be specified at the boundary; the constraints are automatically satisfied everywhere provided they are satisfied on the initial surface.
Example 31. Commonly, one writes Maxwell’s equations as a system of wave equations for the electromagnetic potential in the Lorentz gauge, as discussed in Example 28. By reducing the problem to a first-order symmetric hyperbolic system, one may wonder if it is possible to apply the theory of maximal dissipative boundary conditions and obtain a well-posed IBVP, as in the previous example. As we shall see in Section 5.2.1, the answer is affirmative, but the correct application of the theory is not completely straightforward. In order to illustrate why this is the case, introduce the new independent fields . Then, the set of wave equations can be rewritten as the first-order system for the 20-component vector , ,
At this point, one might ask why we were able to formulate a well-posed IBVP based on the second-order formulation in Example 28, while the first-order reduction discussed here fails. As we shall see, the reason for this is that there exist many first-order reductions, which are inequivalent to each other, and a slightly more sophisticated reduction works, while the simplest choice adopted here does not. See also [354, 14] for well-posed formulations of the IBVP in electromagnetism based on the potential formulation in a different gauge.
Example 32. A generalization of Maxwell’s equations is the evolution system.
Decomposing into its parts parallel and orthogonal to the unit outward normal ,
However, one also needs to control the incoming field at the boundary. This field, which propagates with speed , is related to the constraints in the theory. Like in electromagnetism, the fields and are subject to the divergence constraints , . However, unlike the Maxwell case, these constraints do not propagate trivially. As a consequence of the evolution equations (5.102, 5.103), the constraint fields and obeyfirst derivatives of the fields and , when rewritten as a boundary condition for the main system (5.102, 5.103). Except in some particular cases involving totally-reflecting boundaries, it is not possible to cast these conditions into maximal dissipative form.
A solution to this problem has been presented in  and , where a similar system appears in the context of the IBVP for Einstein’s field equations for solutions with anti-de Sitter asymptotics, or for solutions with an artificial boundary, respectively. The method consists in modifying the evolution system (5.102, 5.103) by using the constraint equations in such a way that the constraint fields for the resulting boundary adapted system propagate along at the boundary surface. In order to describe this system, extend to a smooth vector field on with the property that . Then, the boundary-adapted system reads:
As anticipated in Example 31, the theory of symmetric hyperbolic first-order equations with maximal dissipative boundary conditions can also be used to formulate well-posed IBVP for systems of wave equations, which are coupled through the boundary conditions, as already discussed in Section 5.1.3 based on the Laplace method. Again, the key idea is to show strong well-posedness; that is, an a priori estimate, which controls the first derivatives of the fields in the bulk and at the boundary.
In order to explain how this is performed, we consider the simple case of the Klein–Gordon equation on the half plane . In Example 13 we reduced the problem to a first-order symmetric hyperbolic system for the variables with symmetrizer , and in Example 29 we determined the class of maximal dissipative boundary conditions for this first-order reduction. Consider the particular case of Sommerfeld boundary conditions, where is specified at . Then, Eq. (3.103) gives the following conservation law,
On the other hand, the first-order reduction is not unique, and as we show now, different reductions may lead to stronger estimates. For this, we choose a real constant such that and define the new fields , which yield the symmetric hyperbolic system
Summarizing, we have seen that the most straightforward first-order reduction of the Klein–Gordon equation does not lead to strong well-posedness. However, strong well-posedness can be obtained by choosing a more sophisticated reduction, in which the time-derivative of is replaced by its derivative along the time-like vector , which is pointing outside the domain at the boundary surface. In fact, it is possible to obtain a symmetric hyperbolic reduction leading to strong well-posedness for any future-directed time-like vector field , which is pointing outside the domain at the boundary. Based on the geometric definition of first-order symmetric hyperbolic systems in , it is possible to generalize this result to systems of quasilinear wave equations on curved backgrounds .
In order to describe the result in , let be a vector bundle over with fiber ; let be a fixed, given connection on and let be a Lorentz metric on with inverse , which depends pointwise and smoothly on a vector-valued function , parameterizing a local section of . Assume that each time-slice is space-like and that the boundary is time-like with respect to . We consider a system of quasilinear wave equations of the form
Theorem 8.  The IBVP (5.113, 5.114, 5.115) is well posed. Given and sufficiently small and smooth initial and boundary data , and satisfying the compatibility conditions at the edge , there exists a unique smooth solution on satisfying the evolution equation (5.113), the initial condition (5.114) and the boundary condition (5.115). Furthermore, the solution depends continuously on the initial and boundary data.
Theorem 8 provides the general framework for treating wave systems with constraints, such as Maxwell’s equations in the Lorentz gauge and, as we will see in Section 6.1, Einstein’s field equations with artificial outer boundaries.
Here, we show how to prove the existence of weak solutions for linear, symmetric hyperbolic equations with variable coefficients and maximal dissipative boundary conditions. The method can also be applied to a more general class of linear symmetric operators with maximal dissipative boundary conditions; see [189, 275]. The proof below will shed some light on the maximality condition for the boundary space .
Our starting point is an IBVP of the form (5.1, 5.2, 5.3), where the matrix functions and do not depend on , and where is replaced by , such that the system is linear. Furthermore, we can assume that the initial and boundary data is trivial, , . We require the system to be symmetric hyperbolic with symmetrizer satisfying the conditions in Definition 4(iii), and assume the boundary conditions (5.3) are maximal dissipative. We rewrite the IBVP on as the abstract linear problem
For the following, the adjoint IBVP plays an important role. This problem is defined as follows. First, the symmetrizer defines a natural scalar product on ,
Proof. Fix a boundary point and define the matrix with the unit outward normal to at . Since the system is symmetric hyperbolic, is Hermitian. We decompose into orthogonal subspaces , , on which is positive, negative and zero, respectively. We equip with the scalar products , which are defined by5.94))
The lemma implies that solving the original problem with is equivalent to solving the adjoint problem with , which, since is held fixed at , corresponds to the time-reversed problem with the adjoint boundary conditions. From the a priori energy estimates we obtain:
Proof. Let and set . From the energy estimates in Section 3.2.3 one easily obtains
In particular, Lemma 5 implies that (strong) solutions to the IBVP and its adjoint are unique. Since and are closable operators , their closures and satisfy the same inequalities as in Eq. (5.128). Now we are ready to define weak solutions and to prove their existence:
Definition 10. is called a weak solution of the problem (5.118) iffor all .
In order to prove the existence of such , we introduce the linear space and equip it with the scalar product defined by
If is a weak solution, which is sufficiently smooth, it follows from the Green type identity (5.120) that has vanishing initial data and that it satisfies the required boundary conditions, and hence is a solution to the original IBVP (5.118). The difficult part is to show that a weak solution is indeed sufficiently regular for this conclusion to be made. See [189, 275, 344, 343, 387] for such “weak=strong” results.
Living Rev. Relativity 15, (2012), 9
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