### 3.3 Quasilinear equations

Next, we generalize the theory one more step and consider evolution systems, which are described by quasilinear partial differential equations, that is, by nonlinear partial differential equations, which are linear in their highest-order derivatives. This already covers most of the interesting physical systems, including the Yang–Mills and the Einstein equations. Restricting ourselves to the first-order case, such equations have the form
where all the coefficients of the complex matrices , …, and the nonlinear source term belong to the class of bounded, -functions with bounded derivatives. Compared to the linear case, there are two new features the solutions may exhibit:
• The nonlinear term may induce blowup of the solutions in finite time. This is already the case for the simple example where , all the matrices vanish identically and , in which case Eq. (3.115) reduces to . In the context of Einstein’s equations such a blowup is expected when a curvature singularity forms, or it could also occur in the presence of a coordinate singularity due to a “bad” gauge condition.
• In contrast to the linear case, the matrix functions in front of the derivative operator now depend pointwise on the state vector itself, which implies, in particular, that the characteristic speeds and fields depend on . This can lead to the formation of shocks where characteristics cross each other, like in the simple example of Burger’s equation corresponding to the case , and . In general, shocks may form when the system is not linearly degenerated or genuinely nonlinear [250]. The Einstein vacuum equations, on the other hand, can be written in linearly degenerate form (see, for example, [6, 7, 348, 8]) and are therefore expected to be free of physical shocks.

For these reasons, one cannot expect global existence of smooth solutions from smooth initial data with compact support in general, and the best one can hope for is existence of a smooth solution on some finite time interval , where might depend on the initial data.

Under such restrictions, it is possible to prove well-posedness of the Cauchy problem. The idea is to linearize the problem and to apply Banach’s fixed-point theorem. This is discussed next.

#### 3.3.1 The principle of linearization

Suppose is a (reference) solution of Eq. (3.115), corresponding to initial data . Assuming this solution to be uniquely determined by the initial data , we may ask if a unique solution also exists for the perturbed problem

where the perturbations and belong to the class of bounded, -functions with bounded derivatives. This leads to the following definition:

Definition 5. Consider the nonlinear Cauchy problem given by Eq. (3.115) and prescribed initial data for at . Let be a -solution to this problem, which is uniquely determined by its initial data . Then, the problem is called well posed at , if there are normed vector spaces , , and and constants , such that for all sufficiently-smooth perturbations and lying in and , respectively, with

the perturbed problem (3.116, 3.117) is also uniquely solvable and the corresponding solution satisfies and the estimate

Here, the norms and appearing on both sides of Eq. (3.119) are different from each other because controls the function over the spacetime region while is a norm controlling the function on .

If the problem is well posed at , we may consider a one-parameter curve of initial data lying in that goes through and assume that there is a corresponding solution for each small enough , which lies close to in the sense of inequality (3.119). Expanding

and plugging into the Eq. (3.115) we find, to first order in ,
with
Eq. (3.121) is a first-order linear equation with variable coefficients for the first variation, , for which we can apply the theory described in Section 3.2. Therefore, it is reasonable to assume that the linearized problem is strongly hyperbolic for any smooth function . In particular, if we generalize the definitions of strongly and symmetric hyperbolicity given in Definition 4 to the quasilinear case by requiring that the symmetrizer has coefficients in , it follows that the linearized problem is well posed provided that the quasilinear problem is strongly or symmetric hyperbolic.

The linearization principle states that the converse is also true: the nonlinear problem is well posed at if all the linear problems, which are obtained by linearizing Eq. (3.115) at functions in a suitable neighborhood of are well posed. To prove that this principle holds, one sets up the following iteration. We define the sequence of functions by iteratively solving the linear problems

for starting with the reference solution . If the linearized problems are well posed in the sense of Definition 3 for functions lying in a neighborhood of , one can solve each Cauchy problem (3.123, 3.124), at least for small enough time . The key point then, is to prove that does not shrink to zero when and to show that the sequence of functions converges to a solution of the perturbed problem (3.116, 3.117). This is, of course, a nontrivial task, which requires controlling and its derivatives in an appropriate way. For particular examples where this program is carried through; see [259]. For general results on quasilinear symmetric hyperbolic systems; see [251, 164, 412, 51].