Generalization to variable coefficient and non-linear systems

Systems of this type are called quasi-linear because the derivative appears linearly. This property allows one to use most of the machinery for constant coefficient equations to prove well posedness, thus the local existence is well understood, via linearization techniques. There are few global results, and in general they depend on more refined knowledge of the equation systems for which they apply.

The behavior of solutions to quasi-linear equations is not yet fully understood. Most of the solutions develop singularities in a finite time for most initial data, even if they are in $M0$ . This is the case for convective systems, or more generally for genuinely non-linear systems, see [43

, 53

] for the definition and main results, a class which includes systems like perfect fluids and relativistic dissipative fluids – for they contain as part of the system the perfect fluid equations. This is also the case for general relativity, where singularity theorems (see [38

, 56 ]) tell us about the development of singularities, although of a different type. Thus the concept of well posedness has to be modified to account for the fact that solutions only last for a finite time and this time depends on the initial data. Basically, the most we can pretend to show in the above generality is the same type of well posedness one requires from an ordinary system of equations. Which is quite a lot! The non-linear aspect of the equations implies also that it is not possible to generalize their solutions to be distributions. The minimum differentiability needed to make sense of an equation depends on the particular equation. Furthermore, there are cases (e.g. convection) in which, for some function spaces of low differentiability, the equation makes sense and some solutions exist, but they are not unique³ .

Definition 4 Let $u0(t,x), t ∈ [0,T0)$ , $T0 < + ∞$ be a smooth solution of a quasi-linear evolution system. We shall say the system is well posed at the solution $u0$ with respect to a norm $|| ||$ if given any $δ > 0$ there exists $𝜀 > 0$ such that for any smooth initial data $f(x)$ such that $||f − f0|| < 𝜀$ , with $f0(x) := u0(0,x)$ there exists a smooth solution $u (t,x )$ defined in a strip $0 ≤ t < T$ , with $|u (t,⋅) − u (t,⋅)| < δ 0$ , $|T − T | < δ 0$ .

In order not to worry about the possibility that the smoothness of the solutions be too stringent a requirement, one can smooth out the equation using a one parameter family of mollifiers, and require that the relation $δ(𝜀)$ be independent of that parameter family.

To obtain results about well posedness, we just have to slightly modify the concepts of hyperbolicity already discussed in the constant coefficient case. Since in the constant coefficient case the matrices did not depend on the points of space-time, nor on the solution itself, we had only two cases. In one case, the norm $H$ did not depend on $ωa$ , and so in some base the matrix $a A$ was symmetric. In the other case, the norm $H$ did depend on $ωa$ , and we had a general strongly hyperbolic system. In the latter case, it can be seen that $H (ω)$ is piece-wise continuous and so integrable, which is, in that case, all that is needed to proceed with the proof. In the general case with which we are now dealing, $H$ would in general depend not only on $ωa$ , but also on the point of space-time and on the solution, $H = H (ωa, t,x, u)$ .

This difference has caused terminology to be not uniform in the literature, so I have taken advantage of this and establish terms in the way I consider best suited for the topic.

Certain authors call some systems symmetric hyperbolic and others symmetrizable. They call symmetric hyperbolic only those systems where the symmetrizer does not depend on the unknown variables nor on the space-time variables (or at most depends only on the base space variables $H := H (t,x )$ ); they call the other systems symmetrizable. This is a rather arbitrary distinction, since the methods of proof used are valid for both with no essential difference. Thus, if $H$ does not depend on $ωa$ but depends smoothly on all other variables, $H := H (t,x, u)$ , then we shall still say the system is symmetric hyperbolic. In this case the non-singular transformation which symmetrizes $a A (t,x,u )$ is smooth in all its variables.

The existence and smoothness proof is based, as in the constant coefficient case, on energy norm estimates, but now supplemented by Sobolev inequalities. Since the norm is built out of $H$ and it does not depend on $ωa$ , no passage to Fourier space is needed.

If $H$ does also depend on $ωa$ , and is smooth on all variables, $H := H (t,x,u, ωa)$ , we shall say the system is strongly hyperbolic. The existence and smoothness proof now requires the construction of a pseudo-differential norm out of $H$ , and so pseudo-differential calculus is needed, which implies that $H$ has to be smooth in all its entries, in particular in $ωa$ .

We shall not discuss weak hyperbolic systems, for they are generically unstable under perturbations, nor shall we discuss strictly hyperbolic systems, i.e. systems with strictly different eigenvalues of $Aaω a$ , for they seldom appear in physical processes in more than one dimension.

With this concept of well posedness we have the following theorem [See for instance [54

] p. 123]:

Theorem 4 Let $r > n2 + 1$ , then a strongly hyperbolic system is well posed with respect to the Sobolev norm $|| ||r$ . The solution is in $C ([0,T ),Hr )$ , the time of existence depends only on $||f|| r$ .

2.3 Generalization to variable coefficient and non-linear systems