Quasi-linear System of First Order P.D.E.s of "Mixed" type

Question

In my research work, I am dealing with a quasi-linear system of first order p.d.e.'s with two independent variables (say $x_1$ and $x_2$) and four dependent variables (say $u_1(x_1,x_2)$, $u_2(x_1,x_2)$, $u_3(x_1,x_2)$ and $u_4(x_1,x_2)$) of the form \begin{equation} \mathbf{A}(\mathbf{x},\mathbf{u}(\mathbf{x}))\frac{\partial \mathbf{u}}{\partial x_1}+\mathbf{B}(\mathbf{x},\mathbf{u}(\mathbf{x}))\frac{\partial \mathbf{u}}{\partial x_2}=\mathbf{f}(\mathbf{x}), \end{equation} where $\mathbf{x}=[x_1\,\,x_2]^{T}$ and $\mathbf{u}=[u_1\,\,u_2\,u_3\,u_4]^{T}$ are, respectively, the independent and dependent vector variables, constructed out of $x_\alpha$'s and $u_i$'s, $\alpha=1,2$ and $i=1,2,3,4$. $\mathbf{A}$ and $\mathbf{B}$ are two $4\times 4$ real matrices which are functions of $\mathbf{x}$ and $\mathbf{u}$ (dependence on $\mathbf{u}$ makes the system quasi-linear) and $\mathbf{f}$ is a known vector valued function of $\mathbf{x}$.

To determine the type of the system, we consider the eigenvalues of the matrix $\mathbf{A}^{-1}\mathbf{B}$ (assuming $\mathbf{A}$ to be invertible which it is in my case). Now there are 4 possibilities about the nature of these eigenvalues and there are corresponding "type" of the p.d.e. system defined:

1. All eigenvalues real and distinct, implying hyperbolic system. We have the well-studied system of hyperbolic system of conservation laws falling under this category.

2. All eigenvalues are complex (two pairs of complex conjugate eigenvalues). We have then an elliptic system. (I don't know about any physical example of this type.)

3. All eigenvalues are real but with unequal algebraic and geometric multiplicity (i.e. no set of 4 linearly independent eingenvectors). We then have a parabolic system. (Again, no knowledge of physical example.)

4. Two real and one pair of complex conjugate eigenvalues. This does not fall into the above categories. Probably these are called the quasi-linear system of mixed type.

My first question is about the the above classification scheme which, in fact, imitates the classification scheme of linear system of first order p.d.e.'s. Matrices $\mathbf{A}$ and $\mathbf{B}$ involve unknown function $\mathbf{u}$. So, how to determine the type of the system (from the nature of its eigenvalues) a priori to the determination of the solution?

My second question is about the type 4 systems, because my problem falls into this category. How to deal with this mixed type p.d.e. systems?

My major is mechanical engineering and I am working in theory of elasticity. Any help would be greatly appreciated as I know very little about analytical methods to solve quasi-linear p.d.e. systems of first order.

Just to mention, in a previous question I was concerned about a system of 3 linear first order pdes of mixed type (3 roots of the characteristic polynomial: either all are real, or one real and the others appearing as a complex conjugate pair) which is a special case of this problem. To emphasize, I come up with these systems in a physical problem and I know that there must exist real valued "well-behaved" solutions (bounded). Which I do not know is how to find those solutions which must exist and I am looking for possible methods in this forum.

Thanks in advance for any help.

Answer 1

Let me change your notations. You deal with a 1D quasilinear system with size $N=4$: the standard Cauchy problem is $$ \frac{\partial u}{\partial t}+A(t,x,u)\frac{\partial u}{\partial x}= f(t,x),\quad u(t=0,x)=u_0(x), $$ where $t\in \mathbb R$ (time variable) as well as $x$ (this is a 1D problem), $u$ is valued in $\mathbb R^N$, $A$ is a real-valued $N\times N$ matrix. You may also assume that $A$ depends smoothly (or even analytically) of its arguments.

(1) Your first case: all eigenvalues of $A_0:=A(0,x_0, u_0(x_0))$ are real and distinct, this is indeed the strictly hyperbolic case. In this case, you can guarantee local existence, uniqueness and continuous dependence on the data, i.e. local well-posedness. Of course you cannot expect global existence in general because of the nonlinearity (think about the scalar Burgers).

(2) Let me skip some of your cases and go directly to the case where $A_0$ has a non-real eigenvalue (and thus a pair of non-real eigenvalues). Big trouble ahead: even if the matrix $A$ is analytic, in which case, Cauchy-Kovalevskaya theorem is providing a local (unique) analytic solution, that solution is very unstable in the Hadamard sense. It means that even though $v_0-u_0$ is very small in a very strong topology, such as the $C^\infty$ topology or the $H^s$ topology for a very large $s$, you will not be able to control $u(t)-v(t)$ in a quite weak topology such as $L^2$ (all this is local of course). You will find precise statements in a paper by Métivier, Remarks on the well-posedness of the nonlinear Cauchy problem, with MR number MR2127041.

(3) When all eigenvalues are real, some with multiplicity larger than 2, then instability could or could not occur, depending on other structural factors such as semi-simplicity of $A_0$. Generally speaking, multiple roots will trigger difficulties.

(4) A very important class of theorems, with the name of Lax-Mizohata theorems is establishing a weak converse to (1): if the problem is well-posed (e.g. meaning that you have some Sobolev norm control of $u(t)$ by some Sobolev norm of $u(0)$), then it implies that the system is weakly hyperbolic (case (3)$\cup$(1)). So if you expect your system to be well-behaved in the sense of Hadamard, no choice, the roots must be real-valued, possibly with multiplicity.

Do not think that all physically relevant problem of that type are hyperbolic: to quote just one example, Van der Waals classical system is $$ \partial_t u+\partial_x v=0\quad \partial_t v+\partial_x q(u)=0. $$ When $q'(u)>0$, you are in a hyperbolic region, but when $q'(u)<0$, you have a non-real eigenvalue.

Answer 2

I gave an answer to your earlier question, but although it was technically correct, I see now that since you have only two independent variables my answer wasn't really the right one. Let me try one more time (but I can't guarantee I'm right).

I'm going to address only case 4, since that's the one you care about. Given your assumptions, there exist invertible-matrix-valued functions $P(x,u)$ and $Q(x, u)$ such that the system $$ P(A\partial_1 u + B\partial_2 u) = Pf $$ can be written as $$ \partial_1 v + C(x,v)\partial_2v = g(x,v), $$ where $v = (v_1, v_2, v_3, v_4) = Q(x,u)u$. $$ C(x,v) = \begin{bmatrix} a(x,b)& 0 & 0 & 0 \newline 0 & b(x,v) & 0 & 0 \newline 0 & 0 & 0 & -1 \newline 0 & 0 & 1 & 0 \end{bmatrix} $$ If you fix a rectangular domain, say, $D = [0,T]\times[0,S]$, then this system is well-behaved from the point of view of uniqueness and, with the right additional assumptions, existence if you specify initial data consisting of $v_1$ and $v_2$ on $\{0\} \times [0,S]$ and $v_3$ on the boundary of $D$.

This is a coupled hyperbolic-elliptic system that is hyperbolic in $v_1$ and $v_2$ and elliptic in $v_3$ and $v_4$. My guess is that linear systems like this been analyzed and solved before, but unfortunately I don't know any specific reference. You would handle the quasilinear version using the linear theory using the usual techniques (inverse function theorem, fixed point theory, etc.). If you know the theory of linear first order elliptic systems of PDE's and of linear first order hyperbolic systems of PDE's in two independent variables really well, it's reasonably straightforward to adapt it to a coupled system like this.

Answer 3

I can try to address the part of your question about the PDE-type classification when the matrices $\mathbf{A}$ and $\mathbf{B}$ depend on the unknown variable $\mathbf{u}$. There is simply no way to know the type at each point $(x_1,x_2)$ before an actual solution is obtained. If the equations that you are studying are known to admit a rich family of exact solutions, it is likely that you will find solutions that change type in different regions of the domain of the independent variables. (This may not be relevant for you, but General Relativity is a field theory with quasilinear equations of motion. There are known exact solutions which change type from elliptic to hyperbolic in different regions of the domain. The transitional boundaries are usually considered singularities, where some physical quantity becomes infinite.) The significance of coexistence of different PDE types is impossible to assess without more physical input.

However, what you might be able to conclude is that given a point $(x_1,x_2)$ the solution is of a "stable" type, then it will have the same type in a neighborhood of that point. "Stable" means that the matrix $\mathbf{A}^{-1}\mathbf{B}$ is generic (full rank and all eigenvalues are simple). If this matrix has a degenerate eigenvalue/eigenvector structure, it survive small perturbations of the matrices $\mathbf{A}$ and $\mathbf{B}$, unless there is some special algebraic property of these matrices that makes a degenerate form stable despite their dependence on $\mathbf{u}$ (being independent of $\mathbf{u}$ would do the trick, which also shows why degenerate matrix types are important in the theory of semilinear PDE systems).