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Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs:

A$\dfrac{\partial}{\partial t}\pmb{v}(t,x)=B(t,x,\pmb{v}(t,x))\dfrac{\partial}{\partial x}\pmb{v}(t,x)+N(x,t)$,

with

$\pmb{v}_i(t_0,x_0)=\pmb{v}_{i0}$, $\forall (t_0,x_0)\in\Gamma_i$, $i=1,...,n$

where $\pmb{v}$ is the vector of the variables, $A,\;B$ and $N$ are three matrices, the $\pmb{v}_{i0}$ are continuous in initial curves $\Gamma_i$, the initial curves $\Gamma_i$ are non-characteristic and the coefficients of $A$ are constants. In particular, the equations of the system in what I'm really interested have the form:

$0=-\dfrac{\partial b}{\partial x}(t,x)+f_1(x,b(t,x),c(t,x),d(t,x))$,

$0=-\dfrac{\partial c}{\partial x}(t,x)+f_2(x,b(t,x),c(t,x),d(t,x))$

$\dfrac{\partial d}{\partial t}(t,x)=-b(t,x)\dfrac{\partial d}{\partial x}(t,x)+f_3(x,b(t,x),c(t,x),d(t,x))$.

I have only found results for systems in which the matrix $A$ is invertible, but in the case I have, that condition isn't satisfied. So, any suggestion will be really appreciated.

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  • $\begingroup$ Do I read correctly that you are giving initial data at a point? Then chances are you will not get any uniqueness results. Given what you wrote as your system, you can always find a coordinate system $\tau, \xi$ for which you have a hyperbolic first order PDE. and from which with the prescribed initial data at one point you conclude both existence and non-uniqueness. $\endgroup$
    – Willie Wong
    Commented May 4, 2016 at 21:16
  • $\begingroup$ you are right, this new edit take in account your comment? $\endgroup$
    – Fernando
    Commented May 4, 2016 at 21:35
  • $\begingroup$ Re edit: If you are prescribing data on non-characteristic curves, then you have a standard nonlinear hyperbolic system for which local existence and uniqueness follow directly. Most sources want $A$ invertible because they want the make $t = 0$ a non-characteristic curve for the initial data. $\endgroup$
    – Willie Wong
    Commented May 4, 2016 at 21:39
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    $\begingroup$ @Fernando: for the local problem, as long as $b$ is bounded initially, you can always find a change of variables $\tau = t + \lambda x$ and $\xi = t - \lambda x$ (or something else appropriately chosen) so that the matrix is uniformly invertible along the initial data curve. Then by continuity it will remain uniformly invertible for some small time for the solution to the initial value problem. $\endgroup$
    – Willie Wong
    Commented May 5, 2016 at 13:05
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    $\begingroup$ It is of course possible that there will be singularities forming in finite time so that the solution is not global, and there's nothing you can do about that. The classical theorems only assert local well-posedness anyway. $\endgroup$
    – Willie Wong
    Commented May 5, 2016 at 13:06

1 Answer 1

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(This is mostly an extended comment to answer this comment.)

Yes, you are right to worry about the curves. If you initial data curve is real analytic, then you can find a local, real analytic change of variables, so that the proof using CK goes through identically.

When your initial data curve is only smooth, but not real analytic, you will then need to construct a sequence of approximate change of variables and show that the appropriate limit converges. The sequence of approximate change of variables will be based on first approximating your data curve using something like Weierstrass approximation by a sequence of real analytic curves, and then straighten out these approximate curves using a real analytic change of variables. The fact that hyperbolic equations have finite speeds of propagation means that the inevitable spatial localization in this procedure is relatively harmless.

Unfortunately I am not aware of any place where this argument is written down in full detail. It is one of those folk theorems that "everyone knows is true", and many subparts or special cases of it can be found in many places.

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