# Existence of a solution for a quasilinear hyperbolic system of PDEs with many state variables

I'll star by saying that I am not really familiar with the field of PDEs so this questions may be trivial or ill-possed in that case please let me know.

I am in search of some existence (Global) result regarding a system of first order PDEs with many state-variables and a non-homogeneity that is non-linear in the solutions, i.e.

\begin{align} \begin{cases} \partial_t \mathbf u+\sum_{n=1}^{K} A_n(t)\partial_{x_n} \mathbf u=\mathbf b(t,\mathbf u,x_1,\cdots,x_K)\\ \mathbf u(0,x_1,\cdots,x_K)=u_0\in\mathbb R^N, \end{cases} \end{align}

where the matrices $$A_n, n=1,\cdots, K$$ are diagonal, and $$\mathbf b$$ is Lipschitz continuous in $$\mathbf u$$ uniformly in $$t,x_1,\cdots,x_n$$.

In Bressan's "Conservations laws" book the case with two variables is considered, namely $$t$$ and $$x$$. I have found some articles online considering the case with many state variables but in that case the non-homogeneity was linear in the $$\mathbf u$$ .

As far as I know the method of the characteristics applies only the one-dimensional case, so I am a little bit lost.

Is there some result I can use to show the existence (and uniqueness if possible) of a solution for this system?

• I think the standard local existence result for your system would require higher regularity for $\mathbf{b}$. Your equation is automatically symmetric hyperbolic, and so Kato's result (1975, ARMA) can apply, provided that $\mathbf{b}$ is continuously differentiable $\lfloor K/2\rfloor + 1$ times. Dec 1 '20 at 17:49
• Thanks @WillieWong, well asking for more regularity on $b$ is not an issue, what worries me is the local character of the result. Still do you have the complete name of Kato's article/book? I don't find it Dec 1 '20 at 17:53
• The method of characteristics for first order scalar PDEs approaches by finding a foliation of the domain by hypersurfaces on which the solution is constant. Even though $A$ is diagonal, the corresponding foliations are different for different components of $\mathbf{u}$, and thus in general they are not compatible and cannot be reassembled to a coupled equation. Dec 1 '20 at 20:15
• Kato's paper: link.springer.com/article/10.1007/BF00280740. For the "global" problem: if you let me know more precisely what can be assumed about $\mathbf{b}$, I may be able to help you with it. But there is unlikely any general thing you can say about it; results about global existence of hyperbolic systems typically are very structure dependent. Dec 1 '20 at 20:21
• Wait, hang-on: your $A_n$ are actually diagonal, not just block diagonal? And they are real-valued, right? I seem to have been overthinking your problem because I thought it is more general. So what you have is a coupled system of transport equations; then you don't need Kato. Dec 2 '20 at 16:22

Okay, so I would write your equations instead in the following form: $$\partial_t u_i + v_i(t) \cdot \nabla u_i = b_i (t, \vec{x}, \vec{u})$$ This is a system of transport equations and so can actually be solved by using a variation of the Picard-Lindelof argument.

(I am implicitly assuming that your function $$b$$ is suitably nice in certain ways, which are implied by what you said is allowed in this comment.)

## 1

Consider the iteration scheme $$\vec{w} \mapsto \vec{w}'$$ where $$\vec{v}'$$ solves the linear inhomogeneous equation $$\partial_t w'_i + v_i(t) \cdot \nabla w'_i = b_i(t,\vec{x}, \vec{w})$$ with the fixed, given, initial data $$\mathring{\vec{w}}:\mathbb{R}^K\to \mathbb{R}^N$$.

Given a continuous function $$\vec{w}: [-t_0, t_0] \times\mathbb{R}^K \to \mathbb{R}^N$$, the function $$\vec{w}'$$ can be solved component wise by integrating along the integral curves of $$\partial_t + v_i(t) \cdot \nabla$$ on $$[-t_0, t_0]\times \mathbb{R}^K$$.

It thus suffices to show that this iteration scheme converges.

## 2

Suppose $$\vec{z}$$ and $$\vec{w}$$ are given functions that are uniformly bounded by some constant $$M$$. Then there exists (under what I hope are reasonable assumptions on $$b_i$$) some constant $$M'$$ such that $$b_i(t,\vec{x},\vec{z})$$ and $$b_i(t,\vec{x},\vec{w})$$ are uniformly bounded by $$M'$$. Thus you have that

$$| \vec{z}'|_\infty, |\vec{w}'|_\infty \leq |\mathring{\vec{w}}|_\infty + M' t_0$$

Choose $$M = 2 |\mathring{\vec{w}}|_\infty$$ and there exists some $$t_0 > 0$$ such that the iteration mapping maps the (closed) ball of radius $$M$$ in $$C^0([-t_0, t_0]\times \mathbb{R}^K,\mathbb{R}^N)$$ to itself.

## 3

For the differences you find that $$\partial_t (z'_i - w'_i) + v_i(t)\cdot \nabla (z'_i - w'_i) = b_i(t,\vec{x}, \vec{z}) - b_i(t,\vec{x},\vec{w})$$ Using the Lipschitz continuity in the final slot of $$b_i$$ you get that $$\big|\partial_t (z'_i - w'_i) + v_i(t)\cdot \nabla (z'_i - w'_i) \big| \leq M'' |\vec{z} - \vec{w}|$$ for some $$M''$$. Integrating you get that if $$t_0$$ is chosen sufficiently small, the iteration mapping is a contraction mapping.

## 4

Applying Banach fixed-point theorem you get local existence.

## 5

Having proven local existence, you can upgrade this to global in the same way you argue for ODEs: by showing that the sup norm of the functions involved does not blow-up. This follows from the global Lipschitz property of the functions $$b_i$$, which guarantees that $$|b_i(t,\vec{x},\vec{w})| \leq M''' (1 + |\vec{w}|)$$

So a standard Gronwall's inequality argument well tell you that the sup-norm of the solution cannot grow faster than $$A e^{2 M''' |t|}$$ for some constant $$A$$.

• thank you so much Willie! This is really helpful Dec 2 '20 at 17:08
• +1 Willie Wong and @Chaos: I like the analysis of the posed problem you did, up to get the final, optimal solution. In sum, really a nice piece of real research mathematics. Dec 2 '20 at 21:33
• Ey Willie sorry for bothering you again, in the last step, how can I apply the Gronwall inequality? I am not familiar with the application to PDEs. Dec 3 '20 at 13:37
• THe differential inequality, after integration, gives $$|\vec{w}(t)|_{\infty} \leq |\vec{w}(0)|_{\infty} + \int_0^t M'''(1 + |\vec{w}(s)|_{\infty} ~ds$$ Add 1 to both side and apply Gronwall to the quantity $1 + |\vec{w}(t)|_\infty$. Dec 3 '20 at 15:04
• Why do we end up integrating wrt to $t$ alone, what about the $\vec{x}$? Sorry if this is a stupid question, I know very little about this stuff. Dec 4 '20 at 9:04