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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?

Thanks in advance.

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    $\begingroup$ 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. $\endgroup$ Dec 1, 2020 at 17:49
  • $\begingroup$ 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 $\endgroup$
    – Chaos
    Dec 1, 2020 at 17:53
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    $\begingroup$ 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. $\endgroup$ Dec 1, 2020 at 20:15
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    $\begingroup$ 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. $\endgroup$ Dec 1, 2020 at 20:21
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    $\begingroup$ 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. $\endgroup$ Dec 2, 2020 at 16:22

1 Answer 1

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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.)

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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.

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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.

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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.

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Applying Banach fixed-point theorem you get local existence.

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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 $A$.

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  • $\begingroup$ thank you so much Willie! This is really helpful $\endgroup$
    – Chaos
    Dec 2, 2020 at 17:08
  • $\begingroup$ +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. $\endgroup$ Dec 2, 2020 at 21:33
  • $\begingroup$ 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. $\endgroup$
    – Chaos
    Dec 3, 2020 at 13:37
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    $\begingroup$ 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 $1 + |\vec{w}(t)|_\infty$. $\endgroup$ Dec 3, 2020 at 15:04
  • $\begingroup$ 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. $\endgroup$
    – Chaos
    Dec 4, 2020 at 9:04

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