Three-scale singular limits

Together with Arnaud Duran and Khawla Msheik we have uploaded an arXiv preprint¹ concerning the analysis of multi-scale problems for a class of systems which take the form $$ S_0(U)\partial_t U + \sum_{l=1}^dS_l(U)\partial_{x_l}U = \frac1\epsilon L_0 \ U+\delta\sum_{l=1}^d L_l\ \partial_{x_l} U. $$ Here, $U:(t,x)\in \mathbb{R}\times\mathbb{R}^d\to\mathbb{R}^n$ (with $d,n\in\mathbb{N}^\star$) is the unknown and $S_l$ ($l\in{0,\dots,d})$ are given smooth functions into $n\times n$ real-valued symmetric matrices, while $L_0$ is a skew-symmetric matrix and for all $l\in{1,\dots,d}$, $L_l$ is a symmetric matrix. Hence we study hyperbolic symmetrizable systems of quasilinear first-order balance laws, and our interest lies in the following asymptotic regime prescribed by values of the positive parameters $\delta$ and $\epsilon$: $$ 0<\epsilon\ll\delta\ll 1. $$

Our interest in such problems was triggered by the ``hyperbolization method’’ for dispersive equations —and in particular the Serre–Green–Naghdi equation— that was explained in this post, but the class of systems we are considering also includes for instance problems of rotating fluids. Then our asymptotic regime describes situations with fast-propagating waves and dominant rapid rotation.

With respect to the very large litterature dedicated to singular limits, our analysis stands out from the fact that we do not impose a prescribed relation between the two parameters $\delta$ and $\epsilon$, and at a technical level by allowing the presence of a non-trivial (that is, non-identity) ``symmetrizer’’, $S_0(U)$.

We argue that the combination of these two facts leaves room for a mechanism of rapid small spatial scale development. Starting from a smooth initial data, the associated solution to the above equation in our asymptotic regime may, after a short amount of time, exhibit oscillations on a spatial scale of order $\delta$.

If one is interested to control high-regularity norms of the solution, this phenomenon demands to consider only well-prepared initial data in order to guarantee that these oscillations are of sufficiently small amplitude.

Under some assumptions on the considered systems (and most importantly an assumption of scale-separation between groups of eigenvalues of the symbol of the linear operator of the right-hand side), our study provides suitable notions of well-prepared initial data that secure the propagation on large times of the control of high-regularity norms of the solution (which in turns guarantee the existence and uniqueness of a solution, and strong convergence results in the limit $\epsilon\to 0$).

In order to comply with the possible small spatial scale development, these high-regularity norms should be suitably weighted in terms of $\delta$ and $\epsilon$: the control should be sufficiently stringent to avoid untamed growth due to nonlinear interactions, but also sufficiently loose to allow for small-amplitude/small-scale oscillations. It is a delicate task to find the right balance so as to impose only the necessary conditions on initial data that secures the desired control at a certain level of regularity. Our work is specifically dedicated to that issue.

One outcome of our work is that we considerably reduce with respect to my previous analysis² the assumptions for which ``hyperbolized’’ Serre–Green–Naghdi equations are rigorously justified.