# Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

## Abstract

Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models and their parent Euler systems, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, we show that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single-layer homogeneous fluid with a constant-pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates, and finally, through numerical simulations, to the full Euler stratified system. These demonstrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results.

## Keywords

Boundary-interface contact Channel flows Stratified fluids Two-layer models## Mathematics Subject Classification

76B70 35Q35## Notes

### Acknowledgements

We thank R. Colombo and M. Garavello for discussions and useful comments on the theory of quasilinear PDEs. R.C. and G.O. thank S.L. Gavrilyuk for bringing to their attention, after a first draft of this work had been submitted for publication, reference Ovsyannikov (1979) during the Summer school “Dispersive hydrodynamics and oceanography: from experiments to theory” 27 August - 1 September 2017, Les Houches (France). Support by NSF Grants DMS-0908423, DMS-1009750, DMS-1517879, RTG DMS-0943851, CMG ARC-1025523, ONR Grants N00014-18-1-2490, DURIP N00014-12-1-0749, ERC Grant H2020-MSCA-RISE-2017 PROJECT No. 778010 IPaDEGAN, and the auspices of the GNFM Section of INdAM are all gratefully acknowledged. R.C. and M.P. thank the Dipartimento di Matematica e Applicazioni of Università Milano-Bicocca for its hospitality. In addition, the Istituto Nazionale di Alta Matematica (INdAM) and the Institute for Computational and Experimental Research in Mathematics (ICERM), supported by the National Science Foundation under Grant DMS-1439786, are gratefully acknowledged for hosting the visits of R.C. (INdAM, Summer 16) and R.C. & C.T. (ICERM, Spring 17) while some of this work was carried out. Last, but not least, we would like to thank the anonymous referees whose attentive reading of the manuscript greatly helped improve it.

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