Stratified periodic water waves with singular density gradients
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Stratified periodic water waves with singular density gradients Joachim Escher1 · Patrik Knopf2 · Christina Lienstromberg3 · Bogdan‑Vasile Matioc2 Received: 3 December 2019 / Accepted: 23 January 2020 © The Author(s) 2020
Abstract We consider Euler’s equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct twodimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis. Keywords Euler equations · Traveling waves · Stratified fluid · Singular density gradient Mathematics Subject Classification 35Q35 · 35B32 · 76B47 · 76B70
* Bogdan‑Vasile Matioc [email protected] Joachim Escher [email protected]‑hannover.de Patrik Knopf [email protected] Christina Lienstromberg [email protected]‑bonn.de 1
Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
2
Fakultät für Mathematik, Universität Regensburg, 93053 Regensburg, Germany
3
Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
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1 Introduction Stratification is a phenomenon that is common in ocean flows where in the presence of salinity and under the influence of the gravitational force a heterogeneity in the fluid is produced. Stratification corresponds to the formation of fluid layers, normally arranged horizontally with the less dense layers being located on top of the denser ones. This phenomenon may be caused by many other factors including temperature, pressure, topography and oxygenation. Because of the plethora of effects resulting from stratification, such flows have received much attention, especially in geophysical fluid dynamics. In the setting of traveling stratified waves the problem is modeled by the stationary Euler equations for incompressible fluids, subject to natural boundary conditions, cf. (2.2). The study of two-dimensional stratified flows dates back to the pioneering work of Dubreil–Jacotin. In 1937 Dubreil–Jacotin [23] constructed small-amplitude stratified traveling gravity waves by using power series expansions. Previously in [22] she showed that Gerstner’s explicit solution [10, 27] can be accommodated to describe exact traveling gravity waves with an arbitrary stratification. Furthermore, related to Gerstner’s solution, there is a further exact solution describing an edge wave propagating along a sloping beach [9,
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