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Abstract
In this paper we show a structural stability result for water waves. The main motivation for this result is that we aim to exhibit a water wave whose interface starts as a graph and ends in a splash. Numerical simulations lead to an approximate solution with the desired behaviour. The stability result will conclude that near the approximate solution to water waves there is an exact solution.
Highlights
The water waves problem models the motion of an incompressible fluid with constant density ρ in a domain Ω(t) with a free boundary ∂Ω(t), which satisfies the Euler equation with the presence of gravity and whose flow in potential
Suppose we prove that St−01 ≤ C0 by the previous methods
We developed a tool in 900 lines of C++ code that could do all this and output the collection of terms in Tex
Summary
The water waves problem models the motion of an incompressible fluid with constant density ρ in a domain Ω(t) with a free boundary ∂Ω(t), which satisfies the Euler equation with the presence of gravity and whose flow in potential. We start computing a numerical approximation of a solution to the water waves equation 5 that starts as a splash, turns over and is a graph. In order to overcome this difficulty and be able to prove rigorous results, we use the so-called interval arithmetics, in which instead of working with arbitrary real numbers, we perform computations over intervals which have representable numbers as endpoints On these objects, an arithmetic is defined in such a way that we are guaranteed that for every x ∈ X, y ∈ Y x y ∈ X Y, for any operation.
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