Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations
Preprint
Abstract
We give the first mathematical construction of two-dimensional traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations for a single finite depth layer of constant density fluid. Our construction is based on a rigorous justification of the formal shallow water limit, which postulates that in a certain scaling regime the full free boundary traveling Navier-Stokes system of PDEs reduces to a governing system of ODEs. We find heteroclinic orbits solving these ODEs and, through a delicate fixed point argument employing the Stokes problem in thin domains and a nonautonomous orbital perturbation theory, use these ODE solutions as the germs from which we build bore PDE solutions for sufficiently shallow layers.Keywords
All Related Versions
This publication has 64 references indexed in Scilit:
- Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes EquationsCommunications on Pure and Applied Mathematics, 2022
- Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models2013
- Convexity of Stokes Waves of Extreme FormArchive for Rational Mechanics and Analysis, 2004
- Exact steady periodic water waves with vorticityCommunications on Pure and Applied Mathematics, 2004
- Small internal waves in two-fluid systemsArchive for Rational Mechanics and Analysis, 1989
- Reduction of quasilinear elliptic equations in cylindrical domains with applicationsMathematical Methods in the Applied Sciences, 1988
- On the stokes conjecture for the wave of extreme formActa Mathematica, 1982
- On solitary water-waves of finite amplitudeArchive for Rational Mechanics and Analysis, 1981
- The existence of solitary water wavesCommunications on Pure and Applied Mathematics, 1977
- Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions IICommunications on Pure and Applied Mathematics, 1964