The initial value problem for the navier‐stokes equations with a free surface

Beale, J. Thomas

doi:10.1002/cpa.3160340305

Cited by 227 publications

(348 citation statements)

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“…The viscous surface wave problem in our present setting that describes the motion of a layer of viscous fluid lying above a fixed bottom has attracted the attention of many mathematicians since the pioneering work of Beale [5]. For the problem without surface tension, Beale [5] proved the local well-posedness in the Sobolev spaces and Sylvester [23] studied the global wellposedness by using Beale's method.…”

Section: Resultsmentioning

confidence: 99%

“…To circumvent these, as usual, we will transform the free boundary problem under consideration to a problem with a fixed domain and fixed boundary. We will not use a Lagrangian coordinate transformation as in [5,20], but rather a flattening transformation introduced by Beale in [6]. To this end, we consider the fixed equilibrium domain Ω := {x ∈ Σ × R | − b(x 1 , x 2 ) < x 3 < 0} (1.6)…”

Section: 2mentioning

confidence: 99%

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Zero Surface Tension Limit of Viscous Surface Waves

Tan

Wang

2014

Commun. Math. Phys.

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Abstract. We consider the free boundary problem for a layer of viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom and below the atmosphere. For the "semi-small" initial data, we prove the zero surface tension limit of the problem within a local time interval. The unique local strong solution with surface tension is constructed as the limit of a sequence of approximate solutions to a special parabolic regularization. For the small initial data, we prove the global-in-time zero surface tension limit of the problem.

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Section: Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Zero Surface Tension Limit of Viscous Surface Waves

Tan

Wang

2014

Commun. Math. Phys.

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“…We follow here closely the paper of J.T. Beale [4] and adapt carefully each step of his proof to our context. The main difference lies in the type of boundary conditions.…”

Section: Study Of the Fluid Problemmentioning

confidence: 98%

“…As noted in [4] the real number r has to be large enough in order to define and estimate the nonlinear terms which appear in the Lagrangian formulation of the fluid equations and also in order that the solution in the Lagrangian variables can be transformed into a solution of the original problem, i.e. where the fluid equations are written in the eulerian variables.…”

Section: U(t X(t)) = Wmentioning

confidence: 99%

“…The ideas are the same that one can find in the papers [2-4, 16, 17] where the authors have studied the solvability of the Navier-Stokes equations with free boundary in bounded or unbounded domains. Their approach is the following: the equations are rewritten in Lagrangian coordinates and it is shown that solutions for the initial value problem exist locally in time, in smooth functions spaces, that is to say the same kind of spaces we use here [2][3][4], or spaces of W 1,p -type with p bigger than the spatial dimension [17], or in Hölder spaces [16]. Section 2 is devoted to the study of the fluid equations, and contains some standard lemmas which will be useful (Sect.…”

Section: U(t X(t)) = Wmentioning

confidence: 99%

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Existence for an Unsteady Fluid-Structure Interaction Problem

2000

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Abstract.We study the well-posedness of an unsteady fluid-structure interaction problem. We consider a viscous incompressible flow, which is modelled by the Navier-Stokes equations. The structure is a collection of rigid moving bodies. The fluid domain depends on time and is defined by the position of the structure, itself resulting from a stress distribution coming from the fluid. The problem is then nonlinear and the equations we deal with are coupled. We prove its local solvability in time through two fixed point procedures.Mathematics Subject Classification. 76D05, 35Q30, 73K70.

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Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes Equations

Leoni

Tice

2022

Comm Pure Appl Math

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In this paper we study a finite‐depth layer of viscous incompressible fluid in dimension , modeled by the Navier‐Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity‐capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time‐independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension and without surface tension in dimension , for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well‐known, to the best of our knowledge this is the first construction of viscous traveling wave solutions.Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal‐stress to normal‐Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

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The initial value problem for the navier‐stokes equations with a free surface

Cited by 227 publications

References 10 publications

Zero Surface Tension Limit of Viscous Surface Waves

Zero Surface Tension Limit of Viscous Surface Waves

Existence for an Unsteady Fluid-Structure Interaction Problem

Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes Equations

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