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We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial…

Analysis of PDEs · Mathematics 2020-01-08 Raphaël Danchin , Francesco Fanelli , Marius Paicu

We analyze the surface tension exerted at the interface between an active fluid and a solid boundary in terms of tangential forces. Focusing on active systems known to possess an equation of state for the pressure, we show that interfacial…

We deal with the barotropic compressible Navier-Stokes equations subject to large external potential forces with slip boundary condition in a 3D simply connected bounded domain, whose smooth boundary has a finite number of 2D connected…

Analysis of PDEs · Mathematics 2021-02-26 Guocai Cai , Bin Huang , Xiaoding Shi

In this paper we study a finite-depth layer of viscous incompressible fluid in dimension $n \ge 2$, 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…

Analysis of PDEs · Mathematics 2021-07-22 Giovanni Leoni , Ian Tice

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We are interested in well-posedness at a very low level of regularity. We derive dispersive and…

Analysis of PDEs · Mathematics 2019-12-17 Evgueni Dinvay , Sigmund Selberg , Achenef Tesfahun

We prove a local existence theorem for the free boundary problem for a relativistic fluid in a fixed spacetime. Our proof involves an a priori estimate which only requires control of derivatives tangential to the boundary, which holds also…

Analysis of PDEs · Mathematics 2021-09-07 Daniel Ginsberg , Hans Lindblad

We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is…

Analysis of PDEs · Mathematics 2015-11-04 Lydia Bieri , Shuang Miao , Sohrab Shahshahani , Sijue Wu

We consider the barotropic Navier-Stokes system describing the motion of a compressible viscous fluid confined to a bounded domain driven by time periodic inflow/outflow boundary conditions. We show that the problem admits a time periodic…

Analysis of PDEs · Mathematics 2021-01-20 Anna Abbatiello , Eduard Feireisl

The compressible barotropic Navier-Stokes type system in monodimensional case with Neumann boundary condition given on free boundary is considered. The local and the global existence with uniformly boundedness for small viscosity…

Classical Analysis and ODEs · Mathematics 2009-09-09 Ewelina Kaminska

This paper concerns the global well posedness issue of the Navier-Stokes equations (CNS) describing barotropic compressible fluid flow with free surface occupied in the three dimensional exterior domain. Combining the maximal $L_p$-$L_q$…

Analysis of PDEs · Mathematics 2023-06-21 Yoshihiro Shibata , Xin Zhang

We introduce and analyze a class of Surface Quasi-Geostrophic (SQG) equations in the presence of moving rigid obstacles. The model is motivated both by vortex-wave type asymptotics for singular structures in active scalar equations and by…

Analysis of PDEs · Mathematics 2026-05-25 Ludovic Godard-Cadillac , Arnab Roy

We consider the capillary-gravity water wave equation in two dimensions. We assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. We construct an energy functional and prove a local wellposedness…

Analysis of PDEs · Mathematics 2020-12-25 Siddhant Agrawal

Consider a three-dimensional fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of gravity. We prove that one can estimate its energy by looking only at the motion of the…

Analysis of PDEs · Mathematics 2015-06-30 Thomas Alazard

A pinned or free-floating rigid plate lying on the free surface of a thin film of viscous fluid, which itself lies on top of a horizontal substrate that is moving to the right at a constant speed is considered. The focus of the present work…

Classical Physics · Physics 2022-09-07 Philippe H. Trinh , Stephen K. Wilson , Howard A. Stone

In this paper we consider a layer of incompressible viscous fluid lying above a flat periodic surface in a uniform gravitational field. The upper boundary of the fluid is free and evolves in time. We assume that a mass of surfactants…

Analysis of PDEs · Mathematics 2016-06-10 Chanwoo Kim , Ian Tice

The two-phase horizontally periodic quasistationary Stokes flow in $\mathbb{R}^2$, describing the motion of two immiscible fluids with equal viscosities that are separated by a sharp interface, which is parameterized as the graph of a…

Analysis of PDEs · Mathematics 2024-06-12 Daniel Böhme , Bogdan-Vasile Matioc

Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For…

Analysis of PDEs · Mathematics 2019-02-20 Friedrich Lippoth , Mark A. Peletier , Georg Prokert

In this paper, we investigate steady two-dimensional free-surface flows of an inviscid and incompressible fluid emerging from a nozzle, falling under gravity and impinging onto a horizontal wall. More precisely, for any given atmosphere…

Analysis of PDEs · Mathematics 2018-08-07 Jianfeng Cheng , Lili Du , Zhouping Xin

In this paper we study traveling wave solutions to the free boundary incompressible Navier-Stokes system with generalized Navier-slip conditions. The fluid is assumed to occupy a horizontally infinite strip-like domain that is bounded below…

Analysis of PDEs · Mathematics 2023-11-06 Junichi Koganemaru , Ian Tice

We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local…

Analysis of PDEs · Mathematics 2017-08-16 Mats Ehrnström , Long Pei , Yuexun Wang