We consider the dynamics of a layer of an incompressible electrically conducting fluid interacting with the magnetic field in a two-dimensional horizontally periodic setting. The upper boundary is in contact with the atmosphere, and the lower boundary is a rigid flat bottom. We prove the global well-posedness of the inviscid and resistive problem with surface tension around a non-horizontal uniform magnetic field; moreover, the solution decays to the equilibrium almost exponentially. One of the key observations here is an induced damping structure for the fluid vorticity due to the resistivity and transversal magnetic field.
For the free-boundary incompressible Euler equations, the early works were focused on the irrotational fluids, which began with the local well-posedness for the small initial data @cite_22 @cite_38 @cite_1 and was generalized to the general initial data by the breakthrough of Wu @cite_33 @cite_39 , see also Lannes @cite_0 and Ambrose and Masmoudi @cite_13 @cite_26 . For the general incompressible Euler equations without the irrotational assumption, the first local well-posedness was obtained by Lindblad @cite_5 for the case without surface tension and by Coutand and Shkoller @cite_2 for the case with (and without) surface tension, see also Christodoulou and Lindblad @cite_43 , Schweizer @cite_24 , Shatah and Zeng @cite_14 and Zhang and Zhang @cite_12 . We also refer to the local well-posedness in conormal Sobolev spaces as the inviscid limits of the free-boundary incompressible Navier-Stokes equations, see Masmoudi and Rousset @cite_8 and the authors @cite_11 .
@cite_38 @cite_26 @cite_14 @cite_22 @cite_33 @cite_8 @cite_1 @cite_39 @cite_0 @cite_43 @cite_24 @cite_2 @cite_5 @cite_13 @cite_12 @cite_11
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We consider the nonstationary waves on the surface of an incompressible perfect fluid of finite depth above the almost horizontal bottom in the case of two dimensional irrotational motion. We assume that the density of mass is equal to one, the gravitational field to (0, — 1) and at the time f^O the fluid occupies the domain 0 ( t ) ^ ( y l 9 y 2 ) y l e R l 9 -h + b(yl)^y2^fi(t9 y,) where h is a positive constant. We denote by Fb the bottom y2= — i + Kj'i) and by Fs the free surface y2 = rf(t, J'i)The motion of the fluid occupying at t = 0 the given domain Q is described by the velocity v = (vl9 v2)9 the pressure p of the fluid and rj satisfying the equations In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to 0, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension. © 2007 Wiley Periodicals, Inc. . We consider the motion of the interface of 2-D irrotational, incompressible, inviscid water wave, with air above water and surface tension zero. We show that the interface is always not accelerating into the water region, normal to itself, as rapidly as the normal acceleration of gravity, as long as the interface is nonself-intersect. We therefore obtain the existence and uniqueness of solutions of the full water wave problem, locally in time, for any initial interface which is nonself-intersect, including the case that the interface is of multiple heights. We study the inviscid limit of the free boundary Navier–Stokes equations. We prove the existence of solutions on a uniform time interval by using a suitable functional framework based on Sobolev conormal spaces. This allows us to use a strong compactness argument to justify the inviscid limit. Our approach does not rely on the justification of asymptotic expansions. In particular, we get a new existence result for the Euler equations with free surface from the one for Navier–Stokes. In an apparatus having at least one rotary head for reproducing video signals recorded in successive parallel tracks extending obliquely across a tape which is advanced longitudinally and also has control signals recorded therein in predetermined positional relationship to the respective record tracks to be reproduced by a fixed transducer or head with a repetition rate that is dependent on the speed of advancement of the tape: a speed and phase control system for the rotary head or heads has a rotational speed detector for producing pulse signals representative of the rotational position of each rotary head and having a repetition rate that is a function of the rotational speed of the rotary head; a phase comparator for producing a phase responsive signal having a voltage level that is varied in response to changes in the relative timing of the reproduced control signals and the pulse signals; a digital device for producing a speed responsive signal having a voltage level that generally corresponds to the repetition rate of the pulse signals, but which is varied in response to changes in the repetition rate of the pulse signals only when such changes occur without corresponding changes in the repetition rate of the reproduced control signals; and a brake or other means for controlling the rotation of the rotary head in response to the voltage levels of the phase responsive signal and the speed responsive signal. In a preferred embodiment, the digital device for producing the speed responsive signal includes a source of clock pulses, a counter for counting the clock pulses for time intervals determined by the repetition rate of the pulse signals from the rotational speed detector, a digital-analog converter for converting the count of the counter, at the end of the mentioned time intervals, to a corresponding voltage level of the speed responsive signal, and an arrangement for establishing the initial count of the counter at the commencement of each of the time intervals in dependence on the repetition rate of the reproduced control signals. Abstract We study an incompressible ideal fluid with a free surface that is subject to surface tension; it is not assumed that the fluid is irrotational. We derive a priori estimates for smooth solutions and prove a short-time existence result. The bounds are obtained by combining estimates of energy type with estimates of vorticity type and rely on a careful study of the regularity properties of the pressure function. An adequate artificial coordinate system is used instead of the standard Lagrangian coordinates. Under an assumption on the vorticity, a solution to the Euler equations is obtained as a vanishing viscosity limit of solutions to appropriate Navier–Stokes systems. We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler?s equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a ?physical condition?, related to the fact that the pressure of a fluid has to be positive. We consider two-dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases. c In this paper, we consider in three dimensions the motion of a general inviscid, incompressible fluid with a free interface that separates the fluid region from the vacuum. We assume that the fluid region is below the vacuum and that there is no surface tension on the free surface. Then we prove the local well-posedness of the free boundary problem in Sobolev space provided that there is no self-intersection point on the initial surface and under the stability assumption that with ξ being restricted to the initial surface. © 2007 Wiley Periodicals, Inc. We consider the dynamics of a layer of viscous incompressible fluid, subjected to gravity and surface tension forces. The upper boundary is a free boundary, and the lower boundary is a fixed boundary on which the Navier-slip boundary condition is imposed. It is proved that there is a uniform time interval on which the uniform estimates independent of both viscosity and surface tension coefficients can be established. These allow one to justify the vanishing viscosity and surface tension limits by the strong compactness argument. In particular, a unified local well-posedness of the free-surface incompressible Euler equations with or without surface tension can be obtained by the inviscid limits.