growth rates for the linearized motion of fluid interfaces away from equilibrium we consider the motion of a two dimensional interface separating an inviscid incompressible irrotational fluid influenced by gravity from a region of zero density we show that under certain conditions the equations of motion linearized about a presumed time dependent solution are wellposed that is linear disturbances have a bounded rate of growth if surface tension is neglected the linear equations are well posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity similar to the criterion formulated by g i taylor if surface tension is included the linear equations are well posed without qualifications whether the fluid is above or below the interface an interesting qualitative structure is found for the linear equations a lagrangian approach is used like that of numerical work such as 3 except that the interface is assumed horizontal at infinity certain integral equations which occur involving double layer potentials are shown to be solvable in the present case 1993 john wiley sons inc

stochastic transport of particles in straining flows important features associated with the segregration of particles in turbulent flow are investigated by considering the statistical distribution phase space number density of particles subject to the combined effects of straining flow and stochastic forcing a fokker planck model is used to obtain results for the phase space distributions of particles that are entrained into straining flow fields the analysis shows that in marked contrast to the zero strain case nonsingular steady state distributions are generated and also confirms that the diffusional effect resulting from stochastic forcing is sufficient to offset the otherwise singular distributions that would result from the indefinite accumulation of particles along stagnation lines the influence of particle inertia stokes number on the form of the resulting distributions is considered and several significant results are observed the influence of strain rate on the attenuation of particle kinetic stresses is quantified and explained the development of large third order velocity moments is observed for stokes numbers above a critical value the mechanism underlying this phenomenon is seen to be a generic feature of particle transport in flows where vortex structures induce local counterflows of particles the system therefore provides an ideal test for closure models for third order moments of particle velocities and here the standard chapman enskog approximation is assessed

on a magnetohydrodynamic problem of euler type abstract the magnetohydrodynamic or mhd equations of an incompressible and homogeneous plasma are considered in the limit of vanishing viscosity and resistivity the plasma spreading over a smoothly bounded domain g of r n n 2 the problem is posed and solved as an equation of evolution in a space of divergence free vector fields parallel to the boundary of g main results are existence uniqueness and regularity of maximal solutions and continuous dependence on forcing terms and initial data the proof of existence is based on galerkin s method and a priori estimates the continuity result is obtained by use of a regularization procedure the approach is inspired by the work of r temam and t kato and c y lai on the euler equations of ideal fluid flow

the existence of current vortex sheets in ideal compressible magnetohydrodynamics we prove the local in time existence of solutions with a surface of current vortex sheet tangential discontinuity of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the initial discontinuity this paper is a natural completion of our previous analysis trakhinin in arch ration mech anal 177 331 366 2005 where a sufficient condition for the weak stability of planar current vortex sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities the original nonlinear problem is a free boundary hyperbolic problem since the free boundary is characteristic the functional setting is provided by the anisotropic weighted sobolev spaces h m the fact that the kreiss lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates therefore we prove our existence theorem by a suitable nash moser type iteration scheme

existence and stability of compressible current vortex sheets in three dimensional magnetohydrodynamics compressible vortex sheets are fundamental waves along with shocks and rarefaction waves in entropy solutions to multidimensional hyperbolic systems of conservation laws understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions for the euler equations in two dimensional gas dynamics the classical linearized stability analysis on compressible vortex sheets predicts stability when the mach number m sqrt 2 and instability when m sqrt 2 for the euler equations in three dimensions every compressible vortex sheet is violently unstable and this instability is the analogue of the kelvin helmholtz instability for incompressible fluids the purpose of this paper is to understand whether compressible vortex sheets in three dimensions which are unstable in the regime of pure gas dynamics become stable under the magnetic effect in three dimensional magnetohydrodynamics mhd one of the main features is that the stability problem is equivalent to a free boundary problem whose free boundary is a characteristic surface which is more delicate than noncharacteristic free boundary problems another feature is that the linearized problem for current vortex sheets in mhd does not meet the uniform kreiss lopatinskii condition these features cause additional analytical difficulties and especially prevent a direct use of the standard picard iteration to the nonlinear problem in this paper we develop a nonlinear approach to deal with these difficulties in three dimensional mhd we first carefully formulate the linearized problem for the current vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates especially high order energy estimates in terms of the nonhomogeneous terms and variable coefficients then we exploit these results to develop a suitable iteration scheme of the nash moser hormander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence which leads to the existence and stability of compressible current vortex sheets locally in time in three dimensional mhd

well posedness for the motion of an incompressible liquid with free surface boundary 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

partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations in this paper we study the local behavior of the solutions to the three dimensional magnetohydrodynamic equations we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions it is shown that in some neighborhood of x0 t0 the gradients of the velocity field u and the magnetic field b are locally uniformly bounded in l norm as long as that either the scaled local l2 norm of the gradient or the scaled local total energy of the velocity field is small and the scaled local total energy of the magnetic field is uniformly bounded these estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory as an immediately corollary we can derive an estimates of hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid various partial regularity results are obtained as consequences of our blow up estimates

nonstationary peristaltic transport of a heat conducting fluid abstract we study the peristaltic transport of a heat conducting fluid in a flexible lube with a prescribed pressure drop over a wavelength of the lube and newton s cooling law for the temperature and a prescribed velocity on the boundary using the oberbeck boussinesq equations as the governing equations and introducing a generalized solution we prove existence uniqueness and stability theorems for the nonstationary problem and develop and justify the stokes and long wave asymptotic expansions

well posedness for the linearized motion of a compressible liquid with free surface boundary we study the motion of a compressible perfect liquid body in vacuum this can be through 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 linearized stability in sobolev space assuming a physical condition related to the fact that the pressure of a fluid has to be positive

the zero surface tension limit of two dimensional water waves 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

on the well posedness of a linearized plasma vacuum interface problem in ideal compressible mhd abstract we study the initial boundary value problem resulting from the linearization of the plasma vacuum interface problem in ideal compressible magnetohydrodynamics mhd we suppose that the plasma and the vacuum regions are unbounded domains and the plasma density does not go to zero continuously but jumps for the basic state upon which we perform linearization we find two cases of well posedness of the frozen coefficient problem the gas dynamical case and the purely mhd case in the gas dynamical case we assume that the jump of the normal derivative of the total pressure is always negative in the purely mhd case this condition can be violated but the plasma and the vacuum magnetic fields are assumed to be non zero and non parallel to each other everywhere on the interface for this case we prove a basic a priori estimate in the anisotropic weighted sobolev space h 1 for the variable coefficient problem

a priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows in the present paper we prove the a priori estimates of sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions n 2 and 3 by adopting a geometrical point of view used in christodoulou and lindblad commun pure appl math 53 1536 1602 2000 and estimating quantities such as the second fundamental form and the velocity of the free surface we identify the well posedness condition thattheouternormalderivativeofthetotalpressureincludingthefluidandmagnetic pressuresisnegativeonthefreeboundary whichissimilartothephysicalcondition taylor sign condition for the incompressible euler equations of fluids

well posedness in sobolev spaces of the full water wave problem in 2 d 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

well posedness of the free surface incompressible euler equations with or without surface tension 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

on the free boundary problem of three dimensional incompressible euler equations 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 being restricted to the initial surface 2007 wiley periodicals inc

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

the fixed boundary value problems for the equations of ideal magneto hydrodynamics with a perfectly conducting wall condition the equations of ideal magneto hydrodynamics are investigated concerning initial boundary value problems with a perfectly conducting wall condition the local in time solution is proved to exist uniquely provided that the normal component of the initial magnetic field vanishes everywhere or nowhere on the boundary

geometry and a priori estimates for free boundary problems of the euler s equation 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

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

global strong solution to the three dimensional density dependent incompressible magnetohydrodynamic flows abstract an initial boundary value problem is considered for the density dependent incompressible viscous magnetohydrodynamic flow in a three dimensional bounded domain the homogeneous dirichlet boundary condition is prescribed on the velocity and the perfectly conducting wall condition is prescribed on the magnetic field for the initial density away from vacuum the existence and uniqueness are established for the local strong solution with large initial data as well as for the global strong solution with small initial data furthermore the weak strong uniqueness of solutions is also proved which shows that the weak solution is equal to the strong solution with certain initial data