direct simulation of the motion of a settling ellipsoid in newtonian fluid in this paper we discuss the generalization of a lagrange multiplier based fictitious domain method to the simulation of the motion of general shape particles in newtonian fluid preliminary numerical results of a settling ellipsoid in a narrow channel filled with a newtonian fluid are presented as expected the ellipsoid turns its broadside to the stream in the simulations

a fictitious domain formulation for flows with rigid particles in this paper we present a development of the fictitious domain method proposed in ref c diaz goano p minev k nandakumar a fictitious domain finite element method for particulate flows j comput phys 192 2003 105 the main new feature of the modified method is that after a proper splitting it avoids the need to use lagrange multipliers for imposition of the rigid body motion and instead it resolves the interaction force between the two phases explicitly then the end of step fluid velocity is a solution of an integral equation the most straightforward way to resolve it is via an iteration but a direct extrapolation is also possible if the latter approach is applied then the fictitious domain formulation becomes fully explicit with respect to the rigid body constraint and therefore the corresponding numerical procedure is much cheaper most of the numerical results presented in this article are obtained with such an explicit formulation

a relaxation method for constructing a beltrami flow in a bounded domain it is shown that a time dependent equation generates a beltrami flow or a force free field at time t from non beltrami initial data in an arbitrary bounded domain in three dimensions the generated beltrami flow has a zero normal component on the boundary and its helicity is equal to a prescribed value the existence and uniqueness of a solution to the equation are guaranteed globally in time and no finite time singularity occurs

on the well posedness of the euler equations in the triebel lizorkin spaces we prove local in time unique existence and a blowup criterion for solutions in the triebel lizorkin space for the euler equations of inviscid incompressible fluid flows in n n 2 as a corollary we obtain global persistence of the initial regularity characterized by the triebel lizorkin spaces for the solutions of two dimensional euler equations to prove the results we establish the logarithmic inequality of the beale kato majda type the moser type of inequality as well as the commutator estimate in the triebel lizorkin spaces the key methods of proof used are the littlewood paley decomposition and the paradifferential calculus by j m bony 2002 john wiley sons inc

self stretching of a perturbed vortex filament i the asymptotic equation for deviations from a straight line a new asymptotic equation is derived for the motion of thin vortex filaments in an incompressible fluid at high reynolds numbers this equation differs significantly from the familiar local self induction equation in that it includes self stretching of the filament in a nontrivial but to some extent analytically tractable fashion under the same change of variables as employed by hasimoto 1972 to convert the local self induction equation to the cubic nonlinear schrodinger equation the new asymptotic propagation law becomes a cubic nonlinear schrodinger equation perturbed by an explicit nonlocal linear operator explicit formulae are developed which relate the rate of local self stretch along the vortex filament to a particular quadratic functional of the solution of the perturbed schrodinger equation the asymptotic equation is derived systematically from suitable solutions of the navier stokes equations by the method of matched asymptotic expansions based on the limit of high reynolds numbers the key idea in the derivation is to consider a filament whose core deviates initially from a given smooth curve only by small amplitude but short wavelength displacements balanced so that the axial length scale of these perturbations is small compared to an integral length of the background curve but much larger than a typical core size o re 1 2 of the filament in a particular distinguished limit of wavelength preturbation amplitude and filament core size the nonlocal induction integral has a simplified asymptotic representation and yields a contribution in the schrodinger equation that directly competes with the cubic nonlinearity

a nested iterative scheme for indefinite linear systems in particulate flows abstract high fidelity large scale direct numerical simulation of particulate flows is of great value in a variety of industrial applications it is computationally intensive as it combines time integration solving nonlinear algebraic equations and the associated linear systems the finite element discretization of the coupled system of pdes on an unstructured grid using an arbitrary lagrangian eulerian moving mesh technique leads to very large nonlinear systems that are linearized by a version of newton s method the linear algebraic systems jacobians are sparse nonsymmetric and indefinite for which standard linear system solvers based on krylov subspace methods generally fail to converge without appropriate preconditioners the failure of krylov methods in production codes is currently being addressed by reducing the size of the time step this however leads to a very long simulation time and therefore is not always a viable approach in this study we design a hybrid inner outer iterative scheme for solving these indefinite systems which proves to be both efficient robust and ideally suited for parallel computing platforms even with appropriate large time steps comparisons with krylov subspace methods show the superiority of our proposed class of nested iterative schemes which is also scalable with respect to mesh size and insensitive to changes in properties of the fluid particles system

on the evolution of nearly circular vortex patches recently the classical problem of the evolution of patches of constant vorticity was reformulated as an evolution equation for the boundary of the patch we study this equation in the neighborhood of the circular vortex patch and introduce a hierarchy of area preserving nonlinear approximate equations the first of these equations is shown to have a rich rigid structure it possesses an exhaustive increasing sequence of linear invariant manifolds of arbitrarily large finite dimensions on each of these manifolds the equation can be written as an explicit finite system of ordinary differential equations solutions of these odes starting from arbitrarily small neighborhoods of the circular vortex patch are shown to blow up

a new description of force free magnetic fields the moses curl eigenfunctions are used to describe linear force free magnetic fields enabling a proof that such fields are defined entirely by the value of their curl transform on the unit hemisphere in transform space this change of viewpoint suggests an orderly approach to the exploration and classification of the properties of such fields which is briefly sketched the simplest force free fields defined on zero one and two dimensional sets on the transform sphere are exhibited and the possibility of fields with fractal support sets suggested connections with other mathematical descriptions are pointed out as well as several promising directions for further exploration all results apply equally well to the description of the trkalian subset of beltrami fields in fluid dynamics

numerical modelling of electromechanical coupling using fictitious domain and level set methods we present a finite element formulation for simulation of electromechanical coupling using a combination of fictitious domain and level set methods the electric field is treated with a fixed eulerian like mesh whereas the structure taken as a perfect conductor is modelled with a conventional lagrangian approach the compatibility between the potential of the conductor and of the electric domain is obtained by introducing a lagrange multiplier function defined on the boundary of the conductor the electromechanical forces are obtained using a variational formulation for the coupled electromechanical domain we use a heaviside function on the level set to remove the electric energy in the conductor domain results are presented for an radio frequency switch and an element of a comb drive

on the helicity in 3d periodic navier stokes equations i the non statistical case we study the asymptotic behavior of the statistical solutions to the navier stokes equations using the normalization map 9 it is then applied to the study of mean energy mean dissipation rate of energy and mean helicity of the spatial periodic flows driven by potential body forces the statistical distribution of the asymptotic beltrami flows are also investigated we connect our mathematical analysis with the empirical theory of decaying turbulence with appropriate mathematically defined ensemble averages the kolmogorov universal features are shown to be transient in time we provide an estimate for the time interval in which those features may still be present our collaborator and friend basil nicolaenko passed away in september of 2007 after this work was completed honoring his contribution and friendship we dedicate this article to him

ensuring well posedness by analogy stokes problem and boundary control for the wave equation abstract in this article we give a comparative discussion of the finite element approximation of two partial differential equation problems these two problems which are apparantly quite unrelated are the stokes problem for incompressible viscous flow and an exact boundary controllability problem for the wave equation we show that straightforward discrete approximations to these problems yield approximate problems which are ill posed the analysis of the ill posedness of the above problems shows an identical cause namely the strong damping of the high frequency modes beyond a critical wave number from this analogy a well known cure for the discrete stokes problem i e using more accurate approximations for velocity than for pressure provides a simple way to eliminate the ill posedness of the discrete exact boundary controllability problem numerical examples concerning the control problem testify about the soundness of the new approach to conclude this paper one takes advantage of the previous analysis to give a brief discussion of the wavelet approximation of the stokes problem for dirichlet boundary conditions

on the helicity in 3d periodic navier stokes equations ii the statistical case we study the asymptotic behavior of the statistical solutions to the navier stokes equations using the normalization map 9 it is then applied to the study of mean energy mean dissipation rate of energy and mean helicity of the spatial periodic flows driven by potential body forces the statistical distribution of the asymptotic beltrami flows are also investigated we connect our mathematical analysis with the empirical theory of decaying turbulence with appropriate mathematically defined ensemble averages the kolmogorov universal features are shown to be transient in time we provide an estimate for the time interval in which those features may still be present our collaborator and friend basil nicolaenko passed away in september of 2007 after this work was completed honoring his contribution and friendship we dedicate this article to him

local regularity for the modified sqg patch equation we study the patch dynamics on the whole plane and on the half plane for a family of active scalars called modified surface quasi geostrophic sqg equations these involve a parameter that appears in the power of the kernel in their biot savart laws and describes the degree of regularity of the equation the values 0 and correspond to the two dimensional euler and sqg equations respectively we establish here local in time regularity for these models for all 0 on the whole plane and for all small 0 on the half plane we use the latter result in 16 where we show existence of regular initial data on the half plane that lead to a finite time singularity 2016 wiley periodicals inc

numerical methods for the stokes and navier stokes equations driven by threshold slip boundary conditions abstract in this article we discuss the numerical solution of the stokes and navier stokes equations completed by nonlinear slip boundary conditions of friction type in two and three dimensions to solve the stokes system we first reduce the related variational inequality into a saddle point point problem for a well chosen augmented lagrangian to solve this saddle point problem we suggest an alternating direction method of multiplier together with finite element approximations the solution of the navier stokes system combines finite element approximations time discretization by operator splitting and augmented lagrangian method numerical experiment results for two and three dimensional flow confirm the interest of these approaches

a spectral fictitious domain method with internal forcing for solving elliptic pdes a fictitious domain method is presented for solving elliptic partial differential equations using galerkin spectral approximation the fictitious domain approach consists in immersing the original domain into a larger and geometrically simpler one in order to avoid the use of boundary fitted or unstructured meshes in the present study boundary constraints are enforced using lagrange multipliers and the novel aspect is that the lagrange multipliers are associated with smooth forcing functions compactly supported inside the fictitious domain this allows the accuracy of the spectral method to be preserved unlike the classical discrete lagrange multipliers method in which the forcing is defined on the boundaries in order to have a robust and efficient method equations for the lagrange multipliers are solved directly with an influence matrix technique using a fourier chebyshev approximation the high order accuracy of the method is demonstrated on one and two dimensional elliptic problems of second and fourth order the principle of the method is general and can be applied to solve elliptic problems using any high order variational approximation

axisymmetric solutions to the 3d euler equations abstract here we consider the 3 d incompressible euler equations with axisymmetric velocity without swirl first we will show that if u 0 c s l 2 s 1 and 0 x c x 1 2 x 2 2 then there exists a unique u c 0 c s that solves the equation this conclusion improves on the related results given by majda a l bertozzi a majda vorticity and incompressible flow in cambridge texts in applied mathematics vol 27 2002 a majda vorticity and the mathematical theory of incompressible fluid flow comm pure appl math 39 1986 s187 s220 and by raymond x saint raymond remarks on axisymmetric solutions of the incompressible euler system comm partial differential equations 19 1994 321 334 on the other hand if u 0 l 2 0 l and 0 x 1 2 x 2 2 l then there exists a unique quasilipschitzian solution u c 0 c 1 this improves on the corresponding results due to raymond x saint raymond remarks on axisymmetric solutions of the incompressible euler system comm partial differential equations 19 1994 321 334 and to chae and kim d chae n kim axisymmetric weak solutions of the 3 d euler equations for incompressible fluid flows nonlinear anal 29 12 1997 1393 1404

on the geometric regularity conditions for the 3d navier stokes equations abstract we prove geometrically improved version of prodi serrin type blow up criterion let v and be the velocity and the vorticity of solutions to the 3d navier stokes equations and denote f max f 0 q t r 3 0 t if v v v l x t q t with 3 2 1 for some 3 and 1 2 then the local smooth solution v of the navier stokes equations on 0 t can be continued to 0 t for some 0 we also prove localized version of a special case of this let v be a suitable weak solution to the navier stokes equations in a space time domain containing z 0 x 0 t 0 let q z 0 r b x 0 r t 0 r 2 t 0 be a parabolic cylinder in the domain we show that if either v l x t q z 0 r with 3 2 1 or v v l x t q z 0 r with 3 2 2 2 2 then z 0 is a regular point for v this improves previous local regularity criteria for the suitable weak solutions

strongly coupled dynamics of fluids and rigid body systems with the immersed boundary projection method a strong coupling algorithm is presented for simulating the dynamic interactions between incompressible viscous flows and rigid body systems in both two and three dimensional problems in this work the navier stokes equations for incompressible flow are solved on a uniform cartesian grid by the vorticity based immersed boundary projection method of colonius and taira dynamical equations for arbitrary rigid body systems are also developed the proposed coupling method attempts to unify the treatment of constraints in the fluid and structure the incompressibility of the fluid the linkages in the rigid body system and the conditions at the interface through the use of lagrange multipliers the resulting partitioned system of equations is solved with a simple relaxation scheme based on an identification of virtual inertia from the fluid the scheme achieves convergence in only 2 to 5 iterations per time step for a wide variety of mass ratios the formulation requires that only a subset of the discrete fluid equations be solved in each iteration several two and three dimensional numerical tests are conducted to validate and demonstrate the method including a falling cylinder flapping of flexible wings self excited oscillations of a system of many linked plates in a free stream and passive pivoting of a finite aspect ratio plate under the influence of gravity in a free stream the results from the current method are compared with previous experimental and numerical results and good agreement is achieved

boundary holder and estimates for local solutions of the tangential cauchy riemann equation nous etudions la resolubilite locale de l operateur de cauchy riemann tangentiel sur un voisinage w d un point z 0 d une sous variete cr generique q concave m de codimension quelconque de c n nous construisons une formule d homotopie pour le b sur lorsque est l intersection de m et d un domaine strictement pseudoconvexe nous obtenons ainsi un operateur de resolution pour toute forme b fermee sur nous en deduisons des estimations l p et des estimations holderiennes jusqu au bord pour la solution de l equation de cauchy riemann tangentielle sur

numerical simulation of a multi store separation phenomenon a fictitious domain approach abstract in this article we discuss the numerical simulation of multi store separation phenomena taking place in incompressible newtonian viscous fluids we assume that these phenomena are modeled by the navier stokes equations coupled to the euler newton equations describing solid rigid body motions the numerical methodology relies on the combination of a finite element approximation with time discretization by operator splitting and a lagrange multiplier based fictitious domain method allowing the flow calculations to take place in a fixed simple shape space region the results of numerical experiments are presented they concern the separation and free fall of elongated ellipsoids under an airfoil shaped cylinder

on the motion of a neutrally buoyant ellipsoid in a three dimensional poiseuille flow in previous articles the authors introduced lagrange multiplier based fictitious domain methods their goal in the present article is to apply a generalization of the above methods to simulate the motion of a neutrally buoyant ellipsoid in a three dimensional poiseuille flow and to investigate its rotational and orientational behavior via direct numerical simulations we find distinctive states depending on the reynolds number ranges and the shape of the ellipsoid

dissipative continuous euler flows we show the existence of continuous periodic solutions of the 3d incompressible euler equations which dissipate the total kinetic energy

wavelet and finite element solutions for the neumann problem using fictitious domains this paper presents a new fictitious domain formulation for the solution of a strongly elliptic boundary value problem with neumann boundary conditions for a bounded domain in a finite dimensional euclidean space with a smooth possibly only lipschitz boundary this extends the domain to a larger rectangular domain with periodic boundary conditions for which fast solvers are available the extended solution converges on the original domain in the appropriate function spaces as the penalty parameter approaches zero both wavelet galerkin and finite elements numerical approximation schemes are developed using this methodology the convergence rates of both schemes are comparable and the use of finite elements requires a parameterization of the boundary while the wavelet galerkin method admits an implicit description of the boundary in terms of a wavelet representation of the boundary measure defined as the distributional gradient of the characteristic function of the interior the accuracy of both methods is investigated and compared both theoretically and for numerical test cases the conclusion is that the methods are comparable and that the wavelet method allows the use of more general boundaries which are not explicitly parametrized which would be of greater advantage in higher dimensions the numerical tests are carried out in two dimensions

the asymptotic behavior of the smooth solutions of the euler equations in r2 abstract in this paper we consider the smooth solutions u x t with suitable decay at infinity of the euler equations on r 2 0 we assume that the initial vorticity 0 u 0 has a compact support which leads by marchioro 1994 1996 to a consequence that the support of the vorticity t for any finite time is a bounded set then we will show that the decay growth rate in l p r 2 spaces for 2 p for 1 p 2 of the smooth solutions of the euler equations on r 2 cannot be faster than t 1 3 2 3 p as t we will see that the l p norm of the solution u x t is equivalent to the l q norm for any 1 q p

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

a stable numerical algorithm for the brinkman equations by weak galerkin finite element methods this paper presents a stable numerical algorithm for the brinkman equations by using weak galerkin wg finite element methods the brinkman equations can be viewed mathematically as a combination of the stokes and darcy equations which model fluid flow in a multi physics environment such as flow in complex porous media with a permeability coefficient highly varying in the simulation domain in such applications the flow is dominated by darcy in some regions and by stokes in others it is well known that the usual stokes stable elements do not work well for darcy flow and vice versa the challenge of this study is on the design of numerical schemes which are stable for both the stokes and the darcy equations this paper shows that the wg finite element method is capable of meeting this challenge by providing a numerical scheme that is stable and accurate for both darcy and the stokes dominated flows error estimates of optimal order are established for the corresponding wg finite element solutions the paper also presents some numerical experiments that demonstrate the robustness reliability flexibility and accuracy of the wg method for the brinkman equations

on the euler equations in the critical triebel lizorkin spaces in this paper we study the initial value problem of the incompressible euler equations in n for initial data belonging to the critical triebel lizorkin spaces i e v0 fn 11 q q 1 we prove the blow up criterion of solutions in fn 11 q for n 2 3 for n 2 in particular we prove global well posedness of the euler equations in f31 q q 1 for the proof of these results we establish a sharp moser type inequality as well as a commutator type estimate in these spaces the key methods are the littlewood paley decomposition and the paradifferential calculus by j m bony

remarks on a nonlocal transport abstract we consider a one dimensional nonlocal transport equation and its natural multi dimensional analogues by using a new pointwise inequality for the hilbert transform we give a short proof of a nonlinear inequality first proved by cordoba cordoba and fontelos in 2005 we also prove several new weighted inequalities for the hilbert transform and various nonlinear versions some of these results generalize to a related family of nonlocal models

a fast immersed boundary method for external incompressible viscous flows using lattice green s functions a new parallel computationally efficient immersed boundary method for solving three dimensional viscous incompressible flows on unbounded domains is presented immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original peskin s immersed boundary method unlike peskin s method boundary forces are regarded as lagrange multipliers that are used to satisfy the no slip condition the incompressible navier stokes equations are discretized on an unbounded staggered cartesian grid and are solved in a finite number of operations using lattice green s function techniques these techniques are used to automatically enforce the natural free space boundary conditions and to implement a novel block wise adaptive grid that significantly reduces the run time cost of solutions by limiting operations to grid cells in the immediate vicinity and near wake region of the immersed surface these techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half explicit runge kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation incompressibility constraint and no slip constraint linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation free nested projection technique the algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems e g discrete poisson problems that are compatible with recent parallel fast multipole methods for difference equations numerical experiments on low aspect ratio flat plates and spheres at reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation

distributed lagrange multiplier methods for incompressible viscous flow around moving rigid bodies abstract in this article we discuss the application of a distributed lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modelled by the navier stokes equations around moving bodies we suppose the rigid bodies motion known a priori the solution method combines finite element approximations time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method numerical experiment results for two dimensional flow around a moving disk are presented

beltrami fields with a nonconstant proportionality factor are rare we consider the existence of beltrami fields with a nonconstant proportionality factor f in an open subset u of mathbb r 3 by reformulating this problem as a constrained evolution equation on a surface we find an explicit differential equation that f must satisfy whenever there is a nontrivial beltrami field with this factor this ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the c k topology k geqq 7 in particular there are no nontrivial beltrami fields whenever f has a regular level set diffeomorphic to the sphere this provides an explanation of the helical flow paradox of morgulis et al commun pure appl math 48 571 582 1995

serfati solutions to the 2d euler equations on exterior domains abstract we prove existence and uniqueness of a weak solution to the incompressible 2d euler equations in the exterior of a bounded smooth obstacle when the initial data is a bounded divergence free velocity field having bounded scalar curl this work completes and extends the ideas outlined by p serfati for the same problem in the whole plane case with non decaying vorticity the biot savart integral does not converge and thus velocity cannot be reconstructed from vorticity in a straightforward way the key to circumventing this difficulty is the use of the serfati identity which is based on the biot savart integral but holds in more general settings

an initial boundary value problem for one dimensional shallow water magnetohydrodynamics in the solar tachocline abstract in this article we investigate the shallow water magnetohydrodynamic equations in space dimension one with dirichlet boundary conditions only for the velocity this model has been proposed to study the phenomena in the solar tachocline in this article the local well posedness in time of the model is proven by constructing the approximate solutions and showing the strong convergence of the approximate solutions

a three dimensional computational method for blood flow in the heart 1 immersed elastic fibers in a viscous incompressible fluid abstract this paper describes the numerical solution of the 3 dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers implementation details such as vectorization and the efficient use of external memory are discussed the method is applied to the damped vibrations of a fiber wound toroidal tube and empirical evidence of convergence is presented

splitting method for the combined formulation of the fluid particle problem abstract in this paper a splitting method for the direct numerical simulation of solid liquid mixtures is presented where a symmetric pressure equation is newly proposed through numerical experiment it is found that the newly proposed splitting method works well with a matrix free formulation for some benchmark problems avoiding an erroneous pressure field which appears when using the conventional pressure equation of a splitting method when deriving a typical pressure equation of a splitting method the motion of a solid particle has to be approximated by the intermediate velocity instead of treating it as unknowns since it is necessary as a boundary condition therefore the motion of a solid particle is treated in such an explicit way that a particle moves by the known form drag pressure drag that is calculated from the pressure equation in the previous step from the numerical experiment it was shown that this method gives an erroneous pressure field even for a very small time step size as the particle velocity increases in this paper coupling the unknowns of particle velocities in the pressure equation is proposed where the resulting matrix is reduced to the symmetric one by applying the projector of the combined formulation it has been tested over some benchmark problems and gives reasonable pressure fields

vortices and invariant surfaces generated by symmetries for the 3d navier stokes equations abstract we show that certain infinitesimal operators of the lie point symmetries of the incompressible 3d navier stokes equations give rise to vortex solutions with different characteristics this approach allows an algebraic classification of vortices and throws light on the alignment mechanism between the vorticity and the vortex stretching vector s where s is the strain matrix the symmetry algebra associated with the navier stokes equations turns out to be infinite dimensional new vortical structures generalizing in some cases well known configurations such as for example the burgers and lundgren solutions are obtained and discussed in relation to the value of the dynamic angle arctan s s a systematic treatment of the boundary conditions invariant under the symmetry group of the equations under study is also performed and the corresponding invariant surfaces are recognized

the range of the tangential cauchy riemann operator over a small ball in this paper we study the closed range property of tangential cauchy riemann equations of an abstract cr structure over a small ball let d be the set x e r 1 1x1 4 kuranishi 141 and akahori 11 have studied the l theory

on the spectral accuracy of a fictitious domain method for elliptic operators in multi dimensions this work is a continuation of the authors efforts to develop high order numerical methods for solving elliptic problems with complex boundaries using a fictitious domain approach in a previous paper a new method was proposed based on the use of smooth forcing functions with identical shapes mutually disjoint supports inside the fictitious domain and whose amplitudes play the role of lagrange multipliers in relation to a discrete set of boundary constraints for one dimensional elliptic problems this method shows spectral accuracy but its implementation in two dimensions seems to be limited to a fourth order algebraic convergence rate in this paper a spectrally accurate formulation is presented for multi dimensional applications instead of being specified locally the forcing function is defined as a convolution of a mollifier smooth bump function and a lagrange multiplier function the amplitude of the bump the multiplier function is then approximated by fourier series using a fourier galerkin approximation the spectral accuracy is demonstrated on a two dimensional laplacian problem and on a stokes flow around a periodic array of cylinders in the latter the numerical solution achieves the same high order accuracy as a stokes eigenfunction expansion and is much more accurate than the solution obtained with a classical third order finite element approximation using the same number of degrees of freedom

adjoint based optimization of particle trajectories in laminar flows we present a performant numerical algorithm for the optimal control of particle controls in low reynolds number flows in particular circular particles with mass are considered an optimal control problem for the particle trajectories is presented which is solved by means of a steepest descent algorithm here the derivative information is obtained using adjoint calculus finally numerical results are presented underlining the feasibility of our approach

remarques sur l instabilite du probleme des poches de tourbillon resume nous etudions le probleme des poches de tourbillon pour des contours initiaux poches du cercle unite une approximation quadratique de l equation d evolution se revele instable en temps fini la norme lipschitz du contour devenant arbitrairement grande

l p theory of the navier stokes flow in the exterior of a moving or rotating obstacle in this paper we describe two recent approaches for the lp theory of the navier stokes flow in the exterior of a moving or rotating obstacle

contour dynamics for the euler equations in two dimensions we present a contour dynamics algorithm for the euler equations of fluid dynamics in two dimensions this is applied to regions of piecewise constant vorticity withinfinite area vortexregions favrs essentially this reduces the dimensionality by one and we are treating the interaction of closed polygonal contours whose nodes are advected by the total fluid motion computed self consistently a leapfrog centered scheme is used for temporal advancement computer simulation results are given for two and four like signed interacting favrs in all cases wavelike surface deformations are observed if the distance between favrs is comparable to their extent diameter these surface deformations are large they play an essential role in the observed coalescence of favrs

well posedness theory for aggregation sheets in this paper we consider distribution solutions to the aggregation equation rho t mathrm div rho mathbf u 0 mathbf u nabla v rho in mathbb r d where the density concentrates on a co dimension one manifold we show that an evolution equation for the manifold itself completely determines the dynamics of such solutions we refer to such solutions aggregation sheets when the equation for the sheet is linearly well posed we show that the fully non linear evolution is also well posed locally in time for the class of bi lipschitz surfaces moreover we show that if the initial sheet is c1 then the solution itself remains c1 as long as it remains lipschitz lastly we provide conditions on the kernel g s frac mathrm d v mathrm d s that guarantee the solution remains a bi lipschitz surface globally in time and construct explicit solutions that either collapse or blow up in finite time when these conditions fail

structure of helicity and global solutions of incompressible navier stokes equation in this paper we derive a new energy identity for the three dimensional incompressible navier stokes equations by a special structure of helicity the new energy functional is critical with respect to the natural scalings of the navier stokes equations moreover it is conditionally coercive as an application we construct a family of finite energy smooth solutions to the navier stokes equations whose critical norms can be arbitrarily large

a distributed lagrange multiplier fictitious domain method for the simulation of flow around moving rigid bodies application to particulate flow in this article we discuss the application of a lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modeled by the navier stokes equations around moving rigid bodies the rigid body motion is due to hydrodynamical forces and gravity the solution method combines finite element approximations time discretization by operators splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method we conclude this article by the presentation of numerical results concerning the simulation of an incompressible viscous flow around a naca0012 airfoil with a fixed center but free to rotate then the sedimentation of circular cylinders in 2 d channels and finally the sedimentation of spherical balls in cylinders with square cross sections 2000 elsevier science s a all rights reserved

on the dual systems to euler and navier stokes equations in r3 in this paper we consider two model systems of equations obtained by taking the hodge duals of the euler and the navier stokes systems respectively the vorticity formulation of the new system becomes a nonlinear non local scalar equation having similarities to the quasi geostrophic equations we prove the local existence of classical solutions and derive the beale kato majda type of blow up criterion for the dual euler system moreover we actually show that the finite time blow up occurs for generic non trivial initial data for this system for the dual navier stokes equations we construct an explicitly global in time smooth solution for general initial data using the hopf cole transform in the appendix we formulate the euler system and the dual euler system in terms of the clifford algebra

complex trkalian fields and solutions to euler s equations for the ideal fluid we consider solutions to the complex trkalian equation vec nabla times vc vc where vc is a 3 component vector function with each component in the complex field and may be expressed in the form vc e ig vec nabla f with g real and f complex we find there are precisely two classes of solutions one where g is a cartesian variable and one where g is the spherical radial coordinate we consider these flows to be the simplest of all exact 3 d solutions to the euler s equation for the ideal incompressible fluid the novel approach we use in solving for these classes of solutions to these 3 dimensional vector pdes involves differential geometric techniques one may employ the method to generate solutions to other classes of vector pdes

finite time analyticity for the two and three dimensional kelvin helmholtz instability the well posed property for the finite time vortex sheet problem with analytic initial data was first conjectured by birkhoff in two dimensions and is shown here to hold both in two and three dimensions incompressible inviscid and irrotational flow with a velocity jump across an interface is assumed in two dimensions global existence of a weak solution to the euler equation with such initial conditions is established in three dimensions a lagrangian representation of the vortex sheet analogous to the birkhoff equation in two dimensions is presented

blow up of solutions for a 1d transport equation with nonlocal velocity and supercritical dissipation we study a 1d transport equation with nonlocal velocity and supercritical dissipation we show that for a certain class of initial data the solution blows up in finite time

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

on the local solvability and the local integrability of systems of vector fields on etudie des systemes surdetermines de champs vectoriels dans un sous ensemble ouvert de r n 1 de type l j t j j t x x j 1 m ayant des coefficients analytiques on suppose que les champs de vecteurs l j satisfont la condition de frobenius

nonexistence of self similar singularities for the 3d incompressible euler equations we prove that there exists no self similar finite time blowing up solution to the 3d incompressible euler equations if the vorticity decays sufficiently fast near infinity in mathbb r 3 by a similar method we also show nonexistence of self similar blowing up solutions to the divergence free transport equation in mathbb r n this result has direct applications to the density dependent euler equations the boussinesq system and the quasi geostrophic equations for which we also show nonexistence of self similar blowing up solutions

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

notes on the geometric regularity criterion of 3d navier stokes system in this note we show that if the volume of parallelepiped which is defined by three unit vectors direction of the velocity direction of the vorticity and the direction of the curl of the vorticity is small enough then the solution to the 3d navier stokes equations is classical

local existence theorems with estimates for b on weakly pseudo convex cr manifolds in particular we prove the solvability for 0 1 on co for 1 q n 2 when m is a pseudo convex cr hypersurface of finite type in the sense defined in d angelo 11 and c co is the intersection of m with a flat hypersurface m0 see theorem 1 in sect i near a point zo of finite type we also show that there exists a neighborhood basis co of z0 such that 0 1 is solvable in co for any ct satisfying 0 2 see corol lary 1 1 we also obtain the interior regularity of the solutions for 0 1 in the sobolev spaces and show that the range of b is closed in the frechet norm l p q co loc see theorem 2 and corol lary 2 1 in sect 1 since the existence of the solution u of 0 1 was proved on co for any c satisfying 0 2 in co theorems 1 and 2 are semi global results when we assume that mo is only levi flat and with a stein ne ighborhood basis then we prove that for any relatively compact subset co c co there exists a u satisfying 0 2 in co see theorem 3 we note that the local solvability for 0 1 was not known previously even on a small subset in co on general weakly pseudo convex cr manifolds in this paper we shall prove the

the beltrami spectrum for incompressible fluid flows recently v yakhot s orszag and their co workers have suggested that turbulent flows in various regions of space organize into a coherent hierarchy of weakly interacting superimposed approximate beltrami flows a mathematical framework is developed here to study organized beltrami hierarchies in a systematic fashion this framework is applied to several important classes of examples with universal beltrami hierarchies an analysis of the persistence of such beltrami hierarchies is also presented for general solutions of the navier stokes equations

remarks on a paper by j t beale t kato and a majda we prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3 dimensional euler equations more precisely the loss of regularity in a local smooth solution of the euler equations implies the growth without bound of the deformation tensor as the critical time approaches equivalently if the deformation tensor remains bounded the existence of a smooth solution is guaranteed

an operator splitting scheme with a distributed lagrange multiplier based fictitious domain method for wave propagation problems we propose a novel fictitious domain method based on a distributed lagrange multiplier technique for the solution of the time dependent problem of scattering by an obstacle we study discretizations that include a fully conforming approach as well as mixed finite element formulations utilizing the lowest order nedelec edge elements in 2d on rectangular grids we also present a symmetrized operator splitting scheme for the scattering problem which decouples the operator that propagates the wave from the operator that enforces the dirichlet condition on the boundary of an obstacle a new perfectly matched layer pml model is developed to model the unbounded problem of interest this model is based on a formulation of the wave equation as a system of first order equations and uses a change of variables approach that has been developed to construct pml s for maxwell s equations we present an analysis of our fictitious domain approach for a one dimensional wave problem based on calculations of reflection coefficients we demonstrate the advantages of our fictitious domain approach over the staircase approximation of the finite difference method we also demonstrate some important properties of the distributed multiplier approach that are not shared by a boundary multiplier fictitious domain approach for the same problem numerical results for two dimensional wave problems that validate the effectiveness of the different methods are presented

on the global regularity of a helical decimated version of the 3d navier stokes equations we study the global regularity for all time and all initial data in h 1 2 o f ar e cently introduced decimated version of the incompressible 3d navier stokes dns equa tions the model is based on a projection of the dynamical evolution of navier stokes ns equations into the subspace where helicity the l 2 scalar product of velocity and vorticity is sign definite the presence of a second beside energy sign definite inviscid conserved quadratic quantity which is equivalent to the h 1 2 sobolev norm allows us to demonstrate global existence and uniqueness of space periodic solutions together with continuity with respect to the initial conditions for this decimated 3d model this is achieved thanks to the establishment of two new estimates for this 3d model which show that the h 1 2 and the time average of the square of the h 3 2 norms of the velocity field remain finite such two additional bounds are known in the spirit of the work of h fujita and t kato arch ration mech anal 16 269 315 1964 rend semin mat univ padova 32 243 260 1962 to be sufficient for showing well posedness for the 3d ns equations furthermore they are di rectly linked to the helicity evolution for the dns model and therefore with a clear physical meaning and consequences

some notes on periodic beltrami fields in cartesian geometry some mathematical facts about beltrami fields which are time independent solutions of the three dimensional incompressible euler equations with nontrivial helicity are assembled the linearization about an arbitrary fixed beltrami field is studied in a hamiltonian framework a factorization of this linearization is introduced and used to characterize null spaces for beltrami fields with ergodic streamlines an interesting property of unstable modes is noticed and its consequences concerning the nature of potential instabilities are discussed

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

construction of three dimensional stationary euler flows from pseudo advected vorticity equations some vorticity equations with pseudoadvection terms are proposed they yield threedimensional stationary euler flows at t in particular one of them generates the beltrami flow

on the null asymptotic stabilization of the two dimensional incompressible euler equations in a simply connected domain we study the asymptotic stabilization of the origin for the two dimensional 2 d euler equation of incompressible inviscid fluid in a bounded domain we assume that the controls act on an arbitrarily small nonempty open subset of the boundary we prove the null global asymptotic stabilizability by means of explicit feedback laws if the domain is connected and simply connected

an extension of the immersed boundary method based on the distributed lagrange multiplier approach an extended formulation of the immersed boundary method which facilitates simulation of incompressible isothermal and natural convection flows around immersed bodies and which may be applied for linear stability analysis of the flows is presented the lagrangian forces and heat sources are distributed on the fluid structure interface the method treats pressure the lagrangian forces and heat sources as distributed lagrange multipliers thereby implicitly providing the kinematic constraints of no slip and the corresponding thermal boundary conditions for immersed surfaces extensive verification of the developed method for both isothermal and natural convection 2d flows is provided strategies for adapting the developed approach to realistic 3d configurations are discussed

local extension of boundary holomorphic forms on real hypersurfaces in cn abstract let m be a real hypersurface in c n with n 3 and the levi form at p m has q 1 positive eigenvalues 1 q n 2 we solve one sided local closed extension problem near p for 0 r forms for 1 r q

travelling wave like solutions of the navier stokes and the related equations abstract we present a new family of travelling wave like solutions to the navier stokes equations of incompressible fluid flows and other regularized equations of the euler equations obtain their trend to the solutions of the euler equations as the viscosity tends to zero and estimate the rate of convergence we also find a singularizing effect of the viscosity term in the navier stokes equations i e we have a local moving blow up of unbounded solutions with the blow up s speed depending on viscosity we demonstrate that if the initial function is the beltrami flow then the solution of the navier stokes equations conserves the beltrami flow property for all time

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

the immersed boundary method a projection approach a new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion like previous methods a boundary force is applied at the immersed surface to satisfy the no slip constraint this extra constraint can be added to the incompressible navier stokes equations by introducing regularization and interpolation operators the current method gives prominence to the role of the boundary force acting as a lagrange multiplier to satisfy the no slip condition this role is analogous to the effect of pressure on the momentum equation to satisfy the divergence free constraint the current immersed boundary method removes slip and non divergence free components of the velocity field through a projection the boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger cfl numbers compared to some past methods symmetry and positive definiteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field examples show that the current formulation achieves second order temporal accuracy and better than first order spatial accuracy in l 2 norms for one and two dimensional test problems results from two dimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies

global existence and uniqueness theorem to 2 d incompressible non resistive mhd system with non equilibrium background magnetic field abstract in this article we consider the global existence and uniqueness of the solution to the 2d incompressible non resistive mhd system with non equilibrium background magnetic field our result implies that a strong enough non equilibrium background magnetic field will guarantee the stability of the nonlinear mhd system beside the classical energy method the interpolation inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments

viscosity effect on the degenerate lake equations abstract this paper concerns the effect of viscosity on the degenerate lake equations anelastic limit when the bottom topography vanishes on the shore we establish the existence and uniqueness of a global weak solutions for various choices of viscosity term in weighted sobolev spaces where the weight is assumed to be a power type weight our solution is constructed by adapting carefully the known works on the incompressible navier stokes equations in unweighted context finally we study in one case the vanishing viscosity limit when the solution of the inviscid lake equations is regular enough generalizing the results by martino et al 2001 to the degenerate case

slow convergence to vortex patches in quasigeostrophic balance taking advantage of dispersive effects which supposes a flow extended in the whole space a sluggish convergence of the solutions of the boussinesq equations to a solution of the quasigeostrophic system when the rossby number tends to zero can be proved under weaker assumptions on the initial data than usual in particular no assumption of well preparedness is needed two examples are given both involving fields with striated potential vorticity a result of convergence to a vortex patch is deduced from one of them