about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

global existence and nonexistence of solution for cauchy problem of two dimensional generalized boussinesq equations abstract in this paper we consider the cauchy problem of two dimensional generalized boussinesq type equation u t t u u t t 2 u f u 0 under the assumption that f u is a function with exponential growth at infinity and under some assumptions on the initial data we prove the existence and nonexistence of global weak solution there are very few works on boussinesq equation with nonlinear exponential growth term by potential well theory

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

high energy blow up for two dimensional generalized exponential type boussinesq equation abstract a finite time blow up result for the cauchy problem of two dimensional generalized boussinesq type equation with exponential type source term is considered in this paper based on an adapted concave method we derive a sufficient condition on initial data leading to the arbitrarily positive initial energy finite time blow up

global existence and exponential growth of solution for the logarithmic boussinesq type equation abstract in this paper we consider the boussinesq type equation associated with logarithmic nonlinear term and initial boundary value conditions by using potential well method combined with the logarithmic sobolev inequality we obtain the existence of global solution at last we show that the l 2 norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions for initial data the result shows that the polynomial nonlinearity is a critical condition of blow up in finite time for the solutions of boussinesq equations

an implicit immersed boundary method for three dimensional fluid membrane interactions we present an implicit immersed boundary method for the incompressible navier stokes equations capable of handling three dimensional membrane fluid flow interactions the goal of our approach is to greatly improve the time step by using the jacobian free newton krylov method jfnk to advance the location of the elastic membrane implicitly the most attractive feature of this jacobian free approach is newton like nonlinear convergence without the cost of forming and storing the true jacobian the generalized minimal residual method gmres which is a widely used krylov subspace iterative method is used to update the search direction required for each newton iteration each gmres iteration only requires the action of the jacobian in the form of matrix vector products and therefore avoids the need of forming and storing the jacobian matrix explicitly once the location of the boundary is obtained the elastic forces acting at the discrete nodes of the membrane are computed using a finite element model we then use the immersed boundary method to calculate the hydrodynamic effects and fluid structure interaction effects such as membrane deformation the present scheme has been validated by several examples including an oscillatory membrane initially placed in a still fluid capsule membranes in shear flows and large deformation of red blood cells subjected to stretching force

dispersion of low energy waves for the generalized benjamin bona mahony equation on considere les equations suivantes u t u x f u x u xxx 0 1 et u t u x f u x u xxt 0 2 avec f r r est c on donne le comportement des solutions ondes a basse energie pour certains f

well posedness and large time behaviour for the non cutoff kac equation with a gaussian thermostat we consider here a kac equation with a gaussian thermostat in the case of a non cutoff cross section under the sole assumptions of finite mass and finite energy for the initial data we prove the existence of a global in time solution for which mass and energy are preserved then via fourier transform techniques we show that this solution is smooth unique and converges to the corresponding stationary state

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

on the schrodinger boussinesq system with singular initial data abstract we study the existence of local and global solutions for coupled schrodinger boussinesq systems with initial data in weak l r spaces these spaces contain singular functions with infinite l 2 mass such as homogeneous functions of negative degree moreover we analyze the self similarity and radial symmetry of solutions by considering initial data with the right homogeneity and radially symmetric respectively since functions in weak l r with r 2 have local finite l 2 mass the solutions obtained can be physically realized moreover for initial data in h s local solutions belong to h s which shows that the constructed data solution map in weak l r recovers h s regularity

implicit second order immersed boundary methods with boundary mass abstract the immersed boundary method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures immersed boundary computations typically evaluate the elastic forces explicitly in the configuration of the immersed elastic structure in many applications this results in a severe restriction on the time step we present a semi implicit and a fully implicit second order accurate immersed boundary method the methods provide a natural way to handle mass on the immersed elastic structures we demonstrate their performance for a prototypical fluid structure interaction problem the methods are shown to possess superior stability properties that significantly alleviate the typically severe time step restriction of explicit computations

blow up and scattering of solution for a generalized boussinesq equation in this paper we consider the cauchy problem for a class of boussinesq equation we obtain the existence and uniqueness of the local solutions for a class of nonlinearity of the perturbation blow up solutions are obtained furthermore the global existence and nonlinear scattering for small amplitude solutions are established

an efficient semi implicit immersed boundary method for the navier stokes equations the immersed boundary method is one of the most useful computational methods in studying fluid structure interaction on the other hand the immersed boundary method is also known to require small time steps to maintain stability when solved with an explicit method many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint but none of them give satisfactory performance in this paper we propose an efficient semi implicit scheme to remove this stiffness from the immersed boundary method for the navier stokes equations the construction of our semi implicit scheme consists of two steps first we obtain a semi implicit discretization which is proved to be unconditionally stable this unconditionally stable semi implicit scheme is still quite expensive to implement in practice next we apply the small scale decomposition to the unconditionally stable semi implicit scheme to construct our efficient semi implicit scheme unlike other implicit or semi implicit schemes proposed in the literature our semi implicit scheme can be solved explicitly in the spectral space thus the computational cost of our semi implicit schemes is comparable to that of an explicit scheme our extensive numerical experiments show that our semi implicit scheme has much better stability property than an explicit scheme this offers a substantial computational saving in using the immersed boundary method

low regularity solutions of non homogeneous boundary value problems of a higher order boussinesq equation in a quarter plane abstract in this article we study an initial boundary value problem of the sixth order boussinesq equation on a half line with nonhomogeneous boundary conditions u t t u x x u x x x x u x x x x x x u 2 x x 0 x 0 t 0 u x 0 x u t x 0 x x 0 u 0 t h 1 t u x x 0 t h 2 t u x x x x 0 t h 3 t t 0 where 1 it is shown that the problem is locally well posed in c r l o c h s r with initial condition h s r h s 1 r and boundary condition h 1 h 2 h 3 in the product space h s 1 3 r h s 1 3 r h s 3 3 r for 1 2 s 0

nonlinear small data scattering for the generalized korteweg de vries equation abstract we study the longtime stability of small solutions to the ivp for the generalized korteweg de vries equation we obtain a lower bound for the degrees of nonlinearity of the perturbation which guarantees that the small solutions of the nonlinear problem behave asymptotically like the solutions of the associated linear problem this behavior allows us to establish a nonlinear scattering result for small perturbations with these degrees for a class of small data we improve the value of this lower bound the new crucial ingrediants in our proofs are the l p decay estimates of the half derivative of the airy kernel

on space time means and everywhere defined scattering operators for nonlinear klein gordon equations on demontre l existence d operateurs de diffusion definis partout pour certaines equations de klein gordon non lineaires

existence and scattering of small solutions to a boussinesq type equation of sixth order abstract in this paper we consider the existence and uniqueness of the global small solution as well as the small data scattering result to the cauchy problem for a boussinesq type equation of sixth order with the nonlinear term f u behaving as u p p 1 as u 0 in r n n 1 the main method and techniques used in our paper are the littlewood paley dyadic decomposition the stationary phase estimate and some properties of bessel function

brownian approximation and monte carlo simulation of the non cutoff kac equation the non cutoff boltzmann equation can be simulated using the dsmc method by a truncation of the collision term however even for computing stationary solutions this may be very time consuming in particular in situations far from equilibrium by adding an appropriate diffusion to the dsmc method the rate of convergence when the truncation is removed may be greatly improved we illustrate the technique on a toy model the kac equation as well as on the full boltzmann equation in a special case

on the order of accuracy of the immersed boundary method higher order convergence rates for sufficiently smooth problems the immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a visco elastic structure in m c lai simulations of the flow past an array of circular cylinders as a test of the immersed boundary method ph d thesis courant institute of mathematical sciences new york university 1998 m c lai c s peskin an immersed boundary method with formal second order accuracy and reduced numerical viscosity j comput phys 160 2000 705 719 lai and peskin introduced a formally second order accurate immersed boundary method but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions consequently in practice only first order convergence rates have been observed in the present work we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid structure interaction problem involving an immersed viscoelastic shell of finite thickness studied over a broad range of reynolds numbers we consider two sets of material properties for the viscoelastic structure including a case where the material properties of the coupled system are discontinuous at the fluid structure interface for both sets of material properties the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations

removing the stiffness of elastic force from the immersed boundary method for the 2d stokes equations the immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction on the other hand the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization in this paper we propose several efficient semi implicit schemes to remove this stiffness from the immersed boundary method for the two dimensional stokes flow first we obtain a novel unconditionally stable semi implicit discretization for the immersed boundary problem using this unconditionally stable discretization as a building block we derive several efficient semi implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization our stability analysis and extensive numerical experiments show that our semi implicit schemes offer much better stability property than the explicit scheme unlike other implicit or semi implicit schemes proposed in the literature our semi implicit schemes can be solved explicitly in the spectral space thus the computational cost of our semi implicit schemes is comparable to that of an explicit scheme but with a much better stability property

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

strong instability of solitary wave solutions for a nonlinear boussinesq equation abstract in this paper we obtain the solitary wave of the cauchy problem for a class of nonlinear boussinesq equations the strong instability of the solitary wave is proved by constructing a functional of the solution and establishing the invariant regions furthermore an improved blow up result related to the solitary wave solution is also obtained

inviscid limit for the damped boussinesq equation abstract in this paper we consider the limit behavior of solutions to the cauchy problem for damped boussinesq equation in the regime of small viscosity in r n it is shown that the strong solutions of damped boussinesq equation converge to a strong solution of boussinesq equation in any time interval 0 t the key idea is that we only take into account the effect of dispersion and overlook the effect of dissipation in the damped boussinesq equation which is a kind of new view we analyze the dispersion by the techniques of high low frequency decomposition and littlewood paley dyadic decomposition stationary phase estimate and some properties of bessel function

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

automatic adaptive grid refinement for the euler equations a method of adaptive grid refinement for the solution of the steady euler equations for transonic flow is presented the algorithm automatically decides where the coarse grid accuracy is insufficient and creates locally uniform refined grids in these regions this typically occurs at the leading and trailing edges the solution is then integrated to steady state using the same integrator flo52 in the interior of each grid the boundary conditions needed on the fine grids are examined and the importance of treating the fine coarse grid interface conservatively is discussed numerical results indicate substantial computational savings for the same solution accuracy can be achieved

the kac equation with a thermostatted force field we consider the kac equation with a thermostatted force field and prove the existence of a global in time solution that converges weakly to a stationary state as there is no an obvious candidate for the entropy functional in this case the convergence result is obtained via fourier transform techniques

decay and scattering of solutions for a generalized boussinesq equation abstract in this paper we consider the long time behavior of small solutions of the cauchy problem for a generalized boussinesq equation a scattering operator and the nonlinear scattering for small amplitude solutions of the boussinesq equation are established under certain hypotheses

strong instability of solitary wave solutions of a generalized boussinesq equation abstract in this paper we prove the strong instability of solitary wave solutions with small speed of propagation for a generalized boussinesq equation

global existence and nonexistence of the initial boundary value problem for the dissipative boussinesq equation abstract a class of multidimensional dissipative boussinesq equations is considered under three different cases of initial energy we prove the existence and uniqueness of global solutions in the energy space and provide sufficient conditions of finite time blow up solutions the sharp result is given as e 0 d particular attention is paid to initial energy at supercritical initial energy e 0 d level which is novel contribution of this paper and new methods will be introduced and some analysis techniques are needed in addition the asymptotic behavior of the global solutions is studied

computational methods for continuum models of platelet aggregation platelet aggregation plays an important role in blood clotting robust numerical methods for simulating the behavior of fogelson s continuum models of platelet aggregation have been developed which in particular involve a hybrid finite difference and spectral method for the models link evolution equation this partial differential equation involvesfourspatial dimensions and time the new methods are used to begin investigating the influence of new chemically induced activation link formation and shear induced link breaking in determining when aggregates develop sufficient strength to remain intact and when they are broken apart by fluid stresses

decay and scattering of small solutions of a generalized boussinesq equation abstract we study the long time behavior of small solutions of the initial value problem for a generalized boussinesq equation we obtain a lower bound for the degrees of nonlinearity which allows us to establish a nonlinear scattering result for small perturbations that is the small solutions of the nonlinear problem behave asymptotically like the solution of the associated linear problem under certain hypotheses we can construct a scattering operator for the boussinesq equation which carries a neighborhood of 0 in the energy space x into x

an implicit forcing immersed boundary method for simulating viscous flows in irregular domains abstract we present a method for solving the incompressible navier stokes equations in irregular domains these equations are discretized using finite difference method in a uniform cartesian grid stationary rigid boundaries are embedded in the cartesian grid and singular forces are applied at the rigid boundaries to impose the no slip conditions the singular forces are then distributed to the nearby cartesian grid points using the immersed boundary method in the present work the singular forces are computed implicitly by solving a small system of equations at each time step this system of equations is derived from a second order projection method the main advantage of this method is that it imposes the no slip boundary condition exactly and avoids the need for small time step to maintain stability the ability of the method to simulate viscous flows in irregular domains is demonstrated by applying to 2 dimensional flows past a circular cylinder multiple rigid obstacles and 3 dimensional flow past a sphere

an immersed interface method for stokes flows with fixed moving interfaces and rigid boundaries we present an immersed interface method for solving the incompressible steady stokes equations involving fixed moving interfaces and rigid boundaries irregular domains the fixed moving interfaces and rigid boundaries are represented by a number of lagrangian control points in order to enforce the prescribed velocity at the rigid boundaries singular forces are applied on the fluid at these boundaries the strength of singular forces at the rigid boundary is determined by solving a small system of equations for the deformable interfaces the forces that the interface exerts on the fluid are calculated from the configuration position of the deformed interface the jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed moving interfaces and rigid boundaries these forces are interpolated using cubic splines and applied to the fluid through the jump conditions the positions of the deformable interfaces are updated implicitly using a quasi newton method bfgs within each time step in the proposed method the stokes equations are discretized via the finite difference method on a staggered cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient uzawa type method numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible stokes flows with fixed moving interfaces on irregular domains

on the asymptotic behavior of solution for the generalized ibq equation with hydrodynamical damped term abstract we study the cauchy problem for the generalized ibq equation with hydrodynamical damped term in n dimensional space we observe that the dissipative structure of the linearized equation is of the regularity loss type this means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data under smallness condition on the initial data we prove the global existence and decay of the small amplitude solution in the sobolev space

decay estimates for a dissipative dispersive linear semigroup and application to the viscous boussinesq equation abstract at the core of this article is the new estimate for a class of dissipative dispersive linear semigroup e t i p t arising in the study of the viscous boussinesq equation we combine the decay estimate with introducing a set of time weighted sobolev spaces where the time weights and the regularity of the sobolev spaces are determined by our decay estimate to show the global existence and asymptotic behavior of solutions to the viscous boussinesq equation in r n

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

global existence of small solutions to nonlinear evolution equations abstract we study the global existence and asymptotic behaviour of small solutions of a large class of nonlinear partial differential equations if the nonlinear terms are of high degree the solutions will be asymptotic to solutions of the linear equation

existence and scattering theory for boussinesq type equations with singular data we study the initial value problem for the generalized boussinesq equation and prove existence of local and global solutions with singular initial data in weak lp spaces our class of initial data for global existence is larger than that of cho and ozawa 2007 7 long time behavior results are obtained and a scattering theory is proved in that framework with more structure we show sobolev h1 and lorentz type l p q regularity properties for the obtained solutions the approach employed is unified for all dimensions n 1

an immersed boundary method based on the lattice boltzmann approach in three dimensions with application the immersed boundary ib method originated by peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid the navier stokes n s equations in the ib method are usually solved using numerical methods such as fft and projection methods here in our work the n s equations are solved by an alternative approach the lattice boltzmann method lbm compared to many conventional n s solvers the lbm can be easier to implement and more convenient for modeling additional physics in a problem this alternative approach adds extra versatility to the immersed boundary method in this paper we discuss the use of a 3d lattice boltzmann model d3q19 within the ib method we use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle line in a 3d channel and determine a drag scaling law for the sheet our main conclusions are 1 the hybrid method is convergent with first order accuracy which is consistent with the immersed boundary method in general 2 the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

a jacobian free based iim for incompressible flows involving moving interfaces with dirichlet boundary conditions in this paper a finite difference marker and cell mac scheme is presented for the steady stokes equations with moving interfaces and dirichlet boundary condition the moving interfaces are represented by lagrangian control points and their position is updated implicitly using a jacobian free approach within each time step the forces at the moving interfaces are calculated from the position of the interfaces and interpolated using cubic splines and then applied to the fluid through the related jump conditions the proposed jacobian free newton generalized minimum residual gmres method avoids the need to form and store the matrix explicitly in the computation of the inverse of the jacobian and betters numerical stability the stokes equations are discretized on a mac grid via a second order finite difference scheme with the incorporation of jump contributions and the resulting saddle point system is solved by the conjugate gradient uzawa type method numerical results demonstrate very well the accuracy and effectiveness of the proposed method the present algorithm has been applied to solve incompressible navier stokes flows with moving interfaces copyright 2010 john wiley sons ltd