the fujita kato theorem for some oldroyd b model abstract in this paper we investigate the cauchy problem associated to a system of pdes of oldroyd type the considered model describes the evolution of certain viscoelastic fluids within a corotational framework the non corotational setting is also addressed in dimension two we show that some widespread results concerning the incompressible navier stokes equations can be extended to the considered systems in particular we show the existence and uniqueness of global in time classical solutions for large data in dimension two this result is supported by suitable condition on the initial data to provide a global in time lipschitz regularity for the flow which allows to overcome specific challenging due to the lack of a decay in time of the main forcing terms secondly we address the global in time well posedness in dimension d 3 we prove the propagation of lipschitz regularities for the flow for this result we just assume the initial data to be sufficiently small in a critical lorentz space

on the general ericksen leslie system parodi s relation well posedness and stability in this paper we investigate the role of parodi s relation in the well posedness and stability of the general ericksen leslie system modeling nematic liquid crystal flows first we give a formal physical derivation of the ericksen leslie system through an appropriate energy variational approach under parodi s relation in which we can distinguish the conservative dissipative parts of the induced elastic stress next we prove global well posedness and long time behavior of the ericksen leslie system under the assumption that the viscosity 4 is sufficiently large finally under parodi s relation we show the global well posedness and lyapunov stability for the ericksen leslie system near local energy minimizers the connection between parodi s relation and linear stability of the ericksen leslie system is also discussed

weak solutions for an initial boundary q tensor problem related to liquid crystals abstract the coupled navier stokes and q tensor system is considered in a bounded three dimensional domain under homogeneous dirichlet boundary conditions for the velocity u and either nonhomogeneous dirichlet or homogeneous neumann boundary conditions for the tensor q the corresponding initial value problem in the whole space r 3 was analyzed in paicu and zarnescu 2012 in this paper three main results concerning weak solutions will be proved the existence of global in time weak solutions bounded up to infinite time a uniqueness criteria and a maximum principle for q moreover we identify how to modify the system to deduce symmetry and traceless for q for any weak solution the presence of a stretching term in the q system plays a crucial role in all the analysis

on the rigidity of nematic liquid crystal flow on s2 abstract in this paper we establish the uniformity property of a simplified ericksen leslie system modelling the hydrodynamics of nematic liquid crystals on the two dimensional unit sphere s 2 namely the uniform convergence in l 2 to a steady state exponentially as t tends to infinity the main assumption similar to topping 15 concerns the equation of liquid crystal director d and states that at infinity time a weak limit d and any bubble i 1 i l share a common orientation as a consequence the uniformity property holds under various types of small initial data

averaged lagrangians and the mean effects of fluctuations in ideal fluid dynamics we begin by placing the generalized lagrangian mean glm equations for a compressible adiabatic fluid into the euler poincare ep variational framework of fluid dynamics for an averaged lagrangian we then state the ep averag ing result that glm equations arise from glm hamilton s principles in the ep framework next we derive a new set of approximate small amplitude glm equations gm equations at second order in the fluctuating displacement of a lagrangian trajectory from its mean position these equations express the linear and nonlinear back reaction effects on the eulerian mean fluid quantities by the fluctuating displacements of the lagrangian trajectories in terms of their eulerian second moments the derivation of the gm equations uses the linearized relations between eulerian and lagrangian fluctuations in the tradition of lagrangian stability analysis for fluids the gm derivation also uses the method of averaged lagrangians in the tradition of wave mean flow interaction wmfi the gm ep motion equations for compressible and incompressible ideal fluids are compared with the euler alpha turbulence closure equations an alpha model is a glm or gm fluid theory with a taylor hypothesis closure thc such closures are based on the linearized fluctuation relations that determine the dynamics of the lagrangian statistical quantities in the euler alpha closure equations we use the ep averaging result to bridge between the glm equations and the euler alpha closure equations hence combining the small amplitude approximation with thc yields in new gm turbulence closure equations for compressible fluids in the ep variational framework 2002 elsevier science b v all rights reserved

weak solution to the steady compressible flow of nematic liquid crystals abstract in this study three dimensional equations for the steady compressible flow of nematic liquid crystals are considered in a bounded domain for the adiabatic exponent 1 we establish the existence of a weak solution by three level approximation and adapting the weak convergence method

l3 solutions of the liquid crystals system abstract in this article we study the simplified system of the original ericksen leslie equations for the flow of liquid crystals it is shown that the l 3 solutions of the cauchy problem for the three dimensional incompressible liquid crystal system are smooth

global strong solution to the 2d density dependent liquid crystal flows with vacuum abstract we establish the global existence and uniqueness of strong solutions to the 2d density dependent liquid crystal flows with vacuum in a bounded smooth domain

piecewise linear discretization of symbolic implicit monte carlo radiation transport in the difference formulation we describe a monte carlo solution for time dependent photon transport in the difference formulation with the material in local thermodynamic equilibrium that is piecewise linear in its treatment of the material state variable our method employs a galerkin solution for the material energy equation while using symbolic implicit monte carlo to solve the transport equation in constructing the scheme one has the freedom to choose between expanding the material temperature or the equivalent black body radiation energy density at the material temperature in terms of finite element basis functions the former provides a linear treatment of the material energy while the latter provides a linear treatment of the radiative coupling between zones subject to the conditional use of a lumped material energy in the vicinity of strong gradients possible with a linear treatment of the material energy our approach provides a robust solution for time dependent transport of thermally emitted radiation that can address a wide range of problems it produces accurate results in thick media

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

nonlinear theory of defects in nematic liquid crystals phase transition and flow phenomena this is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals it also includes some new results concerning disclinations line defects which have not been published elsewhere

generalized large scale semigeostrophic approximations for the f plane primitive equations we derive a family of balance models for rotating stratified flow in the primitive equation pe setting by construction the models possess conservation laws for energy and potential vorticity and are formally of the same order of accuracy as hoskins semigeostrophic equations our construction is based on choosing a new coordinate frame for the pe variational principle in such a way that the consistently truncated lagrangian degenerates we show that the balance relations so obtained are elliptic when the fluid is stably stratified and certain smallness assumptions are satisfied moreover the potential temperature can be recovered from the potential vorticity via inversion of a non standard monge ampere problem which is subject to the same ellipticity condition while the present work is entirely formal we conjecture based on a careful rewriting of the equations of motion and a straightforward derivative count that the cauchy problem for the balance models is well posed subject to conditions on the initial data our family of models includes in particular the stratified analog of the l 1 balance model of salmon

retrieving topological information for phase field models the phase field approach has become a popular tool in modeling interface motion microstructure evolution and more recently the shape transformation of vesicle membranes under elastic bending energy while it is advantageous to employ phase field models in numerical simu lations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes detecting topological events may also become important for many applications such as those in the simulation of blood cells motivated by such considerations a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events it can also be used as a control method to avoid unphys ical changes of topology due to the numerical methods should it become necessary for particular practical applications through numerical experiments we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes

global well posedness and decay for a q tensor model of incompressible nematic liquid crystals in rn abstract we consider the motion of a viscous incompressible liquid crystal flow in the n dimensional space r n we study the global well posedness of strong solutions in the l p l q maximal regularity class

global existence and regularity of solutions for active liquid crystals we study the hydrodynamics of active liquid crystals in the beris edwards hydrodynamic framework with the landau de gennes q tensor order parameter to describe liquid crystalline ordering the existence of global weak solutions in two and three spatial dimensions is established in the two dimensional case by the littlewood paley decomposition the higher regularity of the weak solutions and the weak strong uniqueness are also obtained

analysis of a monte carlo method for nonlinear radiative transfer it has recently been proved that solutions of nonlinear radiative transfer problems satisfy a maximum principle andreev kozmanov and rachilov u s s r comput math math phys 23 104 1983 mercier siam j math anal in press in this article it is shown that monte carlo solutions of such problems obtained using the method of fleck and cummings j comput phys 8 313 1971 must satisfy this maximum principle for sufficiently small time steps but can violate it for sufficiently large time steps analyses of the frequency dependent and grey cases are given and a numerical solution violating the maximum principle is discussed

a free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension abstract in this paper we consider the free boundary problem for a simplified version of ericksen leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one we obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum

a linear stability analysis for nonlinear grey thermal radiative transfer problems we present a new linear stability analysis of three time discretizations and monte carlo interpretations of the nonlinear grey thermal radiative transfer trt equations the widely used implicit monte carlo imc equations the carter forest cf equations and the ahrens larsen or semi analog monte carlo smc equations using a spatial fourier analysis of the 1 d implicit monte carlo imc equations that are linearized about an equilibrium solution we show that the imc equations are unconditionally stable undamped perturbations do not exist if the imc time discretization parameter satisfies 0 5 1 this is consistent with conventional wisdom however we also show that for sufficiently large time steps unphysical damped oscillations can exist that correspond to the lowest frequency fourier modes after numerically confirming this result we develop a method to assess the stability of any time discretization of the 0 d nonlinear grey thermal radiative transfer problem subsequent analyses of the cf and smc methods then demonstrate that the cf method is unconditionally stable and monotonic but the smc method is conditionally stable and permits unphysical oscillatory solutions that can prevent it from reaching equilibrium this stability theory provides new conditions on the time step to guarantee monotonicity of the imc solution although they are likely too conservative to be used in practice theoretical predictions are tested and confirmed with numerical experiments

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

global well posedness and long time dynamics for a higher order quasi geostrophic type equation abstract in this paper we study a higher order viscous quasi geostrophic type equation this equation was derived in 11 as the limit dynamics of a singularly perturbed navier stokes korteweg system with coriolis force when the mach rossby and weber numbers go to zero at the same rate the scope of the present paper is twofold first of all we investigate well posedness of such a model on the whole space r 2 we prove that it is well posed in h s for any s 3 globally in time interestingly enough we show that this equation owns two levels of energy estimates for which one gets existence and uniqueness of weak solutions with different regularities namely h 3 and h 4 regularities this fact can be viewed as a remainder of the so called bd entropy structure of the original system in the second part of the paper we investigate the long time behavior of these solutions we show that they converge to the solution of the corresponding linear parabolic type equation with same initial datum and external force our proof is based on dispersive estimates both for the solutions to the linear and non linear problems

global strong solution to the three dimensional liquid crystal flows of q tensor model abstract a complex hydrodynamic system that models the fluid of nematic liquid crystals in a bounded domain in r 3 is studied the system is a forced incompressible navier stokes equation coupled with a parabolic type equation of q tensors we invoke the maximal regularity of the stokes operators and parabolic operators in besov spaces to obtain the local strong solution if the initial q tensor is not too wild in addition it is showed that such solution can be extended to a global one if the initial data is a sufficiently small perturbation around the trivial equilibrium state finally it is proved that the global strong solution obtained here is identical to those weak solutions obtained in paicu and zarnescu 26

from the q tensor flow for the liquid crystal to the harmonic map flow in this paper we consider the solutions of the relaxed q tensor flow in mathbb r 3 with small parameter epsilon we show that the limiting map is the so called harmonic map flow as a consequence we present a new proof for the global existence of a weak solution for the harmonic map flow in three dimensions as in 18 23 where the ginzburg landau approximation approach was used

global well posedness and twist wave solutions for the inertial qian sheng model of liquid crystals abstract we consider the inertial qian sheng model of liquid crystals which couples a hyperbolic type equation involving a second order material derivative with a forced incompressible navier stokes system we study the energy law and prove a global well posedness result we further provide an example of twist wave solutions that is solutions of the coupled system for which the flow vanishes for all times

a global 2d well posedness result on the order tensor liquid crystal theory abstract in 18 paicu and zarnescu have studied an order tensor system which describes the flow of a liquid crystal they have proven the existence of weak solutions the propagation of higher regularity namely h s with s 1 and the weak strong uniqueness in dimension two this paper is devoted to fill the gap of their results namely to propagate the low regularity namely h s for 0 s 1 and to prove the uniqueness of the weak solutions for the completeness of this research we also propose an alternative approach in order to prove the existence of weak solutions

global well posedness of the two dimensional beris edwards system with general laudau de gennes free energy abstract in this paper we consider the beris edwards system for incompressible nematic liquid crystal flows the system under investigation consists of the navier stokes equations for the fluid velocity u coupled with an evolution equation for the order parameter q tensor one important feature of the system is that its elastic free energy takes a general form and in particular it contains a cubic term that possibly makes it unbounded from below in the two dimensional periodic setting we prove that if the initial l norm of the q tensor is properly small then the system admits a unique global weak solution the proof is based on the construction of a specific approximating system that preserves the l norm of the q tensor along the time evolution

symbolic implicit monte carlo radiation transport in the difference formulation a piecewise constant discretization the equations of radiation transport for thermal photons are notoriously difficult to solve in thick media without resorting to asymptotic approximations such as the diffusion limit one source of this difficulty is that in thick absorbing media thermal emission and absorption are almost completely balanced a new formulation for thermal radiation transport called the difference formulation was recently introduced in order to remove the stiff balance between emission and absorption in the new formulation thermal emission is replaced by derivative terms that become small in thick media it was proposed that the difficulties of solving the transport equation in thick media would be ameliorated by the difference formulation while preserving full rigor and accuracy of the transport solution in the streaming limit in this paper the transport equation is solved by the symbolic implicit monte carlo method and comparisons are made between the standard formulation and the difference formulation the method is easily adapted to the derivative source terms of the difference formulation and a remarkable reduction in noise is obtained when the difference formulation is applied to problems involving thick media

global weak solutions to the active hydrodynamics of liquid crystals abstract we consider the incompressible flow of the active hydrodynamics of liquid crystals with inhomogeneous density in the beris edwards hydrodynamics framework the landau de gennes q tensor order parameter is used to describe the liquid crystalline ordering faedo galerkin s method is adopted to construct the solutions for the initial boundary value problem two levels of approximations are used and the weak convergence is obtained through compactness estimates by new techniques due to the active terms the existence of global weak solutions in dimension two and three is established in a bounded domain

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

on the global well posedness of a class of 2d solutions for the rosensweig system of ferrofluids abstract we study a class of 2d solutions of a bloch torrey regularization of the rosensweig system in the whole space which arise when the initial data and the external magnetic field are 2d we prove that such solutions are globally defined if the initial data is in h k r 2 k 1

well posedness for the fene dumbbell model of polymeric flows we prove local and global well posedness for the fene dumbbell model for a very general class of potentials indeed in prior local or global well posedness results conditions on the strength of the singularity or on the parameter b were made here we give a proof in the general case we also prove global existence results if the data is small or if we restrict to the co rotational model in dimension 2 2008 wiley periodicals inc

asymptotic analysis of a computational method for time and frequency dependent radiative transfer we consider a time dependent energy dependent nonlinear radiative transfer problem in which opacities are large o 1 and interior sources are small o an asymptotic analysis of this problem as 0 leads to the equilibrium diffusion equation in the interior of the system along with boundary conditions and initial conditions for this equation we apply the same asymptotic analysis to a discrete version of the problem in which the frequency variable is discretized by the multigroup method the direction variable by the discrete ordinates method the time variable by the fully implicit method and the spatial variable by a subcell balance method we find that as 0 the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation with boundary conditions and initial conditions that are remarkably accurate the analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients then the numerical method obtains an accurate solution in the interior of the system even though the optical thickness of the spatial cells tends to and boundary layers in the transport solution are not resolved we go a step further to analyze problems that are optically thin at some photon frequencies but thick at others and show that once again the discrete solution is remarkably accurate we present numerical results that verify these and other predictions of the analyses

a novel linear second order unconditionally energy stable scheme for a hydrodynamic q tensor model of liquid crystals abstract the hydrodynamic q tensor model has been used for studying flows of liquid crystals and liquid crystal polymers it can be derived from a variational approach together with the generalized onsager principle in which the total energy decreases in time in this paper we develop a novel linear second order semi discrete scheme in time to solve the governing system of equations in the model the scheme is developed following the novel energy quadratization strategy so that it is linear and unconditionally energy stable at the semi discrete level this scheme is further discretized in space using a second order finite difference method and implemented on a gpu for high performance computing the convergence rate in time is established using a mesh refinement test several numerical examples are presented to demonstrate the usefulness of the model and the effectiveness of the numerical scheme in simulating defect dynamics in flows of liquid crystals

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

semi implicit time integration for pn thermal radiative transfer implicit time integration involving the solution of large systems of equations is the current paradigm for time dependent radiative transfer in this paper we present a semi implicit linear discontinuous galerkin method for the spherical harmonics p n equations for thermal radiative transfer in planar geometry our method is novel in that the material coupling terms are treated implicitly via linearizing the emission source and the streaming operator is treated explicitly using a second order accurate runge kutta method the benefit of this approach is that each time step only involves the solution of equations that are local to each cell this benefit comes at the cost of having the time step limited by a cfl condition based on the speed of light to guarantee positivity and avoid artificial oscillations we use a slope limiting technique we present analysis and numerical results that show the method is robust in the diffusion limit when the photon mean free path is not resolved by the spatial mesh also in the diffusion limit the time step restriction relaxes to a less restrictive explicit diffusion cfl condition we demonstrate with numerical results that away from the diffusion limit our method demonstrates second order error convergence as the spatial mesh is refined with a fixed cfl number

asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo we perform an asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo imc a monte carlo technique for simulating nonlinear radiative transfer specifically we examine the approximation of absorption and re emission by a spatially continuous artificial scattering process and either a piecewise constant or piecewise linear emission source within each spatial cell we consider three asymptotic scalings representing i a time step that resolves the mean free time ii a courant limit on the time step size and iii a fixed time step that does not depend on any asymptotic scaling for the piecewise constant approximation we show that only the third scaling results in a valid discretization of the proper diffusion equation which implies that imc may generate inaccurate solutions with optically large spatial cells if time steps are refined however we also demonstrate that for a certain class of problems the piecewise linear approximation yields an appropriate discretized diffusion equation under all three scalings we therefore expect imc to produce accurate solutions for a wider range of time step sizes when the piecewise linear instead of piecewise constant discretization is employed we demonstrate the validity of our analysis with a set of numerical examples

efficient moving mesh methods for q tensor models of nematic liquid crystals this paper describes a robust and efficient numerical scheme for solving the system of six coupled partial differential equations which arises when using q tensor theory to model the behavior of a nematic liquid crystal cell under the influence of an applied electric field the key novel feature is the use of a full moving mesh partial differential equation approach to generate an adaptive mesh which accurately resolves important solution features this includes the use of a new monitor function based on a local measure of biaxiality in addition adaptive time step control is used to ensure the accurate predicting of the switching time which is often critical in the design of liquid crystal cells we illustrate the behavior of the method on a one dimensional time dependent problem in a pi cell geometry which admits two topologically different equilibrium states modeling the order reconstruction which occurs on the application of an electric field our numerical results show that in addition to achie

the anisotropic lagrangian averaged euler and navier stokes equations the purpose of this paper is twofold first we give a derivation of the lagrangian averaged euler lae and navier stokes lans equations this theory involves a spatial scale and the equations are designed to accurately capture the dynamics of the euler and navier stokes equations at length scales larger than while averaging the motion at scales smaller than the derivation involves an averaging procedure that combines ideas from both the material lagrangian and spatial eulerian viewpoints this framework allows the use of a variant of g i taylor s frozen turbulence hypothesis as the foundation for the model equations more precisely the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow in this article we use this hypothesis to derive the averaged lagrangian for the theory and all the terms up to and including order 2 are accounted for

liquid crystal flows in two dimensions this paper is concerned with a simplified hydrodynamic equation proposed by ericksen and leslie modeling the flow of nematic liquid crystals in dimension two we establish both interior and boundary regularity theorems for such a flow under smallness conditions as a consequence we establish the existence of global in time weak solutions on a bounded smooth domain in mathbb r 2 which are smooth everywhere with possible exceptions of finitely many singular times

global weak solution and blow up criterion of the general ericksen leslie system for nematic liquid crystal flows abstract in this paper we investigate the three dimensional general ericksen leslie e l system with ginzburg landau type approximation modeling nematic liquid crystal flows first by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor we prove existence of global in time weak solutions under physically meaningful boundary conditions and suitable assumptions on the leslie coefficients which ensures that the total energy of the e l system is dissipated moreover for the e l system with periodic boundary conditions we prove the local well posedness of classical solutions under the so called parodi s relation and establish a blow up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field

global solutions for incompressible viscoelastic fluids we prove the existence of both local and global smooth solutions to the cauchy problem in the whole space and the periodic problem in the n dimensional torus for the incompressible viscoelastic system of oldroyd b type in the case of near equilibrium initial data the results hold in both two and three dimensional spaces the results and methods presented in this paper are also valid for a wide range of elastic complex fluids such as magnetohydrodynamics liquid crystals and mixture problems

energy dissipation and regularity for a coupled navier stokes and q tensor system we study a complex non newtonian fluid that models the flowof nematic liquid crystals the fluid is described by a system that couples a forced navier stokes system with a parabolic type system we prove the existence of global weak solutions in dimensions two and three we show the existence of a lyapunov functional for the smooth solutions of the coupled system and use cancellations that allow its existence to prove higher global regularity in dimension two we also show the weak strong uniqueness in dimension two

analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid we establish the existence of three new subgroups of the group of volumepreserving diffeomorphisms of a compact n dimensional n 2 riemannian manifold which are associated with the dirichlet neumann and mixed type boundary conditions that arise in second order elliptic pdes we prove that when endowed with the hs hilbert class topologies for s n 2 1 these subgroups are c differential manifolds we consider these new diffeomorphism groups with an h1 equivalent right invariant metric and prove the existence of unique smooth geodesics t of this metric as well as existence and uniqueness of the jacobi equations associated to this metric geodesics on these subgroups are in fact the flows of a timedependent velocity vector field u t x so that t t u t t with 0 x x and remarkably the vector field u t x solves the so called lagrangian averaged euler lae equations on m these equations and their viscous counterparts the lagrangian averaged navier stokes lans equations model the motion of a fluid at scales larger than an a priori fixed parameter 0 while averaging or filtering out the small scale motion and this is achieved without the use of artificial viscosity we prove that for divergence free initial data satisfying u 0 on m the lae equations are well posed globally when n 2 we also find the boundary conditions that make the lans equations well posed globally when n 3 and prove that solutions of the lans equations converge when n 2 3 for almost all t in some fixed time interval 0 t in hs s n 2 1 3 to solutions of the lae equations thus confirming the scaling arguments of barenblatt chorin received april 26 2000 1991 mathematics subject classification 35q35 35q53 58b20 58d05

landau de gennes theory of nematic liquid crystals the oseen frank limit and beyond we study global minimizers of a continuum landau de gennes energy functional for nematic liquid crystals in three dimensional domains subject to uniaxial boundary conditions we analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly in w1 2 to a global minimizer predicted by the oseen frank theory for uniaxial nematic liquid crystals with constant order parameter moreover the convergence is uniform in the interior of the domain away from the singularities of the limiting oseen frank global minimizer we obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the landau de gennes global minimizer we also study the interplay between biaxiality and uniaxiality in landau de gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions

strictly physical global weak solutions of a navier stokes q tensor system with singular potential we study the existence regularity and so called strict physicality of global weak solutions of a beris edwards system which is proposed as a model for the incompressible flow of nematic liquid crystal materials an important contribution to the dynamics comes from a singular potential introduced by john ball and apala majumdar which replaces the commonly employed landau de gennes bulk potential this is built into our model to ensure that a natural physical constraint on the eigenvalues of the q tensor order parameter is respected by the dynamics of this system moreover by a maximum principle argument we are able to construct global strong solutions in dimension two

remarks on the regularity criterion for the nematic liquid crystal flows in r 3 in this paper we investigate the regularity criteria for the 3d nematic liquid crystal flows in terms of the components of the velocity u or the gradient of the velocity field u more precisely the anisotropic integrability regularity criterion to the 3d nematic liquid crystal flows involving i u i 1 2 3 is obtained firstly and then it is proved that the solution is regular if u3 3 or 3u3 3 satisfies the prodi serrin type conditions

existence of solutions for the ericksen leslie system we study the general ericksen leslie system which describes the flow of liquid crystal materials the dissipation property of the system is established and is used to prove the global existence of weak solutions we also study the existence of classical solutions and the asymptotic stability of the solutions

a linear discontinuous spatial differencing scheme forsnradiative transfer calculations various types of linear discontinuous spatial differencing schemes have been developed for thesn discrete ordinates equations approximating the linear boltzmann transport equation it has been shown through an asymptotic analysis that the 1d slab geometry lumped linear discontinuous scheme not only goes over to a convergent and robust differencing of the diffusion equation in the monoenergetic thick diffusion limit but it also yields the correct interior solution even when boundary layers are left unresolved by the spatial mesh in the present work we generalize this scheme to obtain a 1d slab geometry lumped linear discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation we present a full nonlinear energy dependent asymptotic analysis of the behavior of this scheme in the thick equilibrium diffusion limit we find that this scheme goes over to a convergent and robust differencing of the equilibrium diffusion equation on the interior of the mesh but it does not yield the exact interior solution when boundary layers are left unresolved by the spatial mesh nevertheless the interior solution obtained with spatially unresolved boundary layers is always well behaved and fairly accurate computational results are presented which test the predictions of our asymptotic analysis and demonstrate the efficiency of our solution technique

asymptotic equilibrium diffusion analysis of time dependent monte carlo methods for grey radiative transfer the equations of nonlinear time dependent radiative transfer are known to yield the equilibrium diffusion equation as the leading order solution of an asymptotic analysis when the mean free path and mean free time of a photon become small we apply this same analysis to the fleck cummings carter forest and n kaoua monte carlo approximations for grey frequency independent radiative transfer although monte carlo simulation usually does not require the discretizations found in deterministic transport techniques monte carlo methods for radiative transfer require a time discretization due to the nonlinearities of the problem if an asymptotic analysis of the equations used by a particular monte carlo method yields an accurate time discretized version of the equilibrium diffusion equation the method should generate accurate solutions if a time discretization is chosen that resolves temperature changes even if the time steps are much larger than the mean free time of a photon this analysis is of interest because in many radiative transfer problems it is a practical necessity to use time steps that are large compared to a mean free time our asymptotic analysis shows that i the n kaoua method has the equilibrium diffusion limit ii the carter forest method has the equilibrium diffusion limit if the material temperature change during a time step is small and iii the fleck cummings method does not have the equilibrium diffusion limit we include numerical results that verify our theoretical predictions

optimal boundary control of a simplified ericksen leslie system for nematic liquid crystal flows in 2d in this paper we investigate an optimal boundary control problem for a two dimensional simplified ericksen leslie system modelling the incompressible nematic liquid crystal flows the hydrodynamic system consists of the navier stokes equations for the fluid velocity coupled with a convective ginzburg landau type equation for the averaged molecular orientation the fluid velocity is assumed to satisfy a no slip boundary condition while the molecular orientation is subject to a time dependent dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals we first establish the existence of optimal boundary controls then we show that the control to state operator is frechet differentiable between appropriate banach spaces and derive first order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables

on hydrodynamics of viscoelastic fluids in this paper we study a hydrodynamic system describing fluids with viscoelastic properties after a brief examination of the relations between several models we shall concentrate on a few analytical issues concerning them in particular we establish local existence and global existence with small initial data of classical solutions for an oldroyd system without an artificially postulated damping mechanism

lagrangian averaging with geodesic mean this paper revisits the derivation of the lagrangian averaged euler lae or euler equations in the light of an intrinsic definition of the averaged flow map as the geodesic mean on the volume preserving diffeomorphism group under the additional assumption that first order fluctuations are statistically isotropic and transported by the mean flow as a vector field averaging of the kinetic energy lagrangian of an ideal fluid yields the lae lagrangian the derivation presented here assumes a euclidean spatial domain without boundaries

convergence of the 2d euler α to euler equations in the dirichlet case indifference to boundary layers abstract in this article we consider the euler system as a regularization of the incompressible euler equations in a smooth two dimensional bounded domain for the limiting euler system we consider the usual non penetration boundary condition while for the euler regularization we use velocity vanishing at the boundary we also assume that the initial velocities for the euler system approximate in a suitable sense as the regularization parameter 0 the initial velocity for the limiting euler system for small values of this situation leads to a boundary layer which is the main concern of this work our main result is that under appropriate regularity assumptions and despite the presence of this boundary layer the solutions of the euler system converge as 0 to the corresponding solution of the euler equations in l 2 in space uniformly in time we also present an example involving parallel flows in order to illustrate the indifference to the boundary layer of the 0 limit which underlies our work

robust and accurate filtered spherical harmonics expansions for radiative transfer we present a novel application of filters to the spherical harmonics p n expansion for radiative transfer problems in the high energy density regime the filter we use is based on non oscillatory spherical splines and a filter strength chosen to i preserve the equilibrium diffusion limit and ii vanish as the expansion order tends to infinity our implementation is based on modified equations that are derived by applying the filter after every time step in a simple first order time integration scheme the method is readily applied to existing codes that solve the p n equations numerical results demonstrate that the solution to the filtered p n equations are i more robust and less oscillatory than standard p n solutions and ii more accurate than discrete ordinates solutions of comparable order in particular the filtered p 7 solution demonstrates comparable accuracy to an implicit monte carlo solution for a benchmark hohlraum problem in 2d cartesian geometry

global well posedness for a smoluchowski equation coupled with navier stokes equations in 2d we prove global existence for a nonlinear smoluchowski equation a nonlinear fokker planck equation coupled with navier stokes equations in 2d the proof uses a deteriorating regularity estimate in the spirit of 5 see also 1

a regularity criterion for the solution of nematic liquid crystal flows in terms of the b 1 norm abstract in this paper we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions n 3 and in two dimensions n 2 we prove that the solution u d is smooth up to time t provided that there exists a positive constant e 0 0 such that i for n 3 u d l 0 t b 1 e 0 and ii for n 2 d l 0 t b 1 e 0

numerical analysis of time integration errors for nonequilibrium radiation diffusion numerical analysis of time integration errors for nonequilibrium radiation diffusion is considered two first order implicit time integration methods are studied asymptotic analysis and modified equation analysis are applied to both time integration methods numerical experiments are used to highlight the results of the analysis asymptotic analysis is used to highlight the source of temperature spiking when a hot radiation wave propagates into a cold material modified equation analysis is used to provide insight into the thermal wave speed coming from the two different first order methods

the effects of slope limiting on asymptotic preserving numerical methods for hyperbolic conservation laws many hyperbolic systems of equations with stiff relaxation terms reduce to a parabolic description when relaxation dominates an asymptotic preserving numerical method is a discretization of the hyperbolic system that becomes a valid discretization of the parabolic system in the asymptotic limit we explore the consequences of applying a slope limiter to the discontinuous galerkin dg method with linear elements for hyperbolic systems with stiff relaxation terms without a limiter the dg method gives an accurate discretization of the chapman enskog approximation of the system when the relaxation length scale is not resolved it is well known that the first order upwind or step method fails to obtain the proper asymptotic limit we show that using the minmod slope limiter also fails but that using double minmod gives the proper asymptotic limit despite its effectiveness in the asymptotic limit the double minmod limiter allows artificial extrema at cell interfaces referred to as sawteeth we present a limiter that eliminates the sawteeth but maintains the proper asymptotic limit the systems that we analyze are the hyperbolic heat equation and the p n thermal radiation equations numerical examples are used to verify our analysis