the hybridized discontinuous galerkin method for implicit large eddy simulation of transitional turbulent flows abstract we present a high order implicit large eddy simulation iles approach for transitional aerodynamic flows the approach encompasses a hybridized discontinuous galerkin dg method for the discretization of the navier stokes ns equations and a parallel preconditioned newton gmres solver for the resulting nonlinear system of equations the combination of hybridized dg methods with an efficient solution procedure leads to a high order accurate ns solver that is competitive to alternative approaches such as finite volume and finite difference codes in terms of computational cost the proposed approach is applied to transitional flows over the naca 65 18 10 compressor cascade and the eppler 387 wing at reynolds numbers up to 460 000 grid convergence studies are presented and the required resolution to capture transition at different reynolds numbers is investigated numerical results show rapid convergence and excellent agreement with experimental data in short this work aims to demonstrate the potential of high order iles for simulating transitional aerodynamic flows this is illustrated through numerical results and supported by theoretical considerations

multipliers paramultipliers and weak strong uniqueness for the navier stokes equations abstract in this article we describe spaces p such that if u is a weak in the sense of leray j leray sur le mouvement d un fluide visqueux remplissant l espace acta math 63 1934 193 248 solution of the navier stokes system for some initial data u 0 and if u belongs to p then u is unique in the class of weak solutions we say then that weak strong uniqueness holds it turns out that the proof of such results relies on the boundedness of a trilinear functional f l 2 h l 2 h p r where belong to 0 1 in order to find optimal conditions for the boundedness of f we are led to describing spaces of multipliers and of paramultipliers that is functions which map by classical pointwise product or by paraproduct a given sobolev spaces in another given sobolev space the study of these spaces enables us to give conditions for weak strong uniqueness which generalise all previously known results from the famous serrin criterion j serrin the initial value problem for the navier stokes equations in r e langer ed nonlinear problems univ of wisconsin press 1963 pp 69 98 to the recent conditions formulated by lemarie rieusset p g lemarie rieusset recent developments in the navier stokes problem chapman and hall 2003

local existence of the strong solutions for the full navier stokes poisson equations abstract we study the full navier stokes poisson equations for viscous polytropic fluids we prove the existence and uniqueness of local strong solutions for all initial values satisfying some compatibility conditions it is important that we allow the initial density to be vacuum

robust and efficient discontinuous galerkin methods for under resolved turbulent incompressible flows abstract we present a robust and accurate discretization approach for incompressible turbulent flows based on high order discontinuous galerkin methods the dg discretization of the incompressible navier stokes equations uses the local lax friedrichs flux for the convective term the symmetric interior penalty method for the viscous term and central fluxes for the velocity pressure coupling terms stability of the discretization approach for under resolved turbulent flow problems is realized by a purely numerical stabilization approach consistent penalty terms that enforce the incompressibility constraint as well as inter element continuity of the velocity field in a weak sense render the numerical method a robust discretization scheme in the under resolved regime the penalty parameters are derived by means of dimensional analysis using penalty factors of order 1 applying these penalty terms in a postprocessing step leads to an efficient solution algorithm for turbulent flows the proposed approach is applicable independently of the solution strategy used to solve the incompressible navier stokes equations i e it can be used for both projection type solution methods as well as monolithic solution approaches since our approach is based on consistent penalty terms it is by definition generic and provides optimal rates of convergence when applied to laminar flow problems robustness and accuracy are verified for the orr sommerfeld stability problem the taylor green vortex problem and turbulent channel flow moreover the accuracy of high order discretizations as compared to low order discretizations is investigated for these flow problems a comparison to state of the art computational approaches for large eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers

multipliers between sobolev spaces and fractional differentiation abstract we characterize the pointwise multipliers which maps a sobolev space h r r d to a sobolev space h s r d in the case s r d 2

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

time average control of martingale problems a linear programming formulation one studies the average cost for controlled systems given as solutions of the martingale problem for their generator the control problem is reformulated as a linear programming problem and conditions are given for the existence of an optimal solution it is further shown that the optimal control can be taken to depend only on the history of the system and that this cost remains optimal for systems with different initial distributions

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

stationary navier stokes equations with critically singular external forces existence and stability results abstract we show the unique existence of solutions to stationary navier stokes equations with small singular external forces belonging to a critical space to the best of our knowledge this is the largest critical space that is available up until now for this kind of existence this result can be viewed as the stationary counterpart of the existence result obtained by h koch and d tataru for the free non stationary navier stokes equations with initial data in bmo 1 the stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup

on global infinite energy solutions to the navier stokes equations in two dimensions this paper studies the bidimensional navier stokes equations with large initial data in the homogeneous besov space as long as r q global existence and uniqueness of solutions are proved we also prove that weak strong uniqueness holds for the d dimensional equations with data in l 2 d for d r 2 q 1

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

a low dissipation dg method for the under resolved simulation of low mach number turbulent flows abstract in recent years the use of high order discontinuous galerkin dg methods for the under resolved direct numerical simulations udns of turbulent flows has received special attention the suitability of the approach for this kind of applications is related to the dissipation and dispersion proprieties of the scheme while the dispersion errors are small over a broad range of frequencies a relevant dissipation error mainly acts at the smallest under resolved scales resembling a high frequency filter nevertheless it was recognized flad and gassner 2017 mengaldo et al 2018 that the choice of the interface convective numerical flux strongly affects this dissipation behaviour and ultimately the success of the udns approach in this regard the excess of numerical dissipation caused by some upwind numerical convective fluxes must be avoided in particular when dealing with low speed flows since this behaviour is exacerbated approaching the incompressibility limit fixes for the excess of numerical dissipation of these schemes have been proposed by several authors in the context of different numerical methods see for example weiss and smith 1995 in this work a simple modification of the dissipation term of the low mach preconditioned roe scheme proposed by weiss and smith is considered the aim is to reduce further the amount of numerical dissipation with the intent of improving the results of udns a spatial dg discretization coupled with a linearly implicit rosenbrock type time integrator is here considered as a numerical framework perfectly suited for the assessment and comparison of different numerical flux functions results on canonical turbulent flow problems as the taylor green vortex and the flow in a straight sided channel are presented the improved accuracy of the proposed flux function is demonstrated the new low dissipation flux can be useful also in the context of standard lower order finite volume methods

blowup or no blowup the interplay between theory and numerics the question of whether the 3d incompressible euler equations can develop a finite time singularity from smooth initial data has been an outstanding open problem in fluid dynamics and mathematics recent studies indicate that the local geometric regularity of vortex lines can lead to dynamic depletion of vortex stretching guided by the local non blowup theory we have performed large scale computations of the 3d euler equations on some of the most promising blowup candidates our results show that there is tremendous dynamic depletion of vortex stretching the local geometric regularity of vortex lines and the anisotropic solution structure play an important role in depleting the nonlinearity dynamically and thus prevents a finite time blowup

spatial eigensolution analysis of energy stable flux reconstruction schemes and influence of the numerical flux on accuracy and robustness this study focuses on the dispersion and diffusion characteristics of high order energy stable flux reconstruction esfr schemes via the spatial eigensolution analysis framework proposed in 1 the analysis is performed for five esfr schemes where the parameter c dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of esfr methods also referred to as vcjh schemes proposed in 2 in particular we used five values of c two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high order methods namely the esfr recovering two versions of discontinuous galerkin methods and one recovering the spectral difference scheme the performance of each scheme is assessed when using different numerical intercell fluxes e g different levels of upwinding ranging from under to over upwinding in contrast to the more common temporal analysis the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of fr schemes for problems involving non periodic boundary conditions typically found in open flow problems including turbulence unsteady aerodynamics and aeroacoustics

on a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems a representation of the limit occupational measures set of a control system in terms of the vector function defining the system s dynamics is established applications in averaging of singularly perturbed control systems are demonstrated

dealiasing techniques for high order spectral element methods on regular and irregular grids high order methods are becoming increasingly attractive in both academia and industry especially in the context of computational fluid dynamics however before they can be more widely adopted issues such as lack of robustness in terms of numerical stability need to be addressed particularly when treating industrial type problems where challenging geometries and a wide range of physical scales typically due to high reynolds numbers need to be taken into account one source of instability is aliasing effects which arise from the nonlinearity of the underlying problem in this work we detail two dealiasing strategies based on the concept of consistent integration the first uses a localised approach which is useful when the nonlinearities only arise in parts of the problem the second is based on the more traditional approach of using a higher quadrature the main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods thereby reducing aliasing driven instabilities we demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations where in the latter both volumetric and interface approximations must be considered we show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations

multiscale singularly perturbed control systems limit occupational measures sets and averaging an averaging technique for nonlinear multiscale singularly perturbed control systems is developed issues concerning the existence and structure of limit occupational measures sets generated by such systems are discussed general results are illustrated with special cases

discontinuous isogeometric analysis methods for the first order form of the neutron transport equation with discrete ordinate sn angular discretisation in this paper two discontinuous galerkin isogeometric analysis methods are developed and applied to the first order form of the neutron transport equation with a discrete ordinate s n angular discretisation the discontinuous galerkin projection approach was taken on both an element level and the patch level for a given non uniform rational b spline nurbs patch this paper describes the detailed dispersion analysis that has been used to analyse the numerical stability of both of these schemes the convergence of the schemes for both smooth and non smooth solutions was also investigated using the method of manufactured solutions mms for multidimensional problems and a 1d semi analytical benchmark whose solution contains a strongly discontinuous first derivative this paper also investigates the challenges posed by strongly curved boundaries at both the nurbs element and patch level with several algorithms developed to deal with such cases finally numerical results are presented both for a simple pincell test problem as well as the c5g7 quarter core mox uox small light water reactor lwr benchmark problem these numerical results produced by the isogeometric analysis iga methods are compared and contrasted against linear and quadratic discontinuous galerkin finite element dgfem s n based methods

occupation measures for controlled markov processes characterization and optimality for controlled markov processes taking values in a polish space control problems with ergodic cost infinite horizon discounted cost and finite horizon cost are studied each is posed as a convex optimization problem wherein one tries to minimize a linear functional on a closed convex set of appropriately defined occupation measures for the problem these are characterized as solutions of a linear equation asssociated with the problem this characterization is used to establish the existence of optimal markov controls the dual convex optimization problem is also studied

on the eddy resolving capability of high order discontinuous galerkin approaches to implicit les under resolved dns of euler turbulence we present estimates of spectral resolution power for under resolved turbulent euler flows obtained with high order discontinuous galerkin dg methods the 1 rule based on linear dispersion diffusion analysis introduced by moura et al 2015 10 is here adapted for 3d energy spectra and validated through the inviscid taylor green vortex problem the 1 rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra as the original rule relies on standard upwinding different riemann solvers are tested very good agreement is found for solvers which treat the different physical waves in a consistent manner relatively good agreement is still found for simpler solvers the latter however displayed spurious features attributed to the inconsistent treatment of different physical waves it is argued that in the limit of vanishing viscosity such features might have a significant impact on robustness and solution quality the estimates proposed are regarded as useful guidelines for no model dg based simulations of free turbulence at very high reynolds numbers

uniqueness for some leray hopf solutions to the navier stokes equations abstract we exhibit simple sufficient conditions which give weak strong uniqueness for the 3d navier stokes equations the main tools are trilinear estimates and energy inequalities we then apply our result to the framework of lorentz morrey and besov over morrey spaces so as to get new weak strong uniqueness classes and so uniqueness classes for solutions in the leray hopf class in the last section we give a uniqueness and regularity result we obtain new uniqueness classes for solutions in the leray hopf class without energy inequalities but sufficiently regular

on the use of kinetic energy preserving dg schemes for large eddy simulation abstract recently element based high order methods such as discontinuous galerkin dg methods and the closely related flux reconstruction fr schemes have become popular for compressible large eddy simulation les element based high order methods with riemann solver based interface numerical flux functions offer an interesting dispersion dissipation behavior for multi scale problems dispersion errors are very low for a broad range of scales while dissipation errors are very low for well resolved scales and are very high for scales close to the nyquist cutoff in some sense the inherent numerical dissipation caused by the interface riemann solver acts as a filter of high frequency solution components this observation motivates the trend that element based high order methods with riemann solvers are used without an explicit les model added only the high frequency type inherent dissipation caused by the riemann solver at the element interfaces is used to account for the missing sub grid scale dissipation due to under resolution of vortical dominated structures typical for les type setups element based high order methods suffer from stability issues caused by aliasing errors of the non linear flux terms a very common strategy to fight these aliasing issues and instabilities is so called polynomial de aliasing where interpolation is exchanged with projection based on an increased number of quadrature points in this paper we start with this common no model or implicit les iles dg approach with polynomial de aliasing and riemann solver dissipation and review its capabilities and limitations we find that the strategy gives excellent results but only when the resolution is such that about 40 of the dissipation is resolved for more realistic coarser resolutions used in classical les e g of industrial applications the iles dg strategy becomes quite inaccurate we show that there is no obvious fix to this strategy as adding for instance a sub grid scale models on top doesn t change much or in worst case decreases the fidelity even more finally the core of this work is a novel les strategy based on split form dg methods that are kinetic energy preserving the scheme offers excellent stability with full control over the amount and shape of the added artificial dissipation this premise is the main idea of the work and we will assess the les capabilities of the novel split form dg approach when applied to shock free moderate mach number turbulence we will demonstrate that the novel dg les strategy offers similar accuracy as the iles methodology for well resolved cases but strongly increases fidelity in case of more realistic coarse resolutions

singular perturbations of nonlinear degenerate parabolic pdes a general convergence result the main result of the paper is a general convergence theorem for the viscosity solutions of singular perturbation problems for fully nonlinear degenerate parabolic pdes partial differential equations with highly oscillating initial data it substantially generalizes some results obtained previously in 2 under the only assumptions that the hamiltonian is ergodic and stabilizing in a suitable sense the solutions are proved to converge in a relaxed sense to the solution of a limit cauchy problem with appropriate effective hamiltonian and initial data in its formulation our convergence result is analogous to the stability property of barles and perthame it should thus reveal a useful tool for studying general singular perturbation problems by viscosity solutions techniques a detailed exposition of ergodicity and stabilization is given with many examples applications to homogenization and averaging are also discussed

singular perturbations in infinite dimensional control systems in this paper we consider a singularly perturbed control system involving differential inclusions in banach spaces with slow and fast solutions using the averaging approach we obtain sufficient conditions for the hausdorff convergence of the set of slow solutions in the supremum norm we present applications of the theorem to prove convergence of the fast solutions in terms of invariant measures and convergence of equi lipschitz solutions

de aliasing on non uniform grids algorithms and applications we present de aliasing rules to be used when evaluating non linear terms with polynomial spectral methods on non uniform grids analogous to the de aliasing rules used in fourier spectral methods they are based upon the idea of super collocation followed by a galerkin projection of the non linear terms we demonstrate through numerical simulation that both accuracy and stability can be greatly enhanced through the use of this approach we begin by deriving from the numerical quadrature rules used by galerkin type projection methods the number of quadrature points and weights needed for quadratic and cubic non linearities we then present a systematic study of the effects of super collocation when using both a continuous galerkin and a discontinuous galerkin method to solve the one dimensional viscous burgers equation we conclude by examining three direct numerical simulation flow examples incompressible turbulent flow in a triangular duct incompressible turbulent flow in a channel at re 395 and compressible flow past a pitching airfoil at re 45 000

spatial eigenanalysis of spectral hp continuous galerkin schemes and their stabilisation via dg mimicking spectral vanishing viscosity for high reynolds number flows abstract this study considers the spatial eigensolution analysis of spectral hp continuous galerkin cg schemes complementing a recent work by moura et al 2016 15 which addressed cg s temporal analysis while the latter assumes periodic boundary conditions the spatial approach presumes inflow outflow type conditions and therefore provides insights for a different class of problems the linear advection diffusion problem is considered for a wide range of peclet numbers allowing for viscous effects at different intensities the inviscid linear advection case receives particular attention owing to the manifestation of peculiar characteristics previously observed for discontinuous galerkin dg schemes in the limit of strong over upwinding these effects are discussed in detail due to their potential to negatively affect solution quality and numerical stability of under resolved simulations at high reynolds numbers the spectral vanishing viscosity svv technique is subsequently considered as a natural stabilization strategy in the context of linear advection an optimization procedure is employed to match svv diffusion levels to those of dg at appropriate polynomial orders the resulting cg svv discretisations are tested against under resolved computations of spatially developing vortex dominated flows and display excellent robustness at high reynolds numbers along with superior eddy resolving characteristics at higher polynomial orders this highlights the importance of appropriate stabilization techniques to improve the potential of spectral hp cg methods for high fidelity simulations of transitional and turbulent flows including implicit les under resolved dns approaches

stabilisation of spectral hp element methods through spectral vanishing viscosity application to fluid mechanics modelling in this paper we present a formulation of spectral vanishing viscosity svv for the stabilisation of spectral hp element methods applied to the solution of the incompressible navier stokes equations we construct the svv around a filter with respect to an orthogonal expansions and prove that this methodology provides a symmetric semi positive definite svv operator after providing a few simple one and two dimensional examples to demonstrate the utility of the svv we examine how it can be applied to a spectral hp element discretisation of the navier stokes equations using a velocity correction splitting scheme we provide three fluid flow examples to help illustrate the pros and cons of this approach on stability and accuracy 2005 elsevier b v all rights reserved

linear programming approach to deterministic long run average problems of optimal control we establish that deterministic long run average problems of optimal control are asymptotically equivalent to infinite dimensional linear programming problems lpps and we establish that these lpps can be approximated by finite dimensional lpps the solutions of which can be used for construction of the optimal controls general results are illustrated with numerical examples

a spectral vanishing viscosity method for large eddy simulations a new simulation approach for high reynolds number turbulent flows is developed combining concepts of monotonicity in nonlinear conservation laws with concepts of large eddy simulation the spectral vanishing viscosity svv first introduced by e tadmor siam j numer anal 26 30 1989 is incorporated into the navier stokes equations for controlling high wavenumber oscillations unlike hyperviscosity kernels the svv approach involves a second order operator which can be readily implemented in standard finite element codes in the work presented here discretization is performed using hierarchical spectral hp methods accommodating effectively an ab initio intrinsic scale separation the key result is that monotonicity is enforced via svv leading to stable discretizations without sacrificing the formal accuracy i e exponential convergence in the proposed discretization several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddy viscosity spectral les of turbulent channel flow in its current implementation the svv approach for controlling the small scales is decoupled from the large scales but a procedure is proposed that will provide coupling similar to the classical les formulation

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

the second iterate for the navier stokes equation we consider the iterative resolution scheme for the navier stokes equation and focus on the second iterate more precisely on the map from the initial data to the second iterate at a given time t we investigate boundedness properties of this bilinear operator this new approach yields very interesting results a new perspective on koch tataru solutions a first step towards weak strong uniqueness for koch tataru solutions and finally an instability result in b q 1 for q 2

on sets of occupational measures generated by a deterministic control system on an infinite time horizon abstract we give a representation for the closed convex hull of the set of discounted occupational measures generated by control state trajectories of a deterministic control system we also investigate the limit behavior of the latter when the discount factor tends to zero and compare it with the limit behavior of the long run time average occupational measures set the novelty of our results is in that we allow the control set dependence on the state variables that make the results to be applicable to differential inclusions

about weak strong uniqueness for the 3d incompressible navier stokes system this article studies the problem of l2 stability and weak strong uniqueness of solutions of the incompressible navier stokes on the whole space input amssym bbb s 3 constructed by kato s approach in spaces coming from littlewood paley theory and using the l1 smoothing effect for the heat flow 2011 wiley periodicals inc

split form nodal discontinuous galerkin schemes with summation by parts property for the compressible euler equations fisher and carpenter 2013 12 found a remarkable equivalence of general diagonal norm high order summation by parts operators to a subcell based high order finite volume formulation this equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux we show that besides the construction of entropy stable high order schemes a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms thus by changing the subcell finite volume flux to a specific choice we are able to generate in a systematic way all common split forms of the compressible euler advection terms such as the ducros splitting and the kennedy and gruber splitting although these split forms are not entropy stable we present a systematic way to prove which of those split forms are at least kinetic energy preserving with this we construct a unified high order split form dg framework we investigate with three dimensional numerical simulations of the inviscid taylorgreen vortex and show that the new split forms enhance the robustness of high order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows in fact we show that for certain test cases the novel split form discontinuous galerkin schemes are more robust than the discontinuous galerkin scheme with over integration

computing the effective hamiltonian using a variational approach a numerical method for homogenization of hamilton jacobi equations is presented and implemented as an l infty calculus of variations problem solutions are found by solving a nonlinear convex optimization problem the numerical method is shown to be convergent and error estimates are provided one and two dimensional examples are worked in detail comparing known results with the numerical ones and computing new examples the cases of nonstrictly convex hamiltonians and hamiltonians for which the cell problem has no solution are treated

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

weak strong uniqueness for the isothermal navier stokes equations in this paper we are concerned with weak strong uniqueness results for the isothermal navier stokes equations in two space dimension using the methods of relative entropy we obtain some conditions on a weak solution such as the ones built up by plotnikov and weigant siam j math anal 47 626 653 2015 so that it is unique

weak strong uniqueness property for the magnetohydrodynamic equations of three dimensional compressible isentropic flows abstract by means of the concept of relative entropy we establish the weak strong uniqueness property in the class of finite energy weak solutions to the magnetohydrodynamic equations of three dimensional compressible isentropic flows with the adiabatic exponent 1 and constant viscosity coefficients under the assumption that weak solutions exist

a comparison of the dispersion and dissipation errors of gauss and gauss lobatto discontinuous galerkin spectral element methods we examine the dispersion and dissipation properties of the gauss and gauss lobatto discontinuous galerkin spectral element methods dgsems for linear wave propagation problems we show that the inherent underintegration in the gauss lobatto variant can be interpreted as a modal filtering of the highest polynomial mode this in turn has a drastic impact on the dispersion and dissipation relations of the gauss lobatto dgsem compared to the gauss variant we show that the gauss dgsem is typically more accurate than the gauss lobatto variant needing fewer points per wavelength for a given accuracy while on the other hand being more restricted in the explicit time step choice we show that the spectra of the dgsem operators depend on the boundary conditions applied and that the ratio of the time step restrictions of the two schemes depends on the choice of boundary conditions

viscosity solutions methods for singular perturbations in deterministic and stochastic control viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order degenerate parabolic hamilton jacobi bellman equations this characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions the results apply to cases where the usual order reduction method does not give the correct limit and to systems with fast state variables depending nonlinearly on the control some connections with ergodic control and periodic homogenization are discussed

remark on uniqueness of mild solutions to the navier stokes equations abstract we investigate a limiting uniqueness criterion to the navier stokes equations we prove that the mild solution is unique under the class c 0 t bmo 1 l loc 0 t l where bmo 1 is the critical space including l n as an application of uniqueness theorem we also consider the local well posedness of navier stokes equations in bmo 1

optimal control problems over large time intervals abstract the paper focuses on problems of optimal control over a large time interval with end states prescribed for a wide class of systems and performance indices an approximate solution is characterized by piecing together two solutions for two infinite time problems these solutions exhibit similar transient behaviour

a large eddy simulation method for dgsem using non linearly optimized relaxation filters abstract in this paper we apply a specifically designed dissipative spatial filter as sub grid scale model within the increasingly popular discontinuous galerkin methods and the closely related flux reconstruction high order methods for large eddy simulation the parameters of the filter kernel are optimized with data obtained from direct numerical simulation that is filtered and used as a ground truth to fit the overall kinetic energy and dissipation rate over time the optimization is carried out for polynomial degree 3 to 10 the optimal kernels are rigorously tested in the limit of infinite reynolds number flows hit and taylor green vortex flow additionally a brief extension to plane turbulent channel flow is given

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

weak strong uniqueness property for the compressible flow of liquid crystals abstract weak strong uniqueness property in the class of finite energy weak solutions is established for two different compressible liquid crystal systems by the method of relative entropy to overcome the difficulties caused by the molecular direction with inhomogeneous dirichlet boundary condition new techniques are introduced to build up the relative entropy inequalities

on weak strong uniqueness of solutions to the generalized incompressible navier stokes equations in the recent paper li and zhai 2010 proved the well posedness of the cauchy problem to the n dimensional generalized incompressible navier stokes equations with initial data u 0 belonging to the so called q space q 1 r n with 1 2 1 and 0 in this paper by using the littlewood paley theory we prove the weak strong uniqueness between weak solution and li zhai s strong solution for the n dimensional generalized incompressible navier stokes equations

on the weak strong uniqueness of koch tataruʼs solution for the navier stokes equations abstract we prove the weak strong uniqueness between koch tataru s solution and leray s weak solution for three dimensional incompressible navier stokes equations

on the behaviour of fully discrete flux reconstruction schemes abstract in this study we employ von neumann analyses to investigate the dispersion dissipation group velocity and error properties of several fully discrete flux reconstruction fr schemes we consider three fr schemes paired with two explicit runge kutta erk schemes and two singly diagonally implicit rk sdirk schemes key insights include the dependence of high wavenumber numerical dissipation relied upon for implicit large eddy simulation iles on the choice of temporal scheme and time step size also the wavespeed characteristics of fully discrete schemes and the relative dominance of temporal and spatial errors as a function of wavenumber and time step size are investigated salient properties from the aforementioned theoretical analysis are then demonstrated in practice using linear advection test cases finally a burgers turbulence test case is used to demonstrate the importance of the temporal discretization when using fr schemes for iles

linear dispersion diffusion analysis and its application to under resolved turbulence simulations using discontinuous galerkin spectral hp methods we investigate the potential of linear dispersion diffusion analysis in providing direct guidelines for turbulence simulations through the under resolved dns sometimes called implicit les approach via spectral hp methods the discontinuous galerkin dg formulation is assessed in particular as a representative of these methods we revisit the eigensolutions technique as applied to linear advection and suggest a new perspective to the role of multiple numerical modes peculiar to spectral hp methods from this new perspective secondary eigenmodes are seen to replicate the propagation behaviour of a primary mode so that dg s propagation characteristics can be obtained directly from the dispersion diffusion curves of the primary mode numerical dissipation is then appraised from these primary eigencurves and its effect over poorly resolved scales is quantified within this scenario a simple criterion is proposed to estimate dg s effective resolution in terms of the largest wavenumber it can accurately resolve in a given hp approximation space also allowing us to present points per wavelength estimates typically used in spectral and finite difference methods although strictly valid for linear advection the devised criterion is tested against 1d burgers turbulence and found to predict with good accuracy the beginning of the dissipation range on the energy spectra of under resolved simulations the analysis of these test cases through the proposed methodology clarifies why and how the dg formulation can be used for under resolved turbulence simulations without explicit subgrid scale modelling in particular when dealing with communication limited hardware which forces one to consider the performance for a fixed number of degrees of freedom the use of higher polynomial orders along with moderately coarser meshes is shown to be the best way to translate available degrees of freedom into resolution power

existence of markov controls and characterization of optimal markov controls given a solution of a controlled martingale problem it is shown under general conditions that there exists a solution having markov controls which has the same cost as the original solution this result is then used to show that the original stochastic control problem is equivalent to a linear program over a space of measures under a variety of optimality criteria existence and characterization of optimal markov controls then follows an extension of echeverria s theorem characterizing stationary distributions for uncontrolled markov processes is obtained as a corollary in particular this extension covers diffusion processes with discontinuous drift and diffusion coefficients

eigensolution analysis of spectral hp continuous galerkin approximations to advection diffusion problems this study addresses linear dispersion diffusion analysis for the spectral hp continuous galerkin cg formulation in one dimension first numerical dispersion and diffusion curves are obtained for the advection diffusion problem and the role of multiple eigencurves peculiar to spectral hp methods is discussed from the eigencurves behaviour we observe that cg might feature potentially undesirable non smooth dispersion diffusion characteristics for under resolved simulations of problems strongly dominated by either convection or diffusion subsequently the linear advection equation augmented with spectral vanishing viscosity svv is analysed dispersion and diffusion characteristics of cg with svv based stabilization are verified to display similar non smooth features in flow regions where convection is much stronger than dissipation or vice versa owing to a dependency of the standard svv operator on a local peclet number first a modification is proposed to the traditional svv scaling that enforces a globally constant peclet number so as to avoid the previous issues in addition a new svv kernel function is suggested and shown to provide a more regular behaviour for the eigencurves along with a consistent increase in resolution power for higher order discretizations as measured by the extent of the wavenumber range where numerical errors are negligible the dissipation characteristics of cg with the svv modifications suggested are then verified to be broadly equivalent to those obtained through upwinding in the discontinuous galerkin dg scheme nevertheless for the kernel function proposed the full upwind dg scheme is found to have a slightly higher resolution power for the same dissipation levels these results show that improved cg svv characteristics can be pursued via different kernel functions with the aid of optimization algorithms

linear programming approach to deterministic infinite horizon optimal control problems with discounting we investigate relationships between the deterministic infinite time horizon optimal control problem with discounting in which the state trajectories remain in a given compact set y and a certain infinite dimensional linear programming idlp problem we introduce the problem dual with respect to this idlp problem and obtain some duality results we construct necessary and sufficient optimality conditions for the optimal control problem under consideration and we give a characterization of the viability kernel of y we also indicate how one can use finite dimensional approximations of the idlp problem and its dual for construction of near optimal feedback controls the construction is illustrated with a numerical example

duality in linear programming problems related to deterministic long run average problems of optimal control it has been established recently that under mild conditions deterministic long run average problems of optimal control are asymptotically equivalent to infinite dimensional linear programming problems lpps and that these lpps can be approximated by finite dimensional lpps in this paper we introduce the corresponding infinite and finite dimensional dual problems and study duality relationships we also investigate the possibility of using solutions of finite dimensional lpps and their duals for numerical construction of the optimal controls in periodic optimization problems the construction is illustrated with a numerical example

on consistency and rate of convergence of flux reconstruction for time dependent problems this study is directed at a rigorous characterization of the consistency and convergence of discontinuous finite element schemes formulated using flux reconstruction fr we show that the fr formulation is consistent for linear advection and converges to the exact solution for any scheme that is stable in the von neumann sense the numerical solution for a scheme of polynomial order p is composed of p 1p 1 eigenmodes of which one and exactly one is physical such that it exhibits the analytical dispersion behavior in the limit of asymptotic grid resolution the remaining p modes are spurious such that the fraction of energy received by them from the initial condition vanishes in the asymptotic limit on grid refinement the rate of convergence of the numerical solution is a function of time starting from a short time rate at t 0 t 0 associated with interpolation and asymptotically approaching a long time rate as t t associated with numerical differentiation both these rates can be inferred directly from the eigensystem of the numerical derivative operator we verify these analytical expectations using simple experiments in 1 d and 2 d

weak solutions for the navier stokes equations with b 1 ln bxr 1 r 21 r l2 initial data one of the major topics in the study of nonlinear partial differential equations of the evolutionary type is to look for as large as possible initial value spaces so that as many as possible solutions of such equations can be obtained in the book recent developments in the navier stokes problems lemarie rieusset proved that the navier stokes equations have global weak solutions for initial data in the space bxr 1 r 21 r rn l2 rn 0 r 1 where xr is the space of functions whose pointwise products with hr functions belong to l2 xr denotes the closure of c0 rn in xr and bxr 1 r 21 r rn is the besov space over xr in this paper we partially extend this result of lemarie rieusset to the larger initial value space b 1 ln rn bxr 1 r 21 r rn l2 rn 0 r 1 where b 1 ln rn is a logarithmically modified version of the usual besov space b 1 rn

entropy adjoint p adaptive discontinuous galerkin method for the under resolved simulation of turbulent flows this paper presents an approach to mesh adaptation suitable for scale resolving simulations the methodology is based on the entropy adjoint approach which corresponds to a standard output based a

suboptimization of singularly perturbed control systems a technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems the technique is applicable in particular in the case when the optima

convex duality and nonlinear optimal control problems in nonlinear optimal control can be reformulated as convex optimization problems over a vector space of linear functionals in this way methods of convex analysis can be brought to bear on the task of characterizing solutions to such problems the result is a necessary and sufficient condition of optimality that generalizes well known sufficient conditions referred to as verification theorems in dynamic programming as a byproduct we obtain a representation of the minimum cost in terms of the upper envelope of subsolutions to the hamilton jacobi equation it is a striking illustration of the wide range of problems to which convex analysis and in particular convex duality is applicable the approach applied to parametric problems in the calculus of variations was pioneered by l c young lectures on the calculus of variations and optimal control theory w b saunders philadelphia pa 1969 as recent work has shown however it is equally fruitful when applied in optimal control thi

approximation of infinite dimensional linear programming problems which arise in stochastic control we study a general approximation scheme for infinite dimensional linear programming lp problems which arise naturally in stochastic control we prove that the optimal value of the approximating problems converges to the value of the original lp problem for the controls we show that if the approximating optimal controls converge the limiting control is an optimal control for the original lp problem as an application of this theory we present numerical approximations to the lp formulation of stochastic control problems in continuous time we study long term average and discounted control problems for the example for which the theoretical solution is known our approximation results are very accurate

taylor s frozen flow hypothesis in burgers turbulence by detailed analytical treatment of the shock dynamics in the burgers turbulence with large scale forcing we calculate the velocity structure functions between pairs of points displaced both in time and space our analytical treatment verifies the so called taylor s frozen flow hypothesis without relying on any closure and under very general assumptions we discuss the limitation of the hypothesis and show that it is valid up to time scales smaller than the correlation time scale of temporal velocity correlation function we support the analytical calculation by performing numerical simulation of the periodically kicked burgers equation

on the relation between discounted and average optimal value functions abstract we investigate the relation between discounted and average deterministic optimal control problems for nonlinear control systems in particular we are interested in the corresponding optimal value functions using the concepts of viability chain controllability and controllability a global convergence result for vanishing discount rate is obtained basic ingredients for the analysis are an abelian type theorem controllability properties of the system and the morse decomposition of the corresponding control flow

averaging method for discontinuous mayer s problem of singularly perturbed control systems abstract in the present paper we prove that the optimal cost of a mayer s problem for a control system with singular perturbation does converge to the optimal cost of a mayer s problem associated with control system obtained by averaging the main novelty of our result lies on the fact that we did not require any continuity assumption on the final cost

connections between the filtered discontinuous galerkin method and the flux reconstruction approach to high order discretizations the purpose of this paper is to provide new insights on the connections that exist between the discontinuous galerkin method dg the flux reconstruction method fr and the recently identified energy stable flux reconstruction method esfr when solving time dependent conservation laws all these schemes appear to be quite similar and it is important to understand how they are related in this paper we first review results on the stability of the discontinuous galerkin method and extend it to the filtered discontinuous galerkin method we then consider the flux reconstruction approach and show its connections with dg in particular we show how the energy stable flux reconstruction method introduced by vincent et al is equivalent to a filtered dg method hence giving a new proof of its stability also it allows the use of the method without having to know the special form of the flux correction polynomials finally we underline some fundamental differences that exist between fr and dg

limit occupational measures set for a control system and averaging of singularly perturbed control systems abstract we discuss conditions for the existence of the limit occupational measures set for a control system we approximate slow components of the trajectories of a singularly perturbed control system by the solutions of a differential inclusion the differential inclusion is obtained via averaging the slow subsystem over measures from the limit occupational measures set constructed for the associated system describing the fast motions with frozen slow ones