an elliptic parabolic free boundary problem continuity of the interface on etablit la continuite de l interface de la solution faible d un probleme elliptique parabolique lie a la theorie des ecoulements partiellement satures dans les milieux poreux on utilise une methode basee sur le principe du maximum pour les equations paraboliques en supposant l ecoulement unidimensionnel

quasi cohen macaulay properties of local homomorphisms abstract for a large class of local homomorphisms r s including those of finite g dimension studied by avramov and foxby proc london math soc 75 1997 241 270 we assign a new numerical invariant called the quasi cohen macaulay defect of and a local homomorphism is called quasi cohen macaulay if it is of finite g dimension and has trivial quasi cohen macaulay defect we show among other things the following g dim spec spec g dim

convex entropies and hyperbolicity for general euler equations the compressible euler equations possess a family of generalized entropy densities of the form rho f sigma where rho is the mass density sigma is the specific entropy and f is an arbitrary function entropy inequalities associated with convex entropy densities characterize physically admissible shocks for polytropic gases harten has determined which rho f sigma are strictly convex in this paper we extend this determination to gases with an arbitrary equation of state moreover we show that at every state where the sound speed is positive i e where the euler equations are hyperbolic there exist rho f sigma that are strictly convex thereby establishing the converse of the general fact that the existence of a strictly convex entropy density implies hyperbolicity

coercive boundary value problems for regular degenerate differential operator equations abstract this study focuses on nonlocal boundary value problems bvp for degenerate elliptic differential operator equations doe that are defined in banach valued function spaces where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients several conditions obtained that guarantee the maximal l p regularity and fredholmness these results are also applied to nonlocal bvp for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties

numerical defects of the hll scheme and dissipation matrices for the euler equations the harten lax van leer scheme is popularly used in the cfd community however oscillations are observed from shock tube problems for stiffened gases when the adiabatic index is greater than 3 to understand this phenomenon the dissipation effect of the scheme is evaluated quantitatively in terms of dissipation matrices we have proven and numerically demonstrated that lack of positive definiteness is the root of numerical defects

weakly nonlinear dissipative approximations of hyperbolic parabolic systems with entropy hyperbolic parabolic systems have spatially homogenous stationary states when the dissipation is weak one can derive weakly nonlinear dissipative approximations that govern perturbations of these constant states these approximations are quadratically nonlinear when the original system has an entropy the approximation is formally dissipative in a natural hilbert space we show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that hilbert space we also prove a weak strong uniqueness theorem for it in addition we give a kawashima type criterion for this approximation to be strictly dissipative we apply the theory to the compressible navier stokes system

a general approach to stability in free boundary problems abstract every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem the free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both dirichlet and neumann data we establish a rigorous stability analysis of such equilibria including the construction of stable and unstable manifolds for this purpose we transform the free boundary problem to a fully nonlinear and nonlocal parabolic problem on a fixed domain with fully nonlinear lateral boundary conditions and we develop the general theory for such problems as an illustration we give two examples the second being the focussing flame problem in combustion theory

connected graded gorenstein algebras with enough normal elements we generalize 12 1 1 and 1 2 to the following situation theorem 1 let a be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of a contains a homogeneous normal element of positive degree then 1 a is auslander gorenstein and cohen macaulay 2 a has a quasi frobenius quotient ring 3 every minimal prime ideal p is graded andgkdim a p d 4 if moreover a has finite global dimension then a is a domain and a maximal order in its quotient division ring to prove the above we need the following result which is a generalization of 3 2 46 ii theorem 2 let a be a connected graded noetherian as gorenstein algebra of injective dimension d then 1 the last term of the minimal injective resolution of aais isomorphic to a shift of a 2 for every noetherian graded a module m extd m a is finite dimensional over k

homological properties of balanced cohen macaulay algebras a balanced cohen macaulay algebra is a connected algebra a having a balanced dualizing complex a d in the sense of yekutieli 1992 for some integer d and some graded a a bimodule a we study some homological properties of a balanced cohen macaulay algebra in particular we will prove the following theorem theorem 0 1 let a be a noetherian balanced cohen macaulay algebra and m a nonzero finitely generated graded left a module then 1 m has a finite resolution of the form 0 a l mj a l 1j h m 0 where h is a finitely generated maximal cohen macaulay graded left a module 2 m has finite injective dimension if and only if m has a finite resolution of the form 0 a l mj a l 1j a l 0j m 0 as a corollary we will have the following characterizations of as gorenstein algebras and as regular algebras corollary 0 2 let a be a noetherian balanced cohen macaulay algebra 1 a is as gorenstein if and only if a has finite projective dimension as a graded left a module 2 a is as regular if and only if every finitely generated maximal cohen macaulay graded left a module is free

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

exact solution and precise asymptotics of a fisher kpp type front the present work concerns a version of the fisher kpp equation where the nonlinear term is replaced by a saturation mechanism yielding a free boundary problem with mixed conditions following an idea proposed in brunetderrida 2015 we show that the laplace transform of the initial condition is directly related to some functional of the front position mu t we then obtain precise asymptotics of the front position by means of singularity analysis in particular we recover the so called ebert and van saarloos correction ebertvansaarloos 2000 we obtain an additional term of order log t t in this expansion and we give precise conditions on the initial condition for those terms to be present

mathematical treatment of the magnetic confinement in a current carrying stellarator the model studied concerns the case of a stellarator machine and the magnetic confinement is modeled by using averaging methods and suitable vacuum coordinates this is shown to lead to a two dimensional grad shafranov type problem for the averaged poloidal flux function various problems are considered and it is pointed out that corresponding problems for models based on tokamak machines are essentially similar

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

finite difference methods for solution of nonlocal boundary value problems the article is devoted to the construction and investigation of various types of difference schemes relevant to nonlocal boundary value problems stated for multidimensional elliptic equations suitable tools for performing the investigation are developed including difference analogue of green s formula and inequalities corresponding to nonlocal boundary conditions

entropy viscosity method for the single material euler equations in lagrangian frame a new finite element method for solving the euler equations in lagrangian coordinates is proposed the method is stabilized by adding artificial diffusion terms compatible with positivity of mass and internal energy a minimum principle on the specific entropy and all generalized entropy inequalities two options of first order artificial diffusion are considered one is in the spirit of the eulerian based method guermond et al 2011 23 22 zingan et al 2013 and the other is similar to existing viscosity stabilizations in lagrangian frame e g dobrev et al 2012 the method is verified to be high order for smooth solutions even with active viscosity terms this is achieved by using high order finite element spaces and an entropy based viscosity stabilization that degenerates the first order viscous terms this stabilization automatically distinguishes smooth and singular regions the formal accuracy and convergence properties of the proposed methods are tested on a series of benchmark problems this is the first result extending the entropy viscosity methodology to the lagrangian hydrodynamics

homological identities for noncommutative rings abstract the auslander buchsbaum formula and bass s theorem are proved for a class of nonlocal noncommutative algebras including certain affine pi hopf algebras we also show that each filtered noetherian affine pi hopf algebra has finite injective dimension which partially answers a question of brown

symmetrization techniques on unbounded domains application to a chemotaxis system on rn the authors study the parabolic elliptic system on rn u t u u v 0 v v u u 0 u0 a version of the mathematical model of chemotaxis proposed by keller and segel a differential inequality for the quantity s0u t d where u is the decreasing rearrangement of the solution u t with respect to the spatial variable is obtained as a consequence they obtain e g lp bounds of the solution u v on r2 and global in time existence of solutions under the condition r2u0 8 then the solution u v blows up in a finite time compared to the previous work of diaz diaz and nagai adv math sci appl 5 1995 no 2 659 680 mr1361010 96j 35246 where this problem has been considered on bounded domains of rn there are some additional technical difficulties connected with the regularity of the derivative u t

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

on fourier multiplier transformations of banach valued functions we obtain analogues of the mihlin multiplier theorem and littlewood paley inequalities for functions with values in a suitable banach space b the requirement on b is that it have the unconditionality property for martingale difference sequences

existence theorems for dualizing complexes over non commutative graded and filtered rings abstract in this note we prove existence theorems for dualizing complexes over graded and filtered rings thereby generalizing some results by zhang yekutieli and jorgensen

on a third order multi point boundary value problem at resonance in this paper we prove some existence results for a third order multi point boundary value problem at resonance our method is based upon the coincidence degree theory of mawhin

convergence to travelling waves in a reaction diffusion system arising in contaminant transport we consider a nonlinear system of parabolic equations describing the transport of contaminant in porous media for this system travelling wave solutions are known to exist we prove existence and uniqueness of general solutions of the system and convergence to travelling waves

b separable boundary value problems in banach valued function spaces in this paper the nonlocal boundary value problems for anisotropic partial differential operator equations with dependent coefficients in banach valued besov b spaces are studied the principal parts of the appropriate differential operators are non self adjoint several conditions for separability and fredholmness are given these results permit us to establish that the inverse of the corresponding differential operators belong to the schatten q class the spectral properties of the appropriate differential operators are also investigated in addition we study the maximal regularity of nonlocal initial boundary value problems for abstract parabolic equations finite or infinite systems of parabolic equations and the separability of nonlocal boundary value problems for finite or infinite systems of quasi elliptic equations in b spaces

dualizing complexes over noncommutative graded algebras the motivation for the work presented in this paper was two questions posed by m artin let a be a d dimensional regular graded algebra over the field k with augmentation ideal m see def 4 13 denote by hi a resp hi a the dth left resp right local cohomology of a at the ideal m i e hi a lim extd a m a and similarly for hi a as left modules and as right modules hi a and h a are both isomorphic to a hom a k up to a twist in degrees all operations are taking place in the graded category

uniqueness of solution to a free boundary problem from combustion we investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows to find a continuous function u x t 0 defined in a domain d rn 0 t and such that u ai uxi ut 0 in d u 0 we also assume that the interior boundary of the positivity set d u 0 so called free boundary is a regular hypersurface on which the following conditions are satisfied

on the divergence free condition in godunov type schemes for ideal magnetohydrodynamics the upwind constrained transport method we present a general framework to design godunov type schemes for multidimensional ideal magnetohydrodynamic mhd systems having the divergence free relation and the related properties of the magnetic field b as built in conditions our approach mostly relies on the constrained transport ct discretization technique for the magnetic field components originally developed for the linear induction equation which assures b num 0 and its preservation in time to within machine accuracy in a finite volume setting we show that the ct formalism when fully exploited can be used as a general guideline to design the reconstruction procedures of the b vector field to adapt standard upwind procedures for the momentum and energy equations avoiding the onset of numerical monopoles of o 1 size and to formulate approximate riemann solvers for the induction equation this general framework will be named here upwind constrained transport uct to demonstrate the versatility of our method we apply it to a variety of schemes which are finally validated numerically and compared a novel implementation for the mhd case of the second order roetype positive scheme by lin and lax j comput fluid dyn 5 1996 133 and both the second and third order versions of a central type mhd scheme presented by londrillo and del zanna astrophys j 530 2000 508 where the basic uct strategies have been first outlined

a free boundary problem for the heat equation arising in flame propagation we introduce a new free boundary problem for the heat equation of interest in combustion theory it is obtained in the description of laminar flames as an asymptotic limit for high activation energy the problem asks for the determination of a domain in space time q c rn x 0 t and a function u x t 0 defined in q such that ut au in q u takes certain initial conditions u x 0 uo x for x e qo oqfn t 0 and two conditions are satisfied on the free boundary r a2 n t 0 u o and u1 1 where u1 denotes the derivative of u along the spatial exterior normal to r we approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data

invariant domain preserving discretization independent schemes and convex limiting for hyperbolic systems abstract we introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving the method is discretization independent provided elementary symmetry and skew symmetry properties are satisfied by the scheme the method is formally first order accurate in space then we introduce a series of higher order methods when these methods violate the invariant domain properties they are corrected by a limiting technique that we call convex limiting after limiting the resulting methods satisfy all the invariant domain properties that are imposed by the user see theorem 7 24 and is formally high order accurate the two key novelties are that i limiting is done by enforcing bounds on quasiconcave functionals ii the bounds that are enforced on the solution at each time step are necessarily satisfied by the low order approximation

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

bass series of local ring homomorphisms of finite flat dimension nontrivial relations between bass numbers of local commutative rings are established in case there exists a local homomorphism r s which makes s into an r module of finite flat dimension in particular it is shown that an inequality r i depth r s i depth s holds for all i z this is a consequence of an equality involving the bass series i r m t i z r i m t i of a complex m of r modules which has bounded above and finite type homology and the bass series of the complex of s modules m r s where denotes the derived tensor product it is prove that there is an equa lity of formal laurent series i s m rs t i r m t i f t where f is the fiber of considered as a homomorphism of commutative differential graded rings

non commutative castelnuovo mumford regularity and as regular algebras abstract let a be a connected graded k algebra with a balanced dualizing complex we prove that a is a koszul as regular algebra if and only if that the castelnuovo mumford regularity and the ext regularity coincide for all finitely generated a modules this can be viewed as a non commutative version of t romer on the regularity over positively graded algebras j algebra 319 2008 1 15 theorem 1 3 by using castelnuovo mumford regularity we prove that any koszul standard as gorenstein algebra is as regular as a preparation to prove the main result we also prove that the following statements are equivalent 1 a is as gorenstein 2 a has finite left injective dimension 3 the dualizing complex has finite left projective dimension this generalizes i mori homological properties of balanced cohen macaulay algebras trans amer math soc 355 2002 1025 1042 corollary 5 9

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

finite difference method for multipoint nonlocal elliptic parabolic problems a finite difference method for solving the multipoint elliptic parabolic partial differential equation with a nonlocal boundary condition is considered stable difference schemes accurate to first and second orders for this problem are presented stability almost coercive stability and coercive stability for the solution of these difference schemes are obtained the theoretical statements for the solution of these difference schemes are supported by numerical examples

the gaussian moment closure for gas dynamics the moment closure method of levermore applied to the boltzmann equation for rarefied gas dynamics leads to a hierarchy of symmetric hyperbolic systems of partial differential equations the euler system is the first member of this hierarchy of closures in this paper we investigate the next member the 10 moment gaussian closure we first reduce the collision term to an integral which may be explicitly evaluated for the special case of maxwell molecular interaction the resulting collision term for this case is shown to be equivalent to the term obtained by replacing the boltzmann collision operator with the bhatnagar gross and krook bgk approximation we then analyze the gaussian system applied to the canonical flow problem of a stationary planar shock an analytic shock profile for the gaussian closure is derived and compared with the numerical solutions of the boltzmann and navier stokes equations the results show reasonable agreement for weak shocks and close agreement between the downstream ga

ring homomorphisms and finite gorenstein dimension the local structure of homomorphisms of commutative noetherian rings is investigated from the point of view of dualizing complexes a concept of finite gorenstein dimension which substantially weakens the notion of finite flat dimension is introduced for homomorphisms it is shown to impose structural constraints due to a remarkable equivalence of subcategories of the derived category of all modules an essential part of this study is the development of relative notions of dualizing complexes and bass numbers it is proved that the bass numbers of local homomorphisms are rigid extending a known result for local rings quasi gorenstein homomorphisms are introduced as local homomorphisms that base change a dualizing complex for the source ring into one for the target they are shown to have the stability properties of the gorenstein homomorphism that they generalize 1991 mathematics subject classification primary 13h10 13d23 14e40 secondary 13c15

an elliptic parabolic free boundary problem in which c is a continuous nondecreas rg function defined on r such that c u is strictly increasing when u 0 ui 1 if bi 0 and bi 1 if ai 0 for i 0 and i 1 and fo fi and t are prescribed functions problem i arises in the theory of fluid aow through pargaily saturated media then t6 denotes the potential due to capiitary suction and c the moisture content the dependence of c on ie is found empirically to be as in fig i c being bounded above by the saturation value c 1 thus in regions where the medium is saturated the flow is of potential type and described by an ehiptic equation and in regions where the medium is unsaturated the flow is of diffusive type and described by a parabolic equation at the boundary between these regions the interface one expects u 0 and u continuous

maximal b regular boundary value problems with parameters this study focuses on non local boundary value problems bvp for elliptic differential operator equations doe defined in banach valued besov b spaces here equations and boundary conditions contain certain parameters this study found some conditions that guarantee the maximal regularity and fredholmness in banach valued b spaces uniformly with respect to these parameters these results are applied to non local boundary value problems for a regular elliptic partial differential equation with parameters on a cylindrical domain to obtain algebraic conditions that guarantee the same properties

regularity results for an elliptic parabolic free boundary problem on etudie un probleme aux limites libre elliptique parabolique a une dimension d espace on donne plusieurs resultats de regularite pour la solution faible et la frontiere libre on donne des conditions pour que la frontiere libre soit une courbe c 1

nonlocal boundary value problems for the schrodinger equation abstract in the present paper the nonlocal boundary value problem i d u d t a u f t 0 t t u 0 m 1 p m u m 0 1 2 p t for the schrodinger equation in a hilbert space h with the self adjoint operator a is considered stability estimates for the solution of this problem are established two nonlocal boundary value problems are investigated the first and second order of accuracy difference schemes for the approximate solutions of this nonlocal boundary value problem are presented the stability of these difference schemes is established in practice stability inequalities for the solutions of difference schemes for the schrodinger equation are obtained a numerical method is proposed for solving a one dimensional schrodinger equation with nonlocal boundary condition a procedure involving the modified gauss elimination method is used for solving these difference schemes the method is illustrated by giving numerical examples

nonstationary filtration in partially saturated porous media from the mathematical formulation of a one dimensional flow through a partially saturated porous medium we arrive at a nonlinear free boundary problem the boundary being between the saturated and the unsaturated regions in the medium in particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part

on a nonlocal stationary free boundary problem arising in the confinement of a plasma in a stellarator geometry we prove the existence and some qualitative properties of the solution to a two dimensional free boundary problem modeling the magnetic confinement of a plasma in a stellarator configuration the nonlinear elliptic partial differential equation on the plasma region was obtained from the three dimensional magnetohydrodynamic system by hender carreras in 1984 by using averaging arguments and a suitable system of coordinates boozer s vacuum coordinates the free boundary represents the separation between the plasma and vacuum regions and the model is described by an inverse type problem some nonlinear terms of the equation are unknown using the zero net current condition for the stellarator configurations we reformulate the problem with the help of the notion of relative rearrangement leading to a new problem involving nonlocal terms in the equation we use an iterative algorithm and establish some new properties on the relative rearrangement in order to prove the convergence of the algorithm and then the existence of a solution

an elliptic parabolic equation with a nonlocal term for the transient regime of a plasma in a stellarator we prove the existence and the regularity of weak solutions of a nonlocal elliptic parabolic free boundary problem involving the notions of relative rearrangement and monotone rearrangement the problem arises in the study of the dynamics of a magnetically confined fusion plasma in a stellarator device when the dimensional analysis on the characteristic times suggests to neglect the inertial acceleration in presence of a time dependent magnetic field

existence of three dimensional toroidal mhd equilibria with nonconstant pressure we establish an existence result for the three dimensional mhd equations x b x b p b 0 b n t 0 with p const in tori t without symmetry more precisely our theorems insure the existence of sharp boundary solutions for tori whose departure from axisymmetry is sufficiently small they allow for solutions to be constructed with an arbitrary number of pressure jumps

some properties of the relative rearrangement abstract we develop some new properties of the relative rearrangement some of these properties generalize well known results known as the hardy littlewood inequality

second order invariant domain preserving ale approximation of hyperbolic systems abstract in this paper we introduce an invariant domain preserving arbitrary lagrangian eulerian method for solving hyperbolic systems the time stepping is explicit and the approximation in space is done with continuous finite elements the method is second order in space and made invariant domain preserving by using a newly introduced convex limiting technique

viscous regularization of the euler equations and entropy principles this paper investigates a general class of viscous regularizations of the compressible euler equations a unique regularization is identified that is compatible with all the generalized entropies la harten et al siam j numer anal 35 1998 pp 2117 2127 and satisfies the minimum entropy principle a connection with a recently proposed phenomenological model by h brenner phys a 370 2006 pp 190 224 is made

rationality of the poincare series for koszul algebras abstract in this paper we study the rationality of the poincare series of a finitely generated graded left module over a koszul connected algebra in particular we show the following result let a be a koszul connected algebra and a be its koszul dual suppose that a is noetherian and having a balanced dualizing complex if a is either 1 an artinian algebra 2 a graded quotient algebra of a noetherian as regular algebra or 3 an fbn algebra e g a noetherian pi algebra then the poincare series of every finitely generated graded left a module is a rational function over the complex numbers