pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction diffusion model we address the stability of solitary pulses as well as some other traveling structures near the onset of spatiotemporal chaos in a two species reaction diffusion model describing the oxidation of c

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

discrete dynamical systems in l topological spaces in this paper we investigate iteration of zadeh image operators and their discrete dynamical systems in l topological spaces some basic concepts such as limit set invariant set periodic point nonwandering point and recurrent point are discussed in weakly induced topological spaces we also study topological mixing topological transitivity and minimality for such dynamical systems while comparing with those corresponding ones in crisp spaces

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

on uniqueness for the harmonic map heat flow in supercritical dimensions we examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d 3 it is shown that generically singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality we also discuss how uniqueness can be retrieved 2017 wiley periodicals inc

point set lattice theoretic topology this essay attempts to survey in a coherent way certain aspects of point set lattice theoretic or poslat topology by which we mean fuzzy topology grounded in notions of sets functions powersets and powerset operators the latter two being lattice theoretic in nature and examined from a lattice theoretic point of view using methods of category theory connections with other related issues and developments are frequently given

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

solitary wave solutions for compound kdv type and compound kdv burgers type equations with nonlinear terms of any order in this letter based on the idea of adomian s decomposition method and with help of mathematica we obtain the exact solutions for compound kdv type and compound kdv burgers type equations with nonlinear terms of any order in this paper by considering the decomposition scheme we first obtain the exact solutions of these two equation for the initial conditions without using any classical transformations and then their numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions their remarkable accuracy is finally demonstrated in the study of some values p 1 of the compound kdv type and compound kdv burgers type equations

a new fuzzy compactness defined by fuzzy nets compactness is one of the most important notions in topology and is thoroughly investigated in general topology but it seems that the research about this notion in the area of fuzzy topology is not so effective as in general topology in fact it would seem that there was not even a good definition of compactness in this area until recently indeed soon after the famous article written by zadeh 8 was published chang 1 introduced the notion of fuzzy topological spaces and at the same time a certain fuzzy compactness was given from then on a series of different notions of fuzzy compactness have been discussed 2 3 71 and in 5 they were compared in detail but it seems unsatisfactory that all these fuzzy compactness ideas are defined for the whole fuzzy topological space but not for an arbitrary fuzzy subset therefore they are far from the crisp compactness for example it is impossible to discuss the hereditary property with respect to closed subsets of fuzzy compactness in 1977 pu and liu introduced the notion of fuzzy nets and a new concept of so called q neighborhood was given which could reflect the features of neighborhood structure in fuzzy topological spaces and by this new neighborhood structure the moore smith convergence theory was established spendidly 6 it furnishes us a basic tool to investigate the fuzzy compactness in a more general way to investigate the compactness of arbitrary fuzzy subsets in this paper a new fuzzy compactness defined by fuzzy nets we shall call it n compactness is given but we introduce the notion of r neighborhood see section 2 instead of qneighborhood to establish the moore smith convergence theory for it is perhaps more convennient to use n compactness possesses the following properties 1 n compactness is defined for arbitrary fuzzy subsets 2 for n compactness the hereditary property with respect to closed subsets holds 3 n compactness is a good extension 5 4 the continuous images of ncompact sets are n compact 5 for n compactness the tychonoff product theorem holds 6 if an n compact fuzzy topological space is t2 6 then it will satisfy a stronger separation axiom which will be called t4 in this paper 1 0022 247x 83 3 00

existence and stability of viscoelastic shock profiles we investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by antman and malek madani we establish that the resulting equations fall into the class of symmetrizable hyperbolic parabolic systems hence spectral stability implies linearized and nonlinear stability with sharp rates of decay the new contributions are treatment of the compressible case formulation of a rigorous nonlinear stability theory including verification of stability of small amplitude lax shocks and the systematic incorporation in our investigations of numerical evans function computations determining stability of large amplitude and nonclassical type shock profiles

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

explicit exact solutions for new general two dimensional kdv type and two dimensional kdv burgers type equations with nonlinear terms of any order in this paper we have improved the tanh method by means of a proper transformation and a general ansatz applying the improved method and direct assumption method with symbolic computation we consider two kinds of equations general two dimensional kdv type equations with nonlinear terms

space time derivative estimates of the koch tataru solutions to the nematic liquid crystal system in besov spaces abstract in recent paper 7 y du and k wang 2013 proved that the global in time koch tataru type solution u d to the n dimensional incompressible nematic liquid crystal flow with small initial data u 0 d 0 in bmo 1 bmo has arbitrary space time derivative estimates in the so called koch tataru space norms the purpose of this paper is to show that the koch tataru type solution satisfies the decay estimates for any space time derivative involving some borderline besov space norms more precisely for the global in time koch tataru type solution u d to the nematic liquid crystal flow with initial data u 0 d 0 bmo 1 bmo and u 0 bmo 1 d 0 bmo e for some small enough e 0 and for any positive integers k and m one has t k m 2 t k m u t k m d l r b 1 l 1 r b 1 e furthermore we shall give that the solution admits a unique trajectory which is holder continuous with respect to space variables

unsteady fronts in an autocatalytic system travelling waves in a model for autocatalytic reactions have for some parameter regimes been suggested to have oscillatory instabilities this instability is confirmed by various methods including linear stability analysis exploiting evan9s function and direct numerical simulations the front instability sets in when the order of the reaction m exceeds some threshold m c that depends on the inverse of the lewis number the stability boundary m m c is found numerically for m order one in the limit m 1 in which the system becomes similar to combustion systems with arrhenius kinetics the method of matched asymptotic expansions is employed to find the asymptotic front speed and show that m c 1 1 as 1 just beyond the stability boundary the unstable rocking of the front saturates supercritically if the order is increased still further period doubling bifurcations occur and for small tau there is a transition to chaos through intermittency after the disappearance of a period 4 orbit

dynamics of the ericksen leslie equations with general leslie stress ii the compressible isotropic case in this article the non isothermal compressible ericksen leslie system for nematic liquid crystals subject to general leslie stress is considered it is shown that this system is locally well posed within the l q setting and that for initial data close to equilibria points which are identical with the ones for the incompressible situation the solution exists globally moreover any global solution which does not develop singularities converges to an equilibrium in the topology of the natural state manifold note that no structural assumptions on the leslie coefficients are imposed and in particular parodi s relation is not being assumed the results can be viewed as an extension of the studies in hieber and pruss math ann 369 977 996 2017 for the incompressible case to the compressible situation

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

invariance and stability of fuzzy systems the elaboration of a stability theory of fuzzy systems is fully justified by the use in a variety of practical situations of fuzzy controllers 11 27 28 however such a theory may be founded upon ideas that arise from a historical study of dynamical systems theory during the last quarter of the nineteenth century the study of ordinary differential equations underwent some rather radical changes prior to this period the major emphasis in the subject had been on the methods of solving equations it was during this period that peano 20 gave a rigorous proof of the existence theorem for solutions of ordinary differential equations during the same period lipschitz 151 and picard 22 showed that the method of successive approximations led to a proof of the existence and the uniqueness of solutions to the initial value problem these developments represent the dhuement of the attempts to solve differential equations as the works of peano lipschitz and picard were finishing one chapter in the book of differential equations another was being initiated with the research of liapunov 141 and poincare 23 this new chapter was based on an entirely different approach where one did not attempt to compute the solutions rather one tried to exploit their topological properties poincart in his famous m moire 23 was the first to look at an ordinary differential equation from the point of view of the geometry of the trajectories thereby creating the topological theory of ordinary differential equations almost at the same time the stability theory created by liapunov 141 although heavily quantitative in its methods stressed the importance of some topological features of ordinary differential equations during the 50 years following liapunov and poincare many important advances were made it is with the works of birkhoff 4 nemytskii and stepanov 191 and smale 26 that researchers realized that the essence of this theory was based on the notion of dynamical system based on liapunov s original works classical stability theory is concerned with the equilibrium points of the system and the dynamical

remarks on the stability of shock profiles for conservation laws with dissipation two remarks are made the first is to establish the stability of monotone shock profiles of the kdv burgers equation based on an energy method of goodman the second remark illustrates specifically in burgers equation that uniform rates of decay are not to be expected for perturbations of shock profiles in typical norms introduction we will prove two results the first is to establish the orbital stability of monotone shock profile solutions 4 x st of the kdv burgers equation 1 1 ut uux uxx uxxx 0 0 the second is to show that for shock profile solutions of viscous conservation laws and specifically for burgers equation 1 2 ut uux uxx one cannot expect perturbations to decay at any uniform rate algebraic or otherwise in ordinary norms such as lp the first result is related to goodman s conditional stability result in 5 for small amplitude shock profiles x st in systems of genuinely nonlinear conservation laws with viscosity u f u x buxx u e rm the second is intended to be relevant to the question of full stability for shock profiles of such systems the nonlinear stability to small perturbations of constant state solutions u0 x c of the system u f u x b u ux x where b c is a strictly stable viscosity matrix see 9 can be demonstrated based on a uniform rate of decay of solutions to the linearized equation the method used is similar to that developed in 8 and 10 for systems in n 1 space dimensions faster decay rates in higher space dimensions make the method easier to apply when n is large but it does just work for conservation laws with stable viscosities in one space dimension see 13 thus for example assuming c 0 if iiuo iih6 ll is sufficiently small then iiu t llh4 ct 1 4iiuoii h4 n la in this note we want to point out that such uniform decay rates cannot be valid for perturbations of shock profiles in particular for the shock profile solution of burgers equations we construct a family of perturbations which decay to zero but do so at no uniform rate in most ordinary norms including all lp norms thus for received by the editors october 15 1984 and in revised form december 19 1984 1980 mathematics subject classification primary 35q20 35l65 1985 american mathematical society 0002 9947 85 1 00 25 per page

global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions we consider weak solutions to a simplified ericksen leslie system of two dimensional compressible flow of nematic liquid crystals an initial boundary value problem is first studied in a bounded domain by developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity nabla mathbf d 2 mathbf d in the equations of angular momentum on the direction field and adapting the standard three level approximation scheme and the weak convergence arguments for the compressible navier stokes equations we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy then the cauchy problem with large initial data is investigated and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem provided that the second component of initial data of the direction field satisfies some geometric angle condition

regularity and existence of global solutions to the ericksen leslie system in mathbb r 2 in this paper we first establish the regularity theorem for suitable weak solutions to the ericksen leslie system in mathbb r 2 building on such a regularity we then establish the existence of a global weak solution to the ericksen leslie system in mathbb r 2 for any initial data in the energy space under the physical constraints on the leslie coefficients ensuring the dissipation of energy of the system which is smooth away from at most finitely many times this extends earlier works by lin et al arch ration mech anal 197 297 336 2010 on a simplified nematic liquid crystal flow to the general ericksen leslie system

numerical bifurcation and stability analysis of solitary pulses in an excitable reaction diffusion medium abstract we present a systematic computer assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space the reaction diffusion model used is of the activator inhibitor type only the activator diffuses in this medium the control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold this study focuses on travelling pulses their domain of existence and the bifurcations that render them unstable these pulses are approximated as a homoclinic orbits in a travelling wave ode frame and b as solutions of the full partial differential equation pde with periodic boundary conditions in large domains a variety of bifurcations in the travelling wave ode frame are observed including heteroclinic loops so called t points a r champneys and y a kuznetsov numerical detection and continuation of codimension 2 homoclinic bifurcations int j bif chaos 4 1994 785 h kokobu homoclinic and heteroclinic bifurcations of vectorfields japan j appl math 5 1988 455 instabilities in the full pde frame include both hopf bifurcations to modulated travelling waves involving the discrete pulse spectrum as well as transitions involving the continuous spectrum such as the so called backfiring transition m bar m hildebrand m eiswirth m falcke h engel and m neufeld chemical turbulence and standing waves in a surface reaction model the influence of global coupling and wave instabilities chaos 4 1994 499 the stability of modulated pulses is computed through numerical floquet analysis and a cascade of period doubling bifurcations is observed as well as certain global bifurcations these results corroborated by observations from direct numerical integration provide a skeleton around which many features of the overall complex spatiotemporal dynamics of the pde are organized

pointwise topology on completely distributive lattices abstract in this paper the author presents a kind of pointwise topology theory on complete completely distributive lattices which is both a generalization of fuzzy topology and a further generalization of the author s papers 14 15

travelling wave solutions to the korteweg de vries burgers equation the existence and certain qualitative properties of travelling wave solutions to the korteweg de vries burgers equation are established the limiting behaviour of these waves when e tends to zero and when tends to zero is examined together with a singular limit wherein both e and tend to zero

categories isomorphic to the kubiak sostak extension of tml abstract in this paper we study wang guo jun s topological molecular lattices wang s tml in a kubiak sostak sense and construct a category the kubiak sostak extension of tml denoted by ftml then we prove that ftml is topological over cd op and tml is bireflective full subcategory of ftml finally we introduce several important categories to characterize ftml

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

local stability theory of solitary pulses in an active medium abstract if the spatio temporal chaos of an extended active medium is driven by repulsive or attractive interaction among localized coherent structures such as kinks and pulses its dynamics can often be described by a low dimensional dynamical system this welcomed reduction in the degrees of freedom only occurs when the localized structures remain stable during their mutual interaction it is shown that the structures can be destroyed by two types of local disturbance an expanding radiation packet and a localized disturbance which travels with the structure these two instabilities and the bifurcations they induce select the proper coherent structure and specify the dynamics of the generalized kuramoto sivashinsky gks equation in an extended domain the possibility of describing pulse radiation interaction by resonance poles also suggests that a more general coherent structure theory can be developed

global solutions to the 3d incompressible nematic liquid crystal system abstract in this paper we consider the global well posedness of the cauchy problem of the 3d incompressible nematic liquid crystal system with initial data in the critical besov space b 2 1 1 2 r 3 b 2 1 3 2 r 3 in particular we prove that there exists a positive constant c 0 such that the nematic liquid crystal system has a unique global solution with initial data u 0 d 0 u 0 h u 0 3 d 0 which satisfies 1 1 d 0 d 0 b 2 1 3 2 1 u 0 h b 2 1 1 2 exp c 0 2 u 0 3 b 2 1 1 2 1 2 c 0 for some c 0 sufficiently small where d 0 is a constant vector with d 0 1 here and are two positive viscosity constants

global solution to the 3 d density dependent incompressible flow of liquid crystals abstract in this paper we consider the global wellposedness of the 3 d density dependent incompressible flow of liquid crystals in the critical besov spaces we first get the local solution for this system without assumptions of small density variation and then extend this local solution to be global under the smallness assumptions on initial density initial orientation field and the horizontal components of the initial velocity field this result implies the global well posedness of this system with large initial vertical velocity component

incompressible limit of the compressible nematic liquid crystal flows in a bounded domain with perfectly conducting boundary abstract in this paper we study the asymptotic behavior of the regular solution to a simplified ericksen leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in r 2 as the mach number tends to zero the evolution system consists of the compressible navier stokes equations coupled with the transported heat flow for the averaged molecular orientation we suppose that the navier stokes equations are characterized by a navier s slip boundary condition while the transported heat flow is subject to neumann boundary condition by deriving a differential inequality with certain decay property the low mach limit of the solutions is verified for all time provided that the initial data are well prepared

global well posedness and decay of smooth solutions to the non isothermal model for compressible nematic liquid crystals abstract the cauchy problem for the three dimensional non isothermal model for compressible nematic liquid crystals is considered existence of global in time smooth solutions is established provided that the initial datum is close to a steady state 0 d by using the l q l p estimates and the fourier splitting method if the initial perturbation is small in h 3 norm and bounded in l q q 1 6 5 norm we obtain the optimal decay rates for the first and second order spatial derivatives of solutions in addition the third and fourth order spatial derivatives of director field d in l 2 norm are achieved

a model problem for groundwater flow with dynamic capillary pressure stability of travelling waves modelling vertical nonsteady groundwater flow with dynamic capillary pressure leads to a nonlinear diffusion convection equation extended with a nonlinear third order term with two space derivatives and one time derivative we consider the special case in which the original equation is the burgers equation and the third order term is linear we investigate stability of travelling wave solutions of the resulting pseudo parabolic burgers equation for which we first obtain well posedness results

blow up and global existence for heat flows of harmonic maps in this paper it is proved that the solution to the evolution problem for harmonic maps blows up in finite time if the initial map belongs to some nontrivial homotopy class and the initial energy is sufficiently small

some new and general solutions to the compound kdv burgers system with nonlinear terms of any order abstract in this letter a new riccati equation expansion method is presented for constructing exact travelling wave solutions of nonlinear partial differential equations the main idea of this method is to take full advantage of the solutions of the riccati equation to construct exact travelling wave solutions of nonlinear partial differential equations as a result some more generalized solutions which contain triangular periodic solutions exp function solutions and the soliton like solutions are obtained

instability of the hocking stewartson pulse and its implications for three dimensional poiseuille flow the linear stability problem for the hocking stewartson pulse obtained by linearizing the complex ginzburg landau cgl equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to stability exponents a numerical algorithm based on the compound matrix method is developed for computing the evans function using values in the cgl equation associated with spanwise modulation of plane poiseuille flow we show that the hocking stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue implications for the spanwise structure of nonlinear poiseuille problem between parallel plates are also discussed

regularity of the solutions to the liquid crystal equations with small rough data in 30 the author has proved the existence of koch tataru 12 type solution to the incompressible nematic liquid crystal flow with initial data u0 d0 in bmo 1 bmo in this paper we shall present that the solution in 30 has arbitrary spatial regularity furthermore we shall give the decay estimates for any space derivative

global existence of weak solutions of the nematic liquid crystal flow in dimension three for any bounded smooth domain 3 or 3 we establish the global existence of a weak solution u d 0 3 s2 of the initial boundary value or the cauchy problem of the simplified ericksen leslie system llf modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary or cauchy data u0 d0 h h1 s2 with d0 s 2 the upper hemisphere furthermore u d satisfies the global energy inequality 2016 wiley periodicals inc

spectral stability of ideal gas shock layers extending recent results in the isentropic case we use a combination of asymptotic ode estimates and numerical evans function computations to examine the spectral stability of shock wave solutions of the compressible navier stokes equations with ideal gas equation of state our main results are that in appropriately rescaled coordinates the evans function associated with the linearized operator about the wave i converges in the large amplitude limit to the evans function for a limiting shock profile of the same equations for which internal energy vanishes at one end state and ii has no unstable positive real part zeros outside a uniform ball thus the rescaled eigenvalue ode for the set of all shock waves augmented with the nonphysical limiting case form a compact family of boundary value problems that can be conveniently investigated numerically an extensive numerical evans function study yields one dimensional spectral stability independent of amplitude for gas constant in 1 2 3 and ratio of heat conduction to viscosity coefficient within 0 2 5 1 4 1 47 for air other values may be treated similarly but were not considered the method of analysis extends also to the multi dimensional case a direction that we shall pursue in a future work

global solution to the 3d inhomogeneous nematic liquid crystal flows with variable density abstract in this paper we investigate the global existence and uniqueness of solution to the 3d inhomogeneous incompressible nematic liquid crystal flows with variable density in the framework of besov spaces it is proved that there exists a global and unique solution to the nematic liquid crystal flows if the initial data 0 1 u 0 n 0 e 3 m b p 1 3 p 1 r 3 b p 1 3 p 1 r 3 b p 1 3 p r 3 with 1 p 6 and satisfies 0 1 m b p 1 3 p 1 u 0 b p 1 3 p 1 n 0 e 3 b p 1 3 p c for some small c 0 depending only on p here m b p 1 3 p 1 r 3 is the multiplier space of besov space b p 1 3 p r 3 using the lagrangian approach in danchin and mucha 2012 2013 7 8 is the key to our results

finite time singularity of the nematic liquid crystal flow in dimension three in this paper we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singularity the first example is constructed within the class of axisymmetric solutions while the second example is constructed for any generic initial data u 0 d 0 that has sufficiently small energy and d 0 has a nontrivial topology

a further note on the regularity criterion for the 3d nematic liquid crystal flows we give a further investigation on the regularity criteria for the 3d nematic liquid crystal flows it is show that the obtained anisotropic integrability regularity criterion to the 3d nematic liquid crystal flows involving i u j i j 1 2 3 is an improvement of qian 2016 besides we also prove the solution is regular if some components of the velocity field and the vectorial field modeling the orientation of the crystal molecules satisfy the prodi serrin type conditions

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

on multi dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain abstract we study the global existence of weak solutions to a multi dimensional simplified ericksen leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain r n where n 2 or 3 by exploiting a maximum principle nirenberg s interpolation inequality and a smallness condition imposed on the n th component of initial direction field d 0 to overcome the difficulties induced by the supercritical nonlinearity d 2 d in the equations of angular momentum and then adapting a modified three dimensional approximation scheme and the weak convergence arguments for the compressible navier stokes equations we establish the global existence of weak solutions to the initial boundary problem with large initial energy and without any smallness condition on the initial density and velocity

dynamics of the ericksen leslie equations with general leslie stress i the incompressible isotropic case the ericksen leslie model for nematic liquid crystals in a bounded domain with general leslie and isotropic ericksen stress tensor is studied in the case of a non isothermal and incompressible fluid this system is shown to be locally strongly well posed in the l p setting and a dynamical theory is developed the equilibria are identified and shown to be normally stable in particular a local solution extends to a unique global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold in this case the solution converges exponentially to an equilibrium in the topology of the state manifold the above results are proven without any structural assumptions on the leslie coefficients and in particular without assuming parodi s relation

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

global solution to the nematic liquid crystal flows with heat effect abstract the temperature dependent incompressible nematic liquid crystal flows in a bounded domain r n n 2 3 are studied in this paper following danchin s method in 7 we use a localization argument to recover the maximal regularity of stokes equation with variable viscosity by which we first prove the local existence of a unique strong solution then extend it to a global one provided that the initial data is a sufficiently small perturbation around the trivial equilibrium state this paper also generalizes hu wang s result in 21 to the non isothermal case

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

exp function method for the exact solutions of fifth order kdv equation and modified burgers equation abstract in this paper we implemented the exp function method for the exact solutions of the fifth order kdv equation and modified burgers equation by using this scheme we found some exact solutions of the above mentioned equations

traveling waves to a burgers korteweg de vries type equation with higher order nonlinearities abstract in this paper first we survey some recent advances in the study of traveling wave solutions to the burgers korteweg de vries equation and some comments are given then we study a burgers korteweg de vries type equation with higher order nonlinearities a qualitative analysis to a two dimensional autonomous system which is equivalent to the burgers kdv type equation is presented and indicates that under certain conditions the burgers korteweg de vries type equation has neither nontrivial bell profile solitary waves nor periodic waves finally a solitary wave solution is obtained by means of the first integral method which is based on the ring theory of commutative algebra

well posedness of the ericksen leslie system we prove the local well posedness of the ericksen leslie system and the global well posedness for small initial data under a physical constraint condition on the leslie coefficients which ensures that the energy of the system is dissipated instead of the ginzburg landau approximation we construct an approximate system with the dissipated energy based on a new formulation of the system

connectivity in fuzzy topological molecular lattices abstract a new kind of the definitions for connectivity in a fuzzy topological molecular lattice called s connectivity is defined some properties for s connectivity are obtained and the relations between s connectivity and each connectivity given in 1 4 7 are discussed

some remarks on fuzzy powerset operators fuzzy powerset operators related to morphisms of underlying ground categories of categories of fuzzy topological spaces are considered and some properties of such operators related to isomorphisms fuzzy points and order reversing involutions are verified

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

chaos entropy and a generalized extension principle we show a relationship between a very simple criterion positive topological entropy and li yorke chaos a general definition of fuzzification and level set based on t norms conorms and their diagonal functions is introduced the chaos theorem of benhabib and day for set valued mappings is considerably strengthened and generalised

well posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data we investigate the well posedness of 1 the heat flow of harmonic maps from mathbb r n to a compact riemannian manifold n without boundary for initial data in bmo and 2 the hydrodynamic flow u d of nematic liquid crystals on mathbb r n for initial data in bmo 1 bmo

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

econometric methods for agricultural supply under uncertainty fertilizer use and crop response the paper develops general econometric methods for the treatment of modelling decision making under conditions of uncertainty and risk the problem of modelling fertilizer use and crop response in the context of irrigated and or rainfed water supply when there is uncertainty and risk with respect to weather conditions over the crop cycle and the subsequent price at marketing suggested the present approach as a by product the stochastic duality theory associated with the present primal problem is also derived

strong solutions to the density dependent incompressible nematic liquid crystal flows abstract in this paper we investigate the density dependent incompressible nematic liquid crystal flows in n dimensional n 2 or 3 bounded domain the local existence and uniqueness of strong solutions are obtained when the viscosity coefficient of fluid depends on density furthermore one establishes blowup criterions for the regularity of the strong solutions in dimensions two and three respectively in particular we build a blowup criterion just in terms of the gradient of density if the initial direction field satisfies some geometric configuration for these results the initial density need not be strictly positive

theory of fuzzy systems in this paper a theory of dynamical fuzzy systems is proposed it is shown that a fuzzy system can be represented by a flow describing the evolution of this fuzzy system in the space of fuzzy subsets of the state space the topological properties of the flow are then established

global low energy weak solution and large time behavior for the compressible flow of liquid crystals abstract in this paper we consider the weak solution of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in r 3 when the initial data are of small energy and initial density is positive and essentially bounded we prove the existence of a global weak solution in r 3 the large time behavior of a global weak solution is also established

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

fuzzy dynamical systems a fuzzy dynamical system on an underlying complete locally compact metric state space x is defined axiomatically in terms of a fuzzy attainability set mapping on x this definition includes as special cases crisp single and multivalued dynamical systems on x it is shown that the support of such a fuzzy dynamical system on x is a crisp multivalued dynamical system on x and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of x in addition fuzzy trajectories are defined their existence established and various properties investigated

long time behavior of solution for the compressible nematic liquid crystal flows in r 3 abstract in this paper we investigate the global existence and long time behavior of classical solution for the compressible nematic liquid crystal flows in three dimensional whole space first of all the global existence of classical solution is established under the condition that the initial data are close to the constant equilibrium state in h n r 3 n 3 framework then one establishes algebraic time decay for the classical solution by weighted energy method finally the algebraic decay rate of classical solution in l p r 3 norm with 2 p and optimal decay rate of their spatial derivative in l 2 r 3 norm are obtained if the initial perturbation belong to l 1 r 3 additionally

pointwise pseudo metrics in l fuzzy set theory a theory of pointwise pseudo metrics is based on l fuzzy sets a pointwise pseudo metric is characterized by its remote neighborhood maps many theorems in general topology are generalized on l fuzzy sets the l fuzzy real line and the l fuzzy unit interval are pointwise pseudo metrizable a pseudo metrization theorem of pointwise uniformity is obtained relations between erceg s p metric pseudometric and pointwise p metric are discussed

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

on the shock wave spectrum for isentropic gas dynamics with capillarity abstract we consider the stability problem for shock layers in slemrod s model of an isentropic gas with capillarity we show that these traveling waves are monotone in the weak capillarity case and become highly oscillatory as the capillarity strength increases using a spectral energy estimate we prove that small amplitude monotone shocks are spectrally stable we also show that monotone shocks have no unstable real spectrum regardless of amplitude this implies that any instabilities of these monotone traveling waves if they exist must occur through a hopf like bifurcation where one or more conjugate pairs of eigenvalues cross the imaginary axis we then conduct a systematic numerical evans function study which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths in particular we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities

instability of oscillatory shock profile solutions of the generalized burgers kdv equation abstract oscillatory shock profile solutions of the burgers kdv equation are studied in the limit as the viscosity e 0 a rigorous proof of their instability is obtained by showing that the linearized operator about such a solution has many unstable eigenvalues for sufficiently small e the result is obtained by applying a topological method introduced by alexander gardner and jones to extract spectral information about the perturbed wave with slightly positive viscosity from particular pieces of the underlying wave which are approximated by a solitary wave solution of the reduced equation with no viscous dissipation

heat flow of harmonic maps whose gradients belong to l n _ x l infty _ t for any compact n dimensional riemannian manifold m g without boundary a compact riemannian manifold n subset mathbb r k without boundary and 0 t we prove that for n 4 if u m 0 t n is a weak solution to the heat flow of harmonic maps such that nabla u in l n x l infty t m times 0 t then u c m 0 t n as a consequence we show that for n 3 if 0 t is the maximal time interval for the unique smooth solution u c m 0 t n of 1 1 then nabla u t l n m blows up as t t

one dimensional stability of parallel shock layers in isentropic magnetohydrodynamics abstract extending investigations of barker humpherys lafitte rudd and zumbrun for compressible gas dynamics and freistuhler and trakhinin for compressible magnetohydrodynamics we study by a combination of asymptotic ode estimates and numerical evans function computations the one dimensional stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters shock amplitude strength of imposed magnetic field viscosity magnetic permeability and electrical resistivity for a law gas with 1 3 other values may be treated similarly but were not checked numerically depending on magnetic field strength these shocks may be of fast lax intermediate overcompressive or slow lax type however the shock layer is independent of magnetic field consisting of a purely gas dynamical profile in each case our results indicate stability interesting features of the analysis are the need to renormalize the evans function in order to pass continuously across parameter values where the shock changes type or toward the large amplitude limit at frequency 0 and the systematic use of winding number computations on riemann surfaces

h1 topological molecular lattices abstract this paper is devoted to the study of h 1 topological molecular lattices a generalization of the hausdorff fuzzy topological spaces some primary properties of such topological molecular lattices are discussed for any complete and completely distributive lattice l the existence and uniqueness of minimal h 1 co topologies on l are investigated and the characteristic theorem of minimal h 1 co topologies on l is established

a new general algebraic method with symbolic computation and its application to two nonlinear differential equations with nonlinear terms of any order in this paper with the help of symbolic computation system maple we present a new generalized riccati equation rational expansion method and successfully obtain some new rational formal solutions of some nonlinear partial differential equation with nonlinear terms of any order the presented method can also be applied to other nonlinear evolution equations with nonlinear terms of any order

order homomorphisms on fuzzes abstract in this paper basic properties of the concept of order homomorphisms on fuzzes 4 which is a proper generalization of the concept of fuzz functions 1 are studied conditions for fuzz functions to be functions of zadeh s type are obtained moreover the concepts of continuous closed and open order homomorphisms are introduced and studied

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

regularity of weak solutions and the number of singular points to the 3d simplified nematic liquid crystal system abstract in this paper we investigate the regularity of weak solutions to the cauchy problem of the 3d simplified nematic liquid crystal flows with lower bound on pressure and show that if the negative part of the pressure is controlled or if the positive part of quantity u 2 d 2 2 is controlled then the weak solution u d is global in time smooth we also study the singular points of weak solutions and prove that if a weak solution u d on r 3 0 t satisfies u t l p r 3 d t l p r 3 c t t p 3 2 p for some 3 p where c is a positive constant then the number of singular points is finite

regularity of solutions to the liquid crystal flows with rough initial data abstract in this paper we are concerned with the regularity of solutions to the liquid crystal flows with rough initial data in r n we prove that the solution constructed by wang 2011 in 23 has higher regularity moreover we obtain a decay estimate in time for any space derivative

oscillatory instability of traveling waves for a kdv burgers equation the stability of traveling wave solutions of a generalization of the kdv burgers equation o u u p o u 03u a o u is studied as the parameters p and a are varied the eigenvalue problem for the linearized evolution of perturbations is analyzed by numerically computing evans function d a an analytic function whose zeros correspond to discrete eigenvalues in particular the number of unstable eigenvalues in the complex plane is evaluated by computing the winding number of d a analytical and numerical evidence suggests that a hopf bifurcation occurs for oscillatory traveling wave profiles in certain parameter ranges dynamic simulations suggest that the bifurcation is subcritical no stable time periodic solution is found

stability of viscous detonations for majda s model abstract using analytical and numerical evans function techniques we examine the spectral stability of strong detonation wave solutions of a version of majda s scalar model for a reacting gas mixture with an arrhenius type ignition function we introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane and we use a numerical approximation of the evans function to search for possible unstable eigenvalues in this region our results show for the parameter values tested that these waves are spectrally stable combining these numerical results with the pointwise green function analysis of lyng raoofi texier zumbrun g lyng m raoofi b texier k zumbrun pointwise green function bounds and stability of combustion waves j differential equations 233 2 2007 654 698 we conclude that these waves are nonlinearly stable this represents the first demonstration of nonlinear stability for detonation wave solutions of a majda type model without a smallness assumption notably our results indicate that for this simplified scalar model there does not occur either in a normal parameter range or in the limit of high activation energy hopf bifurcation to galloping or pulsating solutions as is observed in the full reactive navier stokes equations this answers in the negative for this model a question posed by majda as to whether such scalar detonation models capture this aspect of detonation behavior