the yang mills equations on the universal cosmos global existence and regularity of solutions for the yang mills equations on the universal cosmos m which has the form r1 s3 for each of an 8 parameter continuum of factorizations of m as time space are treated by general methods the cauchy problem in the temporal gauge is globally soluble in its abstract evolutionary form with arbitrary data for the field potential in l2 r s3 l2 r 1 s3 where r is an integer 1 and l2 r denotes the class of sections whose first r derivatives are square integrable if r 1 the problem is soluble locally in time when r is 3 or more the solution is identifiable with a classical one if infinite the solution is in c m these results extend earlier work and approaches 1 5 solutions of the equations on minkowski space time m0 extend canonically modulo gauge transformations to solutions on m provided their cauchy data are moderately smooth and small near spatial infinity precise asymptotic structures for solutions on m0 follow and in turn imply various decay estimates thus the energy in regions uniformly bounded in direction away from the light cone is o x0 5 where x0 is the minkowski time coordinate analysis solely in m0 8 9 earlier yielded the estimate o x0 2 applicable to the region within the light cone similarly it follows that the action integral for a solution of the yang mills equations in m0 is finite in fact absolutely convergent

global existence of time dependent yang mills higgs monopoles we study the cauchy problem for non abelian yang mills higgs theory in 3 1 dimensional minkowski spacetime with suitable conditions on the background fields and a suitable choice of a sobolev space for the subtracted gauge potentials and the higgs field we establish local existence we then prove global existence by showing that an appropriate norm of the solutions cannot blow up in a finite time

the existence and uniqueness of solutions of yang mills equations with bag boundary conditions the cauchy problem for the yang mills equations in the coulomb gauge is studied on a compact connected and simply connected riemannian manifold with boundary an existence and uniqueness theorem for the evolution equations is proven for fields with cauchy data in an appropriate sobolev space the proof is based the hodge decomposition of the yang mills fields and the theory of non linear semigroups

effective twisted conjugacy separability of nilpotent groups this paper initiates the study of effective twisted conjugacy separability for finitely generated groups which measures the complexity of separating distinct twisted conjugacy classes via finite quotients the focus is on nilpotent groups and our main result shows that there is a polynomial upper bound for twisted conjugacy separability that allows us to study regular conjugacy separability in the case of virtually nilpotent groups where we compute a polynomial upper bound as well as another application we improve the work of the second author by giving a possibly sharp upper bound for the conjugacy separability for finitely generated nilpotent groups of nilpotency class 2

the cauchy problem for coupled yang mills and scalar fields in the temporal gauge we study the cauchy problem for minimally coupled classical yang mills and scalar fields inn 1 dimensional space time in the temporal gauge we prove the existence and uniqueness of solutions for small time intervals and for anyn we then develop a general theory of solutions in local spaces and extend the previous local in time results to this more general setting in space time dimensions two and three we prove the existence of global in time solutions by the method of a priori estimates both in global and local spaces in space time dimension four our estimates yield only partial results on the global existence problem

cauchy problems of the gauged sigma model we study the gauged sigma model in r1 1 the existence of the global smooth solution will be proved furthermore we show that the global weak solutions exist in rn 1 n 2 3

construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension given a blow up solution with a characteristic point we refine the blow up behavior first derived by merle and zaag we also refine the geometry of the blow up set near a charac teristic point and show that except may be for one exceptional situation it is never symmetric with the respect to the characteristic point then we show that all blow up modalities predicted by those authors do occur more precisely given any integer k 2 and 0 we construct a blow up solution with a characteristic point a such that the asymptotic behavior of the solution near a t a shows a decoupled sum of k solitons with alternate signs whose centers in the hyperbolic geometry have 0 as a center of mass for all times

regularity of constraints in the minkowski space yang mills theory for yang mills theory in the minkowski space it is proved that the constraint set is a smooth submanifold of the phase space consisting of square integrable cauchy data

on the profile of energy concentration at blow up points for subconformal focusing nonlinear waves we consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it at noncharacteristic points on the singularity surface merle and zaag have identified the rate of blow up of the h1 norm of the solution inside cones that terminate at the singularity we derive bounds that restrict how this h1 energy can be distributed inside such cones our proof relies on new localized estimates obtained using carlemantype inequalities for such nonlinear waves these bound the lp 1 norm in the interior of timelike cones by their h1 norm near the boundary of the cones such estimates can also be applied to obtain certain integrated decay estimates for globally regular solutions to such equations in the interior of time cones

the yang mills heat semigroup on three manifolds with boundary long time existence and uniqueness of solutions to the yang mills heat equation is proven over a compact 3 manifold with smooth boundary the initial data is taken to be a lie algebra valued connection form in the sobolev space h1 three kinds of boundary conditions are explored dirichlet type neumann type and marini boundary conditions the last is a nonlinear boundary condition specified by setting the normal component of the curvature to zero on the boundary the yang mills heat equation is a weakly parabolic nonlinear equation we use gauge symmetry breaking to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation apriori estimates are developed by first establishing a gauge invariant version of the gaffney friedrichs inequality a gauge invariant regularization procedure for solutions is also established uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity this work is motivated by possible applications to quantum field theory

openness of the set of non characteristic points and regularity of the blow up curve for the 1 d semilinear wave equation we consider here the 1 d semilinear wave equation with a power nonlinearity and with no restriction on initial data we first prove a liouville theorem for that equation then we consider a blow up solution its blow up curve x mapsto t x and i 0 subset mathbb r the set of non characteristic points we show that i0 is open and that t x is c1 on i0 all these results fundamentally use our previous result in 19 showing the convergence in selfsimilar variables for x in i 0

global energy solutions of chern simons higgs equations in one space dimension abstract we study an initial value problem of chern simons higgs equations in one space dimension the global existence of solutions is proved for the finite energy data moreover we study local well posedness below the energy regularity observing null forms of the system and applying bilinear estimates for wave sobolev space h s

wave equations with concentrated nonlinearities in this paper we address the problem of wave dynamics in the presence of concentrated nonlinearities given a vector field v on an open subset of and a discrete set with n elements we define a nonlinear operator v y on which coincides with the free laplacian when restricted to regular functions vanishing at y and which reduces to the usual laplacian with point interactions placed at y when v is linear and represented by a hermitian matrix we then consider the nonlinear wave equation and study the corresponding cauchy problem giving an existence and uniqueness result when v is lipschitz the solution of such a problem is explicitly expressed in terms of the solutions of two cauchy problems one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part the main properties of the solution are given and when y is a singleton the mechanism and details of blow up are studied

solutions blowing up on any given compact set for the energy subcritical wave equation abstract we consider the focusing energy subcritical nonlinear wave equation t t u u u p 1 u in r n n 1 given any compact set k r n we construct finite energy solutions which blow up at t 0 exactly on k the construction is based on an appropriate ansatz the initial ansatz is simply u 0 t x t a x 2 p 1 where a 0 vanishes exactly on k which is a solution of the ode h h p we refine this first ansatz inductively using only ode techniques and taking advantage of the fact that for suitably chosen a space derivatives are negligible with respect to time derivatives we complete the proof by an energy argument and a compactness method

on the computation for blow up solutions of the nonlinear wave equation we consider in this paper the 1 dim nonlinear wave equation frac partial 2 u partial t 2 t x frac partial 2 u partial x 2 t x u 1 alpha t x alpha 0 and its finite difference analogue it is known that the solution of the current equation may become unbounded in finite time a phenomenon which is known as blow up moreover since the nonlinear wave equation enjoys finite speed of propagation even if the solution has become unbounded at certain points the solution continues to exist blows up at later times and forms the so called blow up curve up to the present however numerical approximation for blow up problems gave only the convergence up to the blow up time of vert u t cdot vert infty in fact adaptive time mesh strategy was used to numerically reproduce the phenomenon of finite time blow up nevertheless computation using such schemes can not be extended beyond the blow up time of vert u t cdot vert infty this is a fatal problem when computing blow up solutions for the nonlinear wave equation as a consequence we reconsider a finite difference scheme whose temporal grid size is given uniformly and propose an algorithm with rigorous convergence analysis to numerically reconstruct the blow up curve for the nonlinear wave equation

differentiability of the blow up curve for one dimensional nonlinear wave equations consider the cauchy problem rac artial 2 u artial t 2 umimits i 1 n rac artial 2 u artial x i 2 f u x n athbb r n t 0 ueft x 0 ight feft x ight eft x n athbb r n ight u t eft x 0 ight geft x ight eft x n athbb r n ight with n 3 it is well known that if f u is superlinear i e feft u ight a eft u ight p ifeft u ight o nfty eft a 0 p 1 ight 0 1 then solutions generally blow up in finite time see 2 6

a discrete phenomenon in the c and analytic regularity of the blow up curve of solutions to the liouville equation in one space dimension we give a complete discussion of the c or analytic regularity of blow up curves for cauchy problems or some mixed problems for the liouville equation in one space dimension in the case of mixed problems the regularity results depend on the boundary condition actually we show the existence of a sequence of boundary conditions for which the regularity of the blow up curve is better than in the general case

smoothing effects for schrodinger equations with electro magnetic potentials and applications to the maxwell schrodinger equations abstract we consider schrodinger equations in r 1 2 with electro magnetic potentials the potentials belong to h 1 and typically they are time independent or determined as solutions to inhomogeneous wave equations we prove kato type smoothing estimates for solutions we also apply this result to the maxwell schrodinger equations in the lorentz gauge and prove unique solvability of this system in the energy space

solitary wave solutions of 3 1 dimensional extended zakharov kuznetsov equation by lie symmetry approach abstract in this paper the solitary wave solutions of 3 1 dimensional extended zakharov kuznetsov ezk equation are constructed which appear in the magnetized two ion temperature dusty plasma and quantum physics lie group of transformation method is proposed to investigate the solution of 3 1 dimensional ezk equation via lie symmetry method the optimal system of one dimensional lie subalgebra is constructed by using lie point symmetries the three dimensional ezk equation reduced into number of ordinary differential equations odes by applying similarity reductions consequently solutions so extracted are more general than erstwhile known results we have obtained twenty one solutions in the explicit form some of them are likewise general and some are new for the best study of us eventually single soliton quasi periodic soliton multisoliton lump type soliton traveling wave and solitary wave interaction behavior are illustrated graphically through numerical simulation for physical affirmation of the results please check whether the affiliations are correct

benjamin feir instability in nonlinear dispersive waves in this paper the authors extended the derivation to the nonlinear schrodinger equation in two dimensions modified by the effect of non uniformity the authors derived several classes of soliton solutions in 2 1 dimensions when the solution is assumed to depend on space and time only through a single argument of the function they showed that the two dimensional nonlinear schrodinger equation is reduced either to the sine gordon for the hyperbolic case or sinh gordon equations for the elliptic case moreover the authors extended this method to obtain analytical solutions to the nonlinear schrodinger equation in two space dimensions plus time this contains some interesting solutions such as the plane solitons the n multiple solitons the propagating breathers and quadratic solitons the authors displayed graphically the obtained solutions by using the software mathematica 5

the blow up boundary for nonlinear wave equations consider the cauchy problem for a nonlinear wave equation du f u in n space dimensions n x with x continuously difierentiable introduction consider the nonlinear wave equation 0 1 uu u t au f u for x 6 r i 0 with the initial data 0 2 u x 0 x ul x 0 g x for x g rn it is well known that if f u is nonnegative and superlinear then in general a solution cannot exist for all times furthermore if t is the supremum of all times s such that a classical solution exists for all 0 t s then sup w jc t oo if t t

the global existence in the cauchy problem of the maxwell chern simons higgs system in this article we prove the global existence of solution of the classical maxwell chern simons higgs equations in 2 1 dimensional minkowski space time in the lorentz gauge we also prove that the topological solution of the maxwell chern simons higgs system converges to that of maxwell higgs system as the chern simons constant goes to zero reproducing the classical result by moncrief j math phys 21 2291 1980 on the global existence of the maxwell klein gordon system in 2 1 dimension

existence and classification of characteristic points at blow up for a semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension we first show the existence of a blow up solution with a characteristic point then we consider an arbitrary blow up solution u x t the graph x mapsto t x of its blow up points and scr s subset bbb r the set of all characteristic points and show that scr s has an empty interior finally given x 0 in scr s we show that in selfsimilar variables the solution decomposes into a decoupled sum of at least two solitons with alternate signs and that t x forms a corner of angle pi over 2 at x 0

the global existence of yang mills higgs fields in 4 dimensional minkowski space in this paper we complete the proof of global existence of yang mills higgs fields in 4 dimensional minkowski space by showing that an appropriate norm of the solutions cannot blow up in a finite time a key step in the proof is the demonstration that thel norm of the curvature is boundeda priori our results apply to any compact guage group and to any invariant higgs self coupling which is positive and of no higher than quartic degree

water wave solutions of zufiria s higher order boussinesq type equations and its stability the weakly nonlinear hamiltonian theory for two dimensional irrotational shallow water waves is formulated zufiria s higher order boussinesq model for waves on water of finite depth is obtained the formulation zufiria s higher order boussinesq type equations are studied by transforming them into solvable ordinary differential equations by implementing different types of extended direct algebraic method we show that in fact this model has new exact solutions an extended direct algebraic methods are applied to construct solitary wave solutions

arbitrarily large residual finiteness growth the residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients in this paper we construct groups with arbitrarily large residual finiteness growth we also demonstrate a new relationship between residual finiteness growth and some decision problems in groups which we apply to our new groups let be a residually finite group generated by a finite set x an element is detected by a finite quotient q of if there exists a homomorphism q such that ker set f x n to be the minimal natural number n such that any element of word length n with respect to x can be detected in a quotient q of cardinality n the study of f x the residual finiteness growth is an area of active research see for instance b10 b11 bm1 bk10 and km11 our main result is the following theorem 1 for any function f n n there exists a residually finite group bf and a two element generating set x for bf such that fbf x n f n for all n 8 our proof is constructive elementary and demonstrates the versatility of b h neumann s construction of a continuous family of non isomorphic two generated groups n37 our work is of the same flavor as lps96 and p04 where neumann s construction is modified to find groups with interesting subgroup growth properties our remaining results exhibit how residual finiteness growth may be applied to illicit decision problem data on groups the first of these shows that a variant of theorem 1 does not exist for groups that are finitely presented in particular the main result on residual finiteness growth stated in kms13 is the strongest of its kind for finitely presented groups proposition 1 if is a residually finite finitely presented group then the residual finiteness growth function for is computable o kharlampovich showed that there exists a finitely presented solvable group with undecidable word problem k81 of which many variations have been found see b83 in ls77 iv 4 3 theorem 19 it is shown that f2 f2 does not have decidable membership problem with respect to finitely generated subgroups we address a weaker membership problem for the groups bf for a finitely generated group g we say the finite index normal subgroup membership problem in g is solvable if there exists an algorithm which for every element g g and every finiteindex normal subgroup h g will determine in in a finite amount of time whether g h 1991 mathematics subject classification 20e26

incomplete blowup of solutions of quasilinear hyperbolic balance laws we study incomplete blowup of entropy solutions to first order quasilinear hyperbolic balance laws a general procedure to continue solutions beyond the blowup time which makes use of monotonicity methods is given the continuations thus obtained are possibly unbounded and satisfy suitable generalized entropy and rankine hugoniot conditions then the uniqueness of continuations satisfying such conditions is proved

blow up theorem for semilinear wave equations with non zero initial position abstract one of the features of solutions of semilinear wave equations can be found in blow up results for non compactly supported data in spite of finite propagation speed of the linear wave we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak this was first observed by asakura 1986 2 finding out a critical decay to ensure the global existence of the solution but the blow up result is available only for zero initial position having positive speed in this paper the blow up theorem for non zero initial position by uesaka 2009 22 is extended to higher dimensional case and the assumption on the nonlinear term is relaxed to include an example u p 1 u moreover the critical decay of the initial position is clarified by example

the blow up surface for nonlinear wave equations with small spatial velocity on considere le probleme de cauchy pour u tt e 2 u f u a d dimensions d espace 3 ou f u est superlineaire et non negative la solution explose sur une surface t e x on note par t x la surface d explosion correspondant a v f v on demontre que e x x ce 2 e x x ce 2 dans un voisinage d un point x 0 ou x 0

asymptotic properties of the solutions of linear and nonlinear spin field equations in minkowski space in this paper i will first derive based on energy estimations and geometric invariance the asymptotic behavior of solutions of linear spin field equations in minkowski space it generalizes the result in 3 where it was proved for the spin 1 and spin 2 cases the techniques are then applied to yang mills equations the result improves the previous one in 1 by allowing the initial data to have charge dipole and quadrupole moments the lie derivative operator for spinors and some properties will be also discussed they can be used to simplify some algebraic calculations of 4

a nonlinear nonlocal wave equation arising in combustion theory the initial value problem for the equation 2 t2 2 x2 t t 2 t 2 x2 et 1 is considered it is proved that under some restrictions on the initial data there is a curve denoted by t x which is positive lipschitz continuous and satisfies x 0 the case of 1 is also studied the solution for 1 blows up on t x and it is proved that under certain conditions the solutions for 1 converge to the one for 1 as 1 and lim inf 1 x x

blow up of the su 2 c yang mills fields it is remarked that if the gauge group is not compact finite time blowing up yang mills fields over the minkowski spaces may exist the example taken is g su 2 c it is shown that blow up can occur in an arbitrarily small time interval when the initial data of the fields are suitably chosen

kato type estimates for nls equation in a scalar field and unique solvability of nkgs system in energy space abstract we consider schrodinger equation in r 2 1 with nonlinear scalar potential the potentials are time independent or determined as solutions to inhomogeneous wave equations by constructing a modified propagator we derive kato type smoothing estimates for the nonlinear schrodinger nls equation with the help of these results we prove the unique solvability of the nonlinear klein gordon schrodinger nkgs system for all time in the energy space

a proof of the benjamin feir instability the existence and linear stability problem for the stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal hamiltonian structure of the water wave problem a proof within the hamiltonian framework of instability of the stokes periodic wavetrain is presented a hamiltonian center manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on 4 a projection of the reduced stability problem onto the tangent space of the 2 manifold of periodic stokes waves is used to prove the existence of a dispersion relation i1 i2 0 where e is the stability exponent for the stokes wave with amplitude i1 and mass flux i2 and is the sideband or spatial exponent a rigorous analysis of the dispersion relation proves the result first discovered in the 1960 s that the stokes gravity wavetrain of sufficiently small amplitude is unstable for f e 0 f0 where f0 0 8 and f is the froude number

determination of the curvature of the blow up set and refined singular behavior for a semilinear heat equation we consider u x t a solution of ut u u p 1u which blows up at some time t 0 where u rn 0 t r p 1 and n 2 p n 2 under a nondegeneracy condition we show that the mere hypothesis that the blow up set s is continuous and n 1 dimensional implies that it is c2 in particular we compute the n 1 principal curvatures and directions of s moreover a much more refined blow up behavior is derived for the solution in terms of the newly exhibited geometric objects refined regularity for s and refined singular behavior of u near s are linked through a new mechanism of algebraic cancellations that we explain in detail

determination of the blow up rate for a critical semilinear wave equation in this paper we determine the blow up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance

the blow up rate for a non scaling invariant semilinear wave equations abstract we consider the semilinear wave equation 1 t 2 u u f u x t r n 0 t with f u u p 1 u log a 2 u 2 where p 1 and a r we show an upper bound for any blow up solution of 1 then in the one space dimensional case using this estimate and the logarithmic property we prove that the exact blow up rate of any singular solution of 1 is given by the ode solution associated with 1 namely u u p 1 u log a 2 u 2 unlike the pure power case g u u p 1 u the difficulties here are due to the fact that equation 1 is not scale invariant

the cauchy problem for the coupled maxwell schrodinger equations the cauchy problem for the coupled maxwell schrodinger equations in rd in the lorentz gauge is considered the viscosity method is used to establish local existence in one and two space dimensions global solutions are obtained

local and global solutions of chern simons gauged o 3 sigma equations in one space dimension we study an initial value problem of the chern simons gauged o 3 sigma model in one space dimension the global existence of solutions to the model is proved for high regularity initial data moreover we study low regularity local well posedness observing null forms of the system and applying bilinear estimates for wave sobolev space hs as a by product a finite energy global solution is constructed

some global solutions of the yang mills equations in minkowski space a class of global solutions of the yang mills equations whose cauchy data depend on a pair of arbitrary functions is constructed the asymptotic propagation of the energy in space time is studied the same results are valid if the yang mills field is coupled to a scalar field

quantifying residual finiteness abstract we introduce the notion of quantifying the extent to which a finitely generated group is residually finite we investigate this behavior for examples that include free groups the first grigorchuk group finitely generated nilpotent groups and certain arithmetic groups such as sl n z in the context of finite nilpotent quotients we find a new characterization of finitely generated nilpotent groups

existence and universality of the blow up profile for the semilinear wave equation in one space dimension in this paper we consider the semilinear wave equation with a power nonlinearity in one space dimension we exhibit a universal one parameter family of functions which stand for the blow up profile in self similar variables at a non characteristic point for general initial data the proof is done in self similar variables we first characterize all the solutions of the associated stationary problem as a one parameter family then we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm in the non characteristic case and to a finite decoupled sum of such a solution in the characteristic case finally in the case where this sum is reduced to one element which is the case for noncharacteristic points we use modulation theory coupled with a nonlinear argument to show the exponential convergence in the self similar time variable of the various parameters and conclude the proof this step provides us with a result of independent interest the trapping of the solution in self similar variables near the set of stationary solutions valid also for non characteristic points the proof of these results is based on a new analysis in the self similar variable 2007 elsevier inc all rights reserved

stable blowup for the cubic wave equation in higher dimensions abstract we consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data we prove that there exists an open set of initial data such that the corresponding solution exists in a backward light cone and approaches the ode blowup profile

local and global solutions of the chern simons higgs system abstract we study low regularity solutions of the chern simons higgs equations the lorentz gauge condition makes them hyperbolic equations with the null form under the coulomb gauge condition they are formulated in the hyperbolic equation coupled with elliptic equation the div curl decomposition is used in the temporal gauge

yang mills and dirac fields in a bag existence and uniqueness theorems the cauchy problem for the yang mills dirac system with minimal coupling is studied under the mit quark bag boundary conditions an existence and uniqueness theorem for the free dirac equation is proven under that boundary condition the existence and uniqueness of the classical time evolution of the yang mills dirac system in a bag is shown to ensure sufficient differentiability of the fields we need additional boundary conditions in the proof we use the hodge decomposition of yang mills fields and the theory of non linear semigroups