stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

the local well posedness and global solution for a modified periodic two component camassa holm system abstract the local well posedness and global solution for a modified periodic two component camassa holm system on the circle s are established in the sobolev space h s h s 2 with s 7 2 which are different from that of the two component camassa holm system

the stability in _ and _ of the projection onto finite element function spaces on etablit la stabilite de la l 2 projection sur quelques espaces a elements finis v h consideree comme une application dans l p et w p 1 1 p sous des hypotheses de regularite faibles

large deviations for random walks on galton watson trees averaging and uncertainty in the study of large deviations for random walks in random environment a key distinction has emerged between quenched asymptotics conditional on the environment and annealed asymptotics obtained from averaging over environments in this paper we consider a simple random walk x n on a galton watson tree t i e on the family tree arising from a supercritical branching process denote by x n the distance between the node x n and the root of t our main result is the almost sure equality of the large deviation rate function for x n n under the quenched measure conditional upon t and the rate function for the same ratio under the annealed measure averaging on t according to the galton watson distribution this equality hinges on a concentration of measure phenomenon for the momentum of the walk the momentum at level n for a specific tree t is the average over random walk paths of the forward drift at the hitting point of that level this concentration or certainty is a consequence of the uncertainty in the location of the hitting point we also obtain similar results when x n is a biased walk on a galton watson tree even though in that case there is no known formula for the asymptotic speed our arguments rely at several points on a ubiquity lemma for galton watson trees due to grimmett and kesten 1984

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

on the design of general purpose flux limiters for finite element schemes i scalar convection the algebraic flux correction afc paradigm is extended to finite element discretizations with a consistent mass matrix it is shown how to render an implicit galerkin scheme positivity preserving and remove excessive artificial diffusion in regions where the solution is sufficiently smooth to this end the original discrete operators are modified in a mass conserving fashion so as to enforce the algebraic constraints to be satisfied by the numerical solution a node oriented limiting strategy is employed to control the raw antidiffusive fluxes which consist of a convective part and a contribution of the consistent mass matrix the former offsets the artificial diffusion due to upwinding of the spatial differential operator and lends itself to an upwind biased flux limiting the latter eliminates the error induced by mass lumping and calls for the use of a symmetric flux limiter the concept of a target flux and a new definition of upper lower bounds make it possible to combine the advantages of algebraic fct and tvd schemes introduced previously by the author and his coworkers unlike other high resolution schemes for unstructured meshes the new algorithm reduces to a consistent high order galerkin scheme in smooth regions and is designed to provide an optimal treatment of both stationary and time dependent problems its performance is illustrated by application to the linear advection equation for a number of 1d and 2d configurations

quenched annealed and functional large deviations for one dimensional random walk in random environment suppose that the integers are assigned random variables i taking values in the unit interval which serve as an environment this environment defines a random walk x n called a rwre which when at i moves one step to the right with probability i and one step to the left with probability 1 i when the i sequence is i i d greven and den hollander 1994 proved a large deviation principle for x n n conditional upon the environment with deterministic rate function we consider in this paper large deviations both conditioned on the environment quenched and averaged on the environment annealed for the rwre in the ergodic environment case the annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy we also give a detailed qualitative description of the resulting rate functions our techniques differ from those of greven and den hollander and allow us to present also a trajectorial quenched large deviation principle

coincidence of lyapunov exponents for random walks in weak random potentials we investigate the free energy of nearest neighbor random walks on z d endowed with a drift along the first axis and evolving in a nonnegative random potential given by i i d random variables our main result concerns the ballistic regime in dimensions d 4 at which we show that quenched and annealed lyapunov exponents are equal as soon as the strength of the potential is small enough

blow up and global solutions to a new integrable model with two components abstract we will discuss a new integrable model which describes the motion of fluid the present work is mainly concerned with global existence and blow up phenomena which are largely due to the application of conservation laws for this integrable equations moreover a new blow up criterion for nonperiodic case is also established via the associated potential some interesting examples are also given to illustrate the application of our results the precise blow up rate is also investigated finally we will emphasize the relations of classical camassa holm equation and our model by analyzing the existence of global solutions

persistence properties and unique continuation of solutions of the camassa holm equation it is shown that a strong solution of the camassa holm equation initially decaying exponentially together with its spacial derivative must be identically equal to zero if it also decays exponentially at a later time in particular a strong solution of the cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

convergent finite element discretization of the multi fluid nonstationary incompressible magnetohydrodynamics equations we propose a convergent implicit stabilized finite element discretization of the nonstationary incompressible magnetohydrodynamics equations with variable density viscosity and electric conductivity the discretization satisfies a discrete energy law and a discrete maximum principle for the positive density and iterates converge to weak solutions of the limiting problem for vanishing discretization parameters a simple fixed point scheme together with an appropriate stopping criterion is proposed which decouples the computation of density velocity and magnetic field and inherits the above properties provided a mild mesh constraint holds computational studies are provided

a stabilized finite element method for the incompressible magnetohydrodynamic equations we propose and analyze a stabilized finite element method for the incompressible magnetohydrodynamic equations the numerical results that we present show a good behavior of our approximation in experiments which are relevant from an industrial viewpoint we explain in particular in the proof of our convergence theorem why it may be interesting to stabilize the magnetic equation as soon as the hydrodynamic diffusion is small and even if the magnetic diffusion is large this observation is confirmed by our numerical tests

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

on the global existence and wave breaking criteria for the two component camassa holm system considered herein is a two component camassa holm system modeling shallow water waves moving over a linear shear flow a wave breaking criterion for strong solutions is determined in the lowest sobolev space hs s 32 by using the localization analysis in the transport equation theory moreover an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this sobolev space hs

large deviations for random walk in a space time product environment we consider random walk x n n 0 on z d in a space time product environment a we take the point of view of the particle and focus on the environment markov chain t n xn n 0 where t denotes the shift on q conditioned on the particle having asymptotic mean velocity equal to any given we show that the empirical process of the environment markov chain converges to a stationary process under the averaged measure when d 3 and is sufficiently close to the typical velocity we prove that averaged and quenched large deviations are equivalent and when conditioned on the particle having asymptotic mean velocity the empirical process of the environment markov chain converges to under the quenched measure as well in this case we show that is a stationary markov process whose kernel is obtained from the original kernel by a doob h transform

convergence of the discontinuous galerkin method for discontinuous solutions we consider linear first order scalar equations of the form rho t rm div rho v a rho f with appropriate initial and boundary conditions it is shown that approximate solutions computed using the discontinuous galerkin method will converge in ltwoltwo when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions in particular v need not be lipschitz so characteristics of the equation may not be defined and the solutions being approximated my not have bounded variation

large deviations for random walks in a mixing random environment and other non markov random walks we extend a recent work by s r s varadhan 8 on large deviations for random walks in a product random environment to include more general random walks on the lattice in particular some reinforced random walks and several classes of random walks in gibbs fields are included c 2004 wiley periodicals inc

wave breaking for a periodic shallow water equation the focus of this paper is on the blow up of a recently derived one dimensional shallow water equation which is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equations in the shallow water regime some new criteria guaranteeing the development of singularities in finite time for strong solutions with regular initial data are obtained for the periodic case

mixed finite element methods for stationary incompressible magneto hydrodynamics a new mixed variational formulation of the equations of stationary incompressible magneto hydrodynamics is introduced and analyzed the formulation is based on curl conforming sobolev spaces for the magnetic variables and is shown to be well posed in possibly non convex lipschitz polyhedra a finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf sup stable velocity pressure space pairs and the magnetic ones by a mixed approach using nedelec s elements of the first kind an error analysis is carried out that shows that the proposed finite element approximation leads to quasi optimal error bounds in the mesh size

random walks in random environments random walks in random environments and their diffusion analogues have been a source of surprising phenomena and challenging problems especially in the non reversible situation since they began to be studied in the 1970s we review the model available results and techniques and point out several gaps in the understanding of these processes

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

a law of large numbers for random walks in random environment we derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of kalikow the approach is based on the existence of a renewal structure under an assumption of transience in the direction l this extends to a multidimensional context previous work of kesten our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution

acute type refinements of tetrahedral partitions of polyhedral domains we present a new technique to perform refinements on acute type tetrahedral partitions of a polyhedral domain provided that the center of the circumscribed sphere around each tetrahedron belongs to the tetrahedron the resulting family of partitions is of acute type thus all the tetrahedra satisfy the maximum angle condition both these properties are highly desirable in finite element analysis

uniqueness of conservative solutions to the modified two component camassa holm system via characteristics abstract in this paper we prove the uniqueness of conservative solutions to the modified two component camassa holm system by the analysis of characteristics

conservation laws of the camassa holm equation we use the bi hamiltonian structure of the camassa holm equation to show that its conservation laws h n m are homogeneous with respect to the scaling m lambdam moreover a direct argument is presented proving that h 1 h 2 are of local character finally simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the h n s

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

large deviations for random walks in a random environment we prove two large deviation principles for random walks in a random environment one for quenched walks in general ergodic environments and the other for averaged walks in a product environment 2003 wiley periodicals inc

blow up criteria of solutions for a modified two component hyperelastic rod system we consider a modified two component rod system which is a generalization of the two component camassa holm system to allow a dependence on the average density as well as on the pointwise density sufficient conditions on the initial data to guarantee finite time singularities for strong solutions are established on the real line and the periodic case

large deviations for a random walk in random environment let px x z be an i i d collection of 0 1 valued random variables given let x n n 0 be the markov chain on z defined by x 0 0 and x n 1 x n 1 resp x n n 1 with probability px n resp 1 px n it is shown that x n n satisfies a large deviation principle with a continuous rate function that is lim n 1 n log p x n n n i a s for n 1 1 first we derive a representation of the rate function i in terms of a variational problem second we solve the latter explicitly in terms of random continued fractions this leads to a classification and qualitative description of the shape of i in the recurrent case i is nonanalytic at 0 in the transient case i is nonanalytic at c 0 c for some c 0 with linear pieces in between

blowup criterion and persistent decay for a modified camassa holm system in this paper we study a modified two component camassa holm system modeling shallow water waves moving over a linear shear flow we demonstrate a condition on the initial data that lead to finite time blow up of solutions by moderate weight functions from time frequency analysis we provide persistence results for solutions of the system in weighted lp spaces for a large class of moderate weights

analysis of finite element approximation for time dependent maxwell problems we provide an error analysis of finite element methods for solving time dependent maxwell problem using nedelec and thomas raviart elements we study the regularity of the solution and develop some new error estimates of nedelec finite elements as a result the optimal l 2 error bound for the semidiscrete scheme is obtained

a level set approach to simulate magnetohydrodynamic instabilities in aluminum reduction cells magnetohydrodynamic instabilities at the metal bath interface in aluminum reduction cells is an important and not fully understood topic to simulate the two fluid three dimensional unstationary flow subject to a background magnetic field a level set approach is proposed it features a formulation in terms of the magnetic vector potential to avoid a numerical growth of the divergence of the magnetic field the same exact projection scheme with staggered grids is used for both the velocity field and the magnetic vector potential test simulations show that the overall method behaves well in purely hydrodynamic as well as in fully magnetohydrodynamic regimes in both cases with a single fluid and with two fluids we also simulate with our technique the metal pad roll instability and trace the behavior of coupled interracial modes

wave breaking for a modified two component camassa holm system abstract in this paper we establish sufficient conditions on the initial data to guarantee blow up phenomenon for the modified two component camassa holm mch2 system

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

on the existence uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics the authors consider the equations of stationary incompressible magneto hydrodynamics posed in a bounded domain in three dimensions and treat the full coupled system of equations with inhomogeneous boundary conditions under certain conditions on the data they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed they discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution

singular solutions of a modified two component camassa holm equation the camassa holm ch equation is a well known integrable equation describing the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of singular solutions peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in the dynamics although possessing peakon solutions in the velocity the ch2 equation does not admit singular solutions in the density profile we modify the ch2 system to allow a dependence on the average density as well as the pointwise density the modified ch2 system mch2 does admit peakon solutions in the velocity and average density we analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data numerical results for the mch2 system are given and compared with the pure ch2 case these numerics show that the modification in the mch2 system to introduce the average density has little short time effect on the emergent dynamical properties however an analytical and numerical study of pairwise peakon interactions for the mch2 system shows a different asymptotic feature namely besides the expected soliton scattering behavior seen in overtaking and head on peakon collisions the mch2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes

on a class of transient random walks in random environment we introduce in this article a class of transient random walks in a random environment on z d when d 2 these walks are ballistic and we derive a law of large numbers a central limit theorem and large deviation estimates in the so called nestling situation large deviations in the neighborhood of the segment 0 v v being the limiting velocity are critical they are of special interest in view of their close connection with the presence of traps in the medium that is pockets where a certain spectral parameter takes atypically low values

harmonic functions h transform and large deviations for random walks in random environments in dimensions four and higher we consider large deviations for nearest neighbor random walk in a uniformly elliptic i i d environment on d there exist variational formulae for the quenched and averaged rate functions i q and i a obtained by rosenbluth and varadhan respectively i q and i a are not identically equal however when d 4 and the walk satisfies the so called t condition of sznitman they have been previously shown to be equal on an open set a eq for every a eq we prove the existence of a positive solution to a laplace like equation involving and the original transition kernel of the walk we then use this solution to define a new transition kernel via the h transform technique of doob this new kernel corresponds to the unique minimizer of varadhan s variational formula at it also corresponds to the unique minimizer of rosenbluth s variational formula provided that the latter is slightly modified

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

convergence of numerical approximations of the incompressible navier stokes equations with variable density and viscosity we consider numerical approximations of incompressible newtonian fluids having variable possibly discontinuous density and viscosity since solutions of the equations with variable density and viscosity may not be unique numerical schemes may not converge if the solution is unique then approximate solutions computed using the discontinuous galerkin method to approximate the convection of the density and stable finite element approximations of the momentum equation converge to the solution if the solution is not unique a subsequence of these approximate solutions will converge to a solution

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

finite element methods with matching and nonmatching meshes for maxwell equations with discontinuous coefficients we investigate the finite element methods for solving time dependent maxwell equations with discontinuous coefficients in general three dimensional lipschitz polyhedral domains both matching and nonmatching finite element meshes on the interfaces are considered and optimal error estimates for both cases are obtained the analysis of the latter case is based on an abstract framework for nested saddle point problems along with a characterization of the trace space for h curl d a new extension theorem for h curl d functions in any lipschitz domain d and a novel compactness argument for deriving discrete inf sup conditions

quenched large deviations for random walk in a random environment we take the point of view of a particle performing random walk with bounded jumps on d in a stationary and ergodic random environment we prove the quenched large deviation principle ldp for the pair empirical measure of the so called environment markov chain by an appropriate contraction we deduce the quenched ldp for the mean velocity of the particle and obtain a variational formula for the corresponding rate function we propose an ansatz for the minimizer of this formula when d 1 we verify this ansatz and generalize the nearest neighbor result of comets gantert and zeitouni to walks with bounded jumps 2009 wiley periodicals inc

weighted regularization of maxwell equations in polyhedral domains we present a new method of regularizing time harmonic maxwell equations by a bf grad div term adapted to the geometry of the domain this method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions in the presence of reentrant corners or edges the usual regularization is known to produce wrong solutions due the non density of smooth fields in the variational space we get rid of this undesirable effect by the introduction of special weights inside the divergence integral standard finite elements can then be used for the approximation of the solution this method proves to be numerically efficient

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

wave breaking phenomenon for a modified two component dullin gottwald holm equation this paper is concerned with the wave breaking phenomenon for a modified two component dullin gottwald holm shallow water system we give sufficient conditions on the initial data to guarantee blow up of solutions in finite time

short note an efficient solver for the equations of resistive mhd with spatially varying resistivity we regularize the variable coefficient helmholtz equations arising from implicit time discretizations for resistive mhd in a way that leads to a symmetric positive definite system uniformly in the time step standard centered difference discretizations in space of the resulting pde leads to a method that is second order accurate and that can be used with multigrid iteration to obtain efficient solvers

the cauchy problem for a two component novikov equation in the critical besov space abstract in this paper we consider the cauchy problem for a two component novikov equation in the critical besov space b 2 1 5 2 we first derive a new a priori estimate for the 1 d transport equation in b 2 3 2 which is the endpoint case then we apply this a priori estimate and the osgood lemma to prove the local existence moreover we also show that the solution map u 0 u is holder continuous in b 2 1 5 2 equipped with weaker topology it is worth mentioning that our method is different from the previous one that involves extracting a convergent subsequence from an iterative sequence in critical besov spaces

equality of averaged and quenched large deviations for random walks in random environments in dimensions four and higher we consider large deviations for nearest neighbor random walk in a uniformly elliptic i i d environment it is easy to see that the quenched and the averaged rate functions are not identically equal when the dimension is at least four and sznitman s transience condition t is satisfied we prove that these rate functions are finite and equal on a closed set whose interior contains every nonzero velocity at which the rate functions vanish

lyapounov norms for random walks in low disorder and dimension greater than three we consider a simple random walk on z d d 3 we also consider a collection of i i d positive and bounded random variables left v omega x right x in z d which will serve as a random potential we study the annealed and quenched cost to perform long crossing in the random potential lambda beta v omega x where is positive constant and 0 is small enough these costs are measured by the lyapounov norms we prove the equality of the annealed and the quenched norm

lyapounov exponents and quenched large deviations for multidimensional random walk in random environment assign to the lattice sizes z z d i i d random 2d dimensional vectors z z e e 1 whose entries take values in the open unit interval and add up to one given a realization of this environment let x n n 0 be a markov chain on z d which when at z moves one step to its neighbor z e with transition probability z z e we derive a large deviation principle for x n n by means of a result similar to the shape theorem of first passage percolation and related models this result produces certain constants that are the analogue of the lyapounov exponents known from brownian motion in poissonian potential or random walk in random potential we follow a strategy similar to sznitman

on blow up of solutions to the two component π camassa holm system abstract we investigate cauchy problem of the two component camassa holm system which arises from geodesic flow on a semidirect product lie group of the circle the precise blow up scenarios of strong solutions are derived for the system then several criteria to guarantee blow up of strong solutions are presented finally the exact blow up rate is determined