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

an asymptotic property of the camassa holm equation abstract asymptotic densities are used to study an asymptotic property of the dispersionless camassa holm equation an asymptotic density of a global solution is a weak limit of its scaled momentum density along a sequence of time increasing to infinity for a global solution with non negative compactly supported initial momentum density we show that if the asymptotic density is unique then it is a positive combination of dirac measures supported in a bounded interval in the non negative axis with zero as the only possible accumulation point in other words if the scaled momentum density does not oscillate as time goes to infinity the solution behaves as a combination of peakons moving to the right at different speeds in contrast to many investigations on the topic our approach is not spectral theoretic and hence is independent of the structure of the isospectral problem associated with the equation

long time behaviour for a class of low regularity solutions of the camassa holm equation in this paper we investigate the long time behaviour for a class of low regularity solutions of the camasssa holm equation given by the superposition of infinitely many interacting traveling waves with corners at their peaks

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

blow up solutions of the general b equationa in this short paper we consider the b equation a new precise blow up scenario is described and some new results are obtained for the blow up phenomena

variational derivation of a geophysical camassa holm type shallow water equation abstract we consider two dimensional one layer irrotational equatorial oceanic flows in the f plane approximation in the small amplitude shallow water regime we deduce by a variational method in the lagrangian formalism a camassa holm type equation

fredholm determinants and the camassa holm hierarchy the equation of camassa and holm 2 2 is an approximate description of long waves in shallow water

some new soliton like solutions and periodic wave solutions with loop or without loop to a generalized kdv equation abstract in this paper by using the integral bifurcation method we study a generalized kdv equation which was first derived by fokas from physical considerations via a methodology of fuchssteiner all kinds of soliton like or kink like wave solutions and periodic wave solutions with loop or without loop are obtained smooth compacton like periodic wave solution and non smooth periodic cusp wave solution are also obtained their dynamic properties are investigated and their profiles are given by mathematical software

global weak solutions for a two component camassa holm shallow water system abstract in this paper we prove the existence of global weak solution for an integrable two component camassa holm shallow water system provided the initial data satisfying some certain conditions

a novel generalized dct based jnd profile based on an elaborate cm jnd model for variable block sized transforms in monochrome images in this paper we propose a new dct based just noticeable difference jnd profile incorporating the spatial contrast sensitivity function the luminance adaptation effect and the contrast masking cm effect the proposed jnd profile overcomes two limitations of conventional jnd profiles 1 the cm jnd models in the conventional jnd profiles employed simple texture complexity metrics which are not often highly correlated with perceived complexity especially for unstructured patterns so we proposed a new texture complexity metric that considers not only contrast intensity but also structureness of image patterns called the structural contrast index we also newly found out that as the structural contrast index of a background texture pattern increases the modulation factors for cm jnd show a bandpass property in frequency based on this observation a new cm jnd is modeled as a function of dct frequency and the proposed structural contrast index showing significantly high correlations with measured cm jnd values and 2 while the conventional dct based jnd profiles are only applicable for specific transform block sizes our proposed dct based jnd profile is first designed to be applicable to any size of transform by deriving a new summation effect function which can also be appropriately applied for quad tree transform of high efficiency video coding for the overall performance the proposed dct based jnd profile shows more tolerance for distortions with better perceptual quality than other jnd profiles under comparison

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

dct qm a dct based quality degradation metric for image quality optimization problems recent development of computational image quality assessment methods has shown to give very promising results in measuring perceptual visual quality for distorted images however most of them are difficult to be applied for optimization problems due to the lack of desirable mathematical properties such as differentiability convexity and valid distance metricability this paper proposes a novel discrete cosine transform dct based quality degradation metric called dct qm which is based on the probability summation theory with a psychometric function for neural responses in the receptive fields of visual cortex in psychophysics consequently the dct qm is formulated as a weighted mean text l 2 norm in the dct domain which is very easy to implement and inherits the three desirable mathematical properties that is differentiability convexity and valid distance metricability for image quality optimization problems the extensive experimental results show that the proposed dct qm has promising results for many practical distortion types in image processing problems by showing high consistency with perceived visual quality

a new weighted metric the relative metric ii in the first part of this investigation we generalized a weighted distance function of r c li s and found necessary and sufficient conditions for it being a metric in this paper some properties of this so called m relative metric are established specifically isometries and quasiconvexity results are derived we also illustrate connections between our approach and generalizations of the hyperbolic metric

vsnr a wavelet based visual signal to noise ratio for natural images this paper presents an efficient metric for quantifying the visual fidelity of natural images based on near threshold and suprathreshold properties of human vision the proposed metric the visual signal to noise ratio vsnr operates via a two stage approach in the first stage contrast thresholds for detection of distortions in the presence of natural images are computed via wavelet based models of visual masking and visual summation in order to determine whether the distortions in the distorted image are visible if the distortions are below the threshold of detection the distorted image is deemed to be of perfect visual fidelity vsnr infin and no further analysis is required if the distortions are suprathreshold a second stage is applied which operates based on the low level visual property of perceived contrast and the mid level visual property of global precedence these two properties are modeled as euclidean distances in distortion contrast space of a multiscale wavelet decomposition and vsnr is computed based on a simple linear sum of these distances the proposed vsnr metric is generally competitive with current metrics of visual fidelity it is efficient both in terms of its low computational complexity and in terms of its low memory requirements and it operates based on physical luminances and visual angle rather than on digital pixel values and pixel based dimensions to accommodate different viewing conditions

holder continuity for a modified camassa holm system this paper is concerned with the holder continuity for a modified camassa holm system we show that the date to solution for the modified camassa holm system is holder continuous from bounded set of sobolev spaces with exponent s 3 2 measured in a weaker sobolev norm with index r s on the line our result also holds for the periodic case

blow up phenomena and global existence for a two component camassa holm system with an arbitrary smooth function abstract in the paper we mainly investigate blow up phenomena and global existence of strong solutions to a two component camassa holm systems with an arbitrary smooth function h for three types of smooth functions h by using a conservation law and the sign preserving property of strong solutions we obtain two new blow up results and a new global existence result for the two component system our obtained results generalize and cover the recent results in yan et al 2015 zhang and yin 2015 2016

local well posedness and blow up phenomena of the generalized short pulse equation in this paper we investigate the cauchy problem of the generalized short pulse equation in periodic domain we first establish the local well posedness of the equation in hs s s 2 by taking advantage of the kato method and present some useful conservation laws then we obtain the boundedness of the solution by virtue of the conservation laws finally we give a precise blow up criterion for 0 or 0 and get several blow up results provided the initial data satisfies some certain conditions

the integral bifurcation method and its application for solving a family of third order dispersive pdes abstract in this paper an improved method named the integral bifurcation method is introduced in order to demonstrate its effectiveness for obtaining travelling wave solutions of the nonlinear wave equations a family of third order dispersive partial differential equations which were given by a degasperis d holm and a hone are studied many integral bifurcations are obtained for different parameter conditions by using these integral bifurcations many travelling wave solutions such as loop soliton solutions solitary wave solutions cusp soliton solutions and periodic wave solutions are obtained in particular under the conditions c 1 0 c 2 c 3 1 a very peculiar periodic wave solution is obtained

local well posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude abstract we study the local well posedness of a periodic nonlinear equation for surface waves of moderate amplitude in shallow water we use an approach proposed by kato which is based on semigroup theory for quasi linear equations we also show that singularities for the model equation can occur only in the form of wave breaking in particular surging breakers

a note on the camassa holm equation abstract we consider a generalized camassa holm equation we improve the compactness argument of 5 6 7 17 establishing the existence of global weak solutions in h 1 without the use of an ole nik type estimate the potential interest of this improvement is linked to convergence analysis of numerical schemes for which it may not be easy to verify an ole nik type estimate

color invariance this paper presents the measurement of colored object reflectance under different general assumptions regarding the imaging conditions we exploit the gaussian scale space paradigm for color images to define a framework for the robust measurement of object reflectance from color images object reflectance is derived from a physical reflectance model based on the kubelka munk theory for colorant layers illumination and geometrical invariant properties are derived from the reflectance model invariance and discriminative power of the color invariants is experimentally investigated showing the invariants to be successful in discounting shadow illumination highlights and noise extensive experiments show the different invariants to be highly discriminative while maintaining invariance properties the presented framework for color measurement is well founded in the physics of color as well as in measurement science hence the proposed invariants are considered more adequate for the measurement of invariant color features than existing methods

well posedness and blow up phenomena for a higher order shallow water equation this paper deals with the cauchy problem for a higher order shallow water equation yt auxy buyx 0 where y 2ku i x2 ku and k 2 the local well posedness of solutions for the cauchy problem in sobolev space hs r with s 7 2 is obtained under some assumptions the existence and uniqueness of the global solutions to the equation are shown and conditions that lead to the development of singularities in finite time for the solutions are also acquired finally the weak solution for the equation is considered

well posedness and blow up solution for a new coupled camassa holm equations with peakons a new camassa holm system with two component admitting peakon solitons is proposed the local well posedness for the system is established a criterion and a condition on the initial data guaranteeing the development of singularities in finite time for strong solutions are obtained and an existence result for a class of local weak solution is also given

local in space blow up criteria for a class of nonlinear dispersive wave equations abstract this paper is concerned with blow up phenomena for the nonlinear dispersive wave equation on the real line u t u x x t f u x f u x x x g u f u 2 u x 2 x 0 that includes the camassa holm equation as well as the hyperelastic rod wave equation f u k u 2 2 and g u 3 k u 2 2 as special cases we establish some a local in space blow up criterion i e a criterion involving only the properties of the data u 0 in a neighborhood of a single point simplifying and precising earlier blow up criteria for this equation

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

on some wave breaking for the nonlinear integrable shallow water wave equations mathematicians in this paper based on the conservation laws and the blow up scenario some new blow up phenomena are derived for the camassa holm equation and degasperis procesi equation which have attracted much attention due to their structure c 2015 elsevier ltd all rights reserved

on the cauchy problem for a two component b family system with high order nonlinearity abstract this paper is devoted to the cauchy problem of two component b family system with high order nonlinearity which includes the well known camassa holm equation degasperis procesi equation novikov equation and their two component extension systems as special cases firstly we establish the local well posedness of the system in the non homogeneous besov space by using the littlewood paley decomposition and the transport equation theory secondly we investigate the blow up of strong solutions to the system thirdly we study the gevrey regularity and analyticity of the solutions to the system in the gevrey sobolev spaces by using the generalized ovsyannikov theorem moreover we find a lower bound of the lifespan and prove the continuity of the data to solution mapping

on the mathematical properties of the structural similarity index since its introduction in 2004 the structural similarity ssim index has gained widespread popularity as a tool to assess the quality of images and to evaluate the performance of image processing algorithms and systems there has been also a growing interest of using ssim as an objective function in optimization problems in a variety of image processing applications one major issue that could strongly impede the progress of such efforts is the lack of understanding of the mathematical properties of the ssim measure for example some highly desirable properties such as convexity and triangular inequality that are possessed by the mean squared error may not hold in this paper we first construct a series of normalized and generalized vector valued metrics based on the important ingredients of ssim we then show that such modified measures are valid distance metrics and have many useful properties among which the most significant ones include quasi convexity a region of convexity around the minimizer and distance preservation under orthogonal or unitary transformations the groundwork laid here extends the potentials of ssim in both theoretical development and practical applications

the local well posedness existence and uniqueness of weak solutions for a model equation for shallow water waves of moderate amplitude abstract this paper deals with the cauchy problem for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime which was proposed by constantin and lannes 2009 20 applying the pseudoparabolic regularization technique the local well posedness of strong solutions in sobolev space h s r with s 3 2 is established via a limiting procedure moreover a sufficient condition for the existence of weak solutions of the equation in lower order sobolev space h s r with 1 s 3 2 is obtained

an operator splitting method for the degasperis procesi equation an operator splitting method is proposed for the degasperis procesi dp equation by which the dp equation is decomposed into the burgers equation and the benjamin bona mahony bbm equation then a second order tvd scheme is applied for the burgers equation and a linearized implicit finite difference method is used for the bbm equation furthermore the strang splitting approach is used to construct the solution in one time step the numerical solutions of the dp equation agree with exact solutions e g the multipeakon solutions very well the proposed method also captures the formation and propagation of shockpeakon solutions and reveals wave breaking phenomena with good accuracy

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

the cauchy problem for a generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem of the generalized camassa holm equation using a galerkin type approximation scheme it is shown that this equation is well posed in sobolev spaces h s s 3 2 for both the periodic and the nonperiodic case in the sense of hadamard that is the data to solution map is continuous furthermore it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous the nonuniform dependence is proved using the method of approximate solutions and well posedness estimates moreover it is shown that the solution map for the generalized camassa holm equation is holder continuous in h r topology finally with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time

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

on permanent and breaking waves in hyperelastic rods and rings we prove that the only global strong solution of the periodic rod equation vanishing in at least one point t0 x0 r s1 is the identically zero solution such conclusion holds provided the physical parameter of the model related to the finger deformation tensor is outside some neighborhood of the origin and applies in particular for the camassa holm equation corresponding to 1 we also establish the analogue of this unique continuation result in the case of non periodic solutions defined on the whole real line with vanishing boundary conditions at infinity our analysis relies on the application of new local in space blowup criteria and involves the computation of several best constants in convolution estimates and weighted poincare inequalities

a novel image quality assessment with globally and locally consilient visual quality perception computational models for image quality assessment iqa have been developed by exploring effective features that are consistent with the characteristics of a human visual system hvs for visual quality perception in this paper we first reveal that many existing features used in computational iqa methods can hardly characterize visual quality perception for local image characteristics and various distortion types to solve this problem we propose a new iqa method called the structural contrast quality index sc qi by adopting a structural contrast index sci which can well characterize local and global visual quality perceptions for various image characteristics with structural distortion types in addition to sci we devise some other perceptually important features for our sc qi that can effectively reflect the characteristics of hvs for contrast sensitivity and chrominance component variation furthermore we develop a modified sc qi called structural contrast distortion metric sc dm which inherits desirable mathematical properties of valid distance metricability and quasi convexity so it can effectively be used as a distance metric for image quality optimization problems extensive experimental results show that both sc qi and sc dm can very well characterize the hvs s properties of visual quality perception for local image characteristics and various distortion types which is a distinctive merit of our methods compared with other iqa methods as a result both sc qi and sc dm have better performances with a strong consilience of global and local visual quality perception as well as with much lower computation complexity compared with the state of the art iqa methods the matlab source codes of the proposed sc qi and sc dm are publicly available online at https sites google com site sunghobaecv iqa

exact controllability of scalar conservation laws with an additional control in the context of entropy solutions in this paper we study the exact controllability problem for nonlinear scalar conservation laws on a compact interval with a regular convex flux and in the framework of entropy solutions with the boundary data and a source term depending only on the time as controls we provide sufficient conditions for a state to be reachable in arbitrary small time to do so we introduce a slightly modified wave front tracking algorithm

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

a convergent finite difference scheme for the camassa holm equation with general h 1 initial data we suggest a finite dfference scheme for the camassa holm equation that can handle general h 1 initial data the form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality we prove that the difference scheme converges strongly in h 1 towards a dissipative weak solution of the camassa holm equation

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

global conservative solutions of the generalized hyperelastic rod wave equation abstract we prove existence of global and conservative solutions of the cauchy problem for the nonlinear partial differential equation u t u x x t f u x f u x x x g u 1 2 f u u x 2 x 0 where f is strictly convex or concave and g is locally uniformly lipschitz this includes the camassa holm equation f u u 2 2 and g u u u 2 as well as the hyperelastic rod wave equation f u u 2 2 and g u 3 u 2 2 as special cases it is shown that the problem is well posed for initial data in h 1 r if one includes a radon measure that corresponds to the energy of the system with the initial data the solution is energy preserving stability is proved both with respect to initial data and the functions f and g the proof uses an equivalent reformulation of the equation in terms of lagrangian coordinates

well posedness and analytic solutions of a two component water wave equation abstract we first establish the local well posedness for the cauchy problem of a two component water waves system in a critical nonhomogeneous besov space then we derive a continuation criterion for strong solutions to the system finally we prove the existence of analytic solutions to the system

on traveling wave solutions of the θ equation of dispersive type abstract traveling wave solutions to a class of dispersive models u t u t x x u u x u u x x x 1 u x u x x are investigated in terms of the parameter including two integrable equations the camassa holm equation 1 3 and the degasperis procesi equation 1 4 as special models it was proved in h liu and z yin 2011 39 that when 1 2 1 smooth solutions persist for all time and when 0 1 2 strong solutions of the equation may blow up in finite time yielding rich traveling wave patterns this work therefore restricts to only the range 0 1 2 it is shown that when 0 only periodic travel wave is permissible and when 1 2 traveling waves may be solitary periodic or kink like waves for 0 1 2 traveling waves such as periodic solitary peakon peaked periodic cusped periodic or cusped soliton are all permissible

conservative weak solutions of the periodic cauchy problem for μhs equation we prove the existence and uniqueness of conservative weak solutions of the periodic cauchy problem for an integrable evolution equation from mathematical physics our method is to first prove the existence and uniqueness of solutions for all time in lagrangian coordinates and construct the maps to carry this result to the eulerian coordinates

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

image quality assessment based on a degradation model we model a degraded image as an original image that has been subject to linear frequency distortion and additive noise injection since the psychovisual effects of frequency distortion and noise injection are independent we decouple these two sources of degradation and measure their effect on the human visual system we develop a distortion measure dm of the effect of frequency distortion and a noise quality measure nqm of the effect of additive noise the nqm which is based on peli s 1990 contrast pyramid takes into account the following 1 variation in contrast sensitivity with distance image dimensions and spatial frequency 2 variation in the local luminance mean 3 contrast interaction between spatial frequencies 4 contrast masking effects for additive noise we demonstrate that the nonlinear nqm is a better measure of visual quality than peak signal to noise ratio psnr and linear quality measures we compute the dm in three steps first we find the frequency distortion in the degraded image second we compute the deviation of this frequency distortion from an allpass response of unity gain no distortion finally we weight the deviation by a model of the frequency response of the human visual system and integrate over the visible frequencies we demonstrate how to decouple distortion and additive noise degradation in a practical image restoration system

asymptotic variational wave equations we investigate the equation ut f u x x f u ux 2 2 where f u is a given smooth function typically f u u 2 2 or u 3 3 this equation models unidirectional and weakly nonlinear waves for the variational wave equation utt c u c u ux x 0 which models some liquid crystals with a natural sinusoidal c the equation itself is also the euler lagrange equation of a variational problem two natural classes of solutions can be associated with this equation a conservative solution will preserve its energy in time while a dissipative weak solution loses energy at the time when singularities appear conservative solutions are globally defined forward and backward in time and preserve interesting geometric features such as the hamiltonian structure on the other hand dissipative solutions appear to be more natural from the physical point of view we establish the well posedness of the cauchy problem within the class of conservative solutions for initial data having finite energy and assuming that the flux function f has lipschitz continuous second order derivative in the case where f is convex the cauchy problem is well posed also within the class of dissipative solutions however when f is not convex we show that the dissipative solutions do not depend continuously on the initial data

non uniform continuity of periodic holm staley b family of equations abstract we consider a family of non evolutionary partial differential equations known as the holm staley b family which includes the integrable camassa holm and degasperis procesi equations we show that the solution map is not uniformly continuous the proof relies on a construction of smooth periodic traveling waves with small amplitude

the periodic b equation and euler equations on the circle in this note we show that the periodic b equation can only be realized as a euler equation on the lie group diff s1 of all smooth and orientation preserving diffeomorphisms on the circle if b 2 i e for the camassa holm equation in this case the inertia operator generating the metric on diff s1 is given by a 1 x2 in contrast the degasperis procesi equation for which b 3 is not a euler equation on diff s1 for any inertia operator our result generalizes a recent result of kolev some geometric investigations on the degasperis procesi shallow water equation wave motion 46 412 419 2009

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

blow up zero α limit and the liouville type theorem for the euler poincare equations in this paper we study the euler poincare equations in mathbb r n we prove local existence of weak solutions in w 2 p mathbb r n p n and local existence of unique classical solutions in h k mathbb r n k n 2 3 as well as a blow up criterion for the zero dispersion equation 0 we prove a finite time blow up of the classical solution we also prove that as the dispersion parameter vanishes the weak solution converges to a solution of the zero dispersion equation with sharp rate as 0 provided that the limiting solution belongs to c 0 t h k mathbb r n with k n 2 3 for the stationary weak solutions of the euler poincare equations we prove a liouville type theorem namely for 0 any weak solution mathbf u in h 1 mathbb r n is u 0 for 0 any weak solution mathbf u in l 2 mathbb r n is u 0