note a note on the hardy hilbert inequality abstract in this note it is shown that the hardy hilbert inequality for double series can be improved by introducing a proper weight function of the form sin p o r n n 1 1 r with o r n 0 into either of the two single summations when r 2 the classical hilbert inequality is improved

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

the rigorous approximation of long wavelength capillary gravity waves in a previous paper we proved that long wavelength solutions of the water wave problem in the case of zero surface tension split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries kdv equation in this paper we examine the effect of surface tension on this scenario we find that we obtain three different physical regimes depending on the strength of the surface tension for weak surface tension the propagation of the wave packets is very similar to that in the zero surface tension case for strong surface tension the evolution is again governed by a pair of kdv equations but the coefficients in these equations have changed in such a way that the kdv soliton now represents a wave of depression on the fluid surface finally at a special intermediate value of the surface tension where the bond number equals the kdv description breaks down and it is necessary to introduce a new approximating equation the kawahara equation which is a fifth order nonlinear partial differential equation in each of these regimes we give rigorous estimates of the difference between the solution of the appropriate modulation equation and the solution of the true water wave problem

the validity of the kdv approximation in case of resonances arising from periodic media abstract it is the purpose of this short note to discuss some aspects of the validity question concerning the korteweg de vries kdv approximation for periodic media for a homogeneous model possessing the same resonance structure as it arises in periodic media we prove the validity of the kdv approximation with the help of energy estimates

on hardy s inequality abstract in this paper some new inequalities on the constant e are deduced as an application we build a strengthened hardy s inequality

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

phase dynamics in the complex ginzburg landau equation abstract for 1 stable time periodic solutions a x t aqeiqx i qt are the locally preferred planform for the complex ginzburg landau equation t a a 1 i x 2 a 1 i a a 2 in order to describe the spatial global behavior an evolution equation for the local wave number q can be derived formally the local wave number q satisfies approximately a conservation law q h q it is the purpose of this paper to explain the extent to which the conservation law is valid by proving estimates for this formal approximation

a class of local classical solutions for the one dimensional perona malik equation we consider the cauchy problem for the one dimensional perona malik equation u t 1 u 2 x 1 u 2 x 2 u xx in the interval 1 1 with homogeneous neumann boundary conditions we prove that the set of initial data for which this equation has a local in time classical solution u 1 1 x 0 t is dense in c 1 1 1 here classical solution means that u u t u x and u xx are continuous functions in 1 1 0 t

whitham modulation equations coalescing characteristics and dispersive boussinesq dynamics abstract whitham modulation theory with degeneracy in wave action is considered the case where all components of the wave action conservation law when evaluated on a family of periodic travelling waves have vanishing derivative with respect to wavenumber is considered it is shown that whitham modulation equations morph on a slower time scale into the two way boussinesq equation both the 1 1 and 2 1 cases are considered the resulting boussinesq equation arises in a universal form in that the coefficients are determined from the abstract properties of the lagrangian and do not depend on particular equations one curious by product of the analysis is that the theory can be used to confirm that the two way boussinesq equation is not a valid model in shallow water hydrodynamics modulation of nonlinear travelling waves of the complex klein gordon equation is used to illustrate the theory

two phase entropy solutions of a forward backward parabolic equation this article deals with the cauchy problem for a forward backward parabolic equation which is of interest in physical and biological models considering such an equation as the singular limit of an appropriate pseudoparabolic third order regularization we consider the framework of entropy solutions namely weak solutions satisfying an additional entropy inequality inherited by the higher order equation moreover we restrict the attention to two phase solutions that is solutions taking values in the intervals where the parabolic equation iswell posed proving existence and uniqueness of such solutions

the perona malik paradox the perona malik equation is a formally ill posed parabolic equation for which simple discretizations are nevertheless numerically found to be stable after discussing the background of this parad

the validity of modulation equations for extended systems with cubic nonlinearities modulation equations play an essential role in the understanding of complicated systems near the threshold of instability here we show that the modulation equation dominates the dynamics of the full problem locally at least over a long time scale for systems with no quadratic interaction term we develop a method which is much simpler than previous ones it involves a careful bookkeeping of errors and an estimate of gronwall type as an example for the dissipative case we find that the ginzburg landau equation is the modulation equation for the swift hohenberg problem moreover the method also enables us to handle hyperbolic problems the nonlinear schrodinger equation is shown to describe the modulation of wave packets in the sine gordon equation

upper bounds on the coarsening rate of discrete ill posed nonlinear diffusion equations we prove a weak upper bound on the coarsening rate of the discrete in space version of an ill posed nonlinear diffusion equation the continuum version of the equation violates parabolicity and lacks a complete well posedness theory in particular numerical simulations indicate very sensitive dependence on initial data nevertheless models based on its discrete in space version which we study are widely used in a number of applications including population dynamics chemotactic movement of bacteria granular flow formation of shear bands and computer vision image denoising and segmentation our bounds have implications for all three applications 2008 wiley periodicals inc

on new generalizations of hilbert s inequality and their applications this paper deals with new generalizations of hilbert s inequality and their applications it is shown that results of this paper are significant extensions and improvements of many known results

an analysis of the perona malik scheme the perona malik scheme is a numerical procedure for de noising digital pictures in this work we investigate how it evolves piecewise smooth initial data in one space dimension by scaling a natural parameter that appears in the scheme in an appropriate way with respect to the grid size we obtain a meaningful continuum limit the resulting evolution is unusual it involves solving a system of heat equations coupled to each other through nonlinear boundary conditions that become singular at special times leading to gradient blow up for the solutions however the scheme suggests a natural continuation beyond each of the singular times that involves a change in the p d e system we give a convergence proof that applies under some technical restrictions on any bounded interval of time which might include singular times furthermore the proposed continuum limit has some of the essential features observed in applications of the numerical scheme

a backward forward regularization of the perona malik equation abstract it is shown that the perona malik equation pme admits a natural regularization by forward backward diffusions possessing better analytical properties than pme itself well posedness of the regularizing problem along with a complete understanding of its long time behavior can be obtained by resorting to weak young measure valued solutions in the spirit of kinderlehrer and pedregal 1992 1 and demoulini 1996 2 solutions are unique to an extent to be specified but can exhibit micro oscillations in the sense of minimizing sequences and in the spirit of material science between preferred gradient states in the limit of vanishing regularization the preferred gradients have size 0 or thus explaining the well known phenomenon of staircasing the theoretical results do completely confirm and or predict numerical observations concerning the generic behavior of solutions

phase dynamics of periodic waves leading to the kadomtsev petviashvili equation in 3 1 dimensions the kadomstev petviashvili kp equation is a well known modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation in this paper it is shown that the kp equation is also the relevant modulation equation for bifurcation from periodic travelling waves when the wave action flux has a critical point moreover the emergent kp equation arises in a universal form with the coefficients determined by the components of the conservatio n of wave action the theory is derived for a general class of partial differential equations generated by a lagrangian using phase modulation the theory extends to any space dimension and time but the emphasis in the paper is on the ca se of 3 1 motivated by light bullets and quantum vortex dynamics the theory is illustrated by showing how defocussing nls in 3 1 bifurcates to kp in 3 1 at criticality the generalization to n 3 is also discussed

kawahara dynamics in dispersive media abstract in a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled kdv equations here we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes the long wave limit can be rigorously approximated by two decoupled kawahara equations

non uniqueness results for entropy two phase solutions of forward backward parabolic problems with unstable phase abstract this paper studies the well posedness of the entropy formulation given by plotnikov 1994 in 23 for forward backward parabolic problem obtained as singular limit of a proper pseudoparabolic approximation it was proved in mascia et al 2009 20 that such a formulation gives uniqueness when the solution takes values in the stable phases here we consider the situation in which unstable phase is taken in account proving that in general uniqueness does not hold

long wave approximations for water waves in this paper we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions these systems are symmetric and conservative rigorous convergence results are provided showing that solutions of the complete free surface euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large using this result as a tool a rigorous justification of all the two dimensional approximate systems recently put forward and analysed by bona chen and saut is obtained in the two dimensional context our methods also allows a significant improvement of the convergence estimate obtained by schneider and wayne in their justification of the decoupled korteweg de vries approximation of the two dimensional euler equations it also follows from our theory that coupled models provide a better description than the decoupled ones over short time scales results are obtained both on an unbounded domain for solutions that evanesce at infinity as well as for solutions that are spatially periodic

a new approach to variational problems with multiple scales we introduce a new concept the young measure on micropatterns to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter e this allows one to extract in the limit e 0 the relevant information at the macroscopic scale as well as the coarsest microscopic scale say e and to eliminate all finer scales to achieve this we consider rescaled functions rx t x s e t viewed as maps of the macroscopic variable s with values in a suitable function space the limiting problem can then be formulated as a variational problem on the young measures generated by rex as an illustration we study a one dimensional model that describes the competition between formation of microstructure and highest gradient regularization we show that the unique minimizer of the limit problem is a young measure supported on sawtooth functions with a given period 2001 john wiley sons inc

justification and failure of the nonlinear schrodinger equation in case of non trivial quadratic resonances abstract the nonlinear schrodinger nls equation can be derived as an amplitude equation describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet the purpose of this paper is to prove estimates between the formal approximation obtained via the nls equation and true solutions of the original system in case of non trivial quadratic resonances it turns out that the approximation property app holds if the approximation is stable in the system for the three wave interaction twi associated to the resonance we construct a counterexample showing that the nls equation can fail to approximate the original system if instability occurs for the approximation in the twi system in the unstable case we give some arguments why the validity of the app can be expected for spatially localized solutions and why it cannot be expected for non localized solutions although we restrict ourselves to a nonlinear wave equation as original system we believe that the results hold in more general situations too

stability properties of the perona malik scheme the perona malik scheme is a numerical technique for denoising digital images without blurring object boundaries edges in general solutions generated by this scheme do not satisfy a comparison principle we identify conditions under which two solutions initially ordered remain ordered and we state restricted comparison principles these allow us to study stability properties of the scheme we also explore what these results say in the limit as the discretization size goes to 0

validity of the kdv equation for the modulation of periodic traveling waves in the nls equation abstract bethuel et al 1 2 and chiron and rousset 3 gave very nice proofs of the fact that slow modulations in time and space of periodic wave trains of the nls equation can approximately be described via solutions of the kdv equation associated with the wave train here we give a much shorter proof of a slightly weaker result avoiding the very detailed and fine analysis of 1 2 3 our error estimates are based on a suitable choice of polar coordinates a cauchy kowalevskaya like method and energy estimates

validity of the korteweg de vries approximation for the two dimensional water wave problem in the arc length formulation we consider the two dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension it has been proven by several authors that long wavelength solutions to this problem can be approximated over a physically relevant timespan by solutions of the korteweg de vries equation or for certain values of the surface tension by solutions of the kawahara equation these proofs are formulated either in lagrangian or in eulerian coordinates in this paper we provide a new proof which is simpler more elementary and shorter moreover the rigorous justification of the kdv approximation can be given for the cases with and without surface tension together by one proof in our proof we parametrize the free surface by arc length and use some geometrically and physically motivated variables with good regularity properties this formulation of the water wave problem has already been of great usefulness for ambrose and masmoudi to simplify the proof of the local well posedness of the water wave problem in sobolev spaces 2011 wiley periodicals inc

gradient estimates for the perona malik equation we consider the cauchy problem for the perona malik equation u t mathrm div left frac nabla u 1 nabla u 2 right in a bounded open set omega subseteq mathbb r n with neumann boundary conditions

convergence of discrete schemes for the perona malik equation abstract we prove the convergence up to a subsequence of the spatial semidiscrete scheme for the one dimensional perona malik equation u t u x x p 1 2 log 1 p 2 when the initial datum u is 1 lipschitz out of a finite number of jump points and we characterize the problem satisfied by the limit solution in the more difficult case when u has a whole interval where u x is negative we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space time neighbourhood of the jump points the limit solution u we obtain is the same as the one obtained by replacing with the truncated function min 1 and it turns out that u solves a free boundary problem the free boundary consists of the points dividing the region where u x 1 from the region where u x 1 finally we consider the full space time discretization implicit in time of the perona malik equation and we show that if the time step is small with respect to the spatial grid h then the limit is the same as the one obtained with the spatial semidiscrete scheme on the other hand if the time step is large with respect to h then the limit solution equals u i e the standing solution of the convexified problem

existence of infinitely many solutions for a forward backward heat equation let 4 be a piecewise linear function which satisfies the condition s40 s 2 cs2 c 0 s e r and which is monotone decreasing on an interval a b c r it is shown that for f e c2 0 11 with max f a there exists a t 0 such that the initial boundary value problem ut 00 x x ux os t u ls t u 0 f has infinitely many solutions u satisfying 11ul u x u x u 1l 2 6 c f 4 on 0 1i x 0 t 0 introduction consider the initial boundary value problem ut u x x t e 0 1 x 0 t 1 u t uj l t 0 t e 0 t u x 0 f x x e 0 1 if 4 is strictly monotone increasing with 4 c 0 1 has a unique solution which is roughly speaking as smooth as the function o on the other hand if 4 c 0 this assumption allows 4 to have monotone decreasing parts e g the model cubic et s 1s3 s2 2s 4 s 3 42 2s a natural and interesting question is whether problem 1 has a solution if the usual assumption 4 0 is replaced by the weaker coercivity condition c under this hypothesis j bona j nohel and l wahlbin bnw hn obtained several a received by the editors april 2 1982 and in revised form may 14 1982 1980 mathematics subject classification primary 35k55 35k65

a stability criterion for two fluid interfaces and applications we derive a new stability criterion for two fluid interfaces that ensures the existence of stable local solutions that do not break down too fast due to kelvin helmholtz instabilities it can be seen both as a two fluid generalization of the rayleigh taylor criterion and as a nonlinear version of the kelvin stability condition we show that gravity can control the inertial effects of the shear up to frequencies that are high enough for the surface tension to play a relevant role this explains why surface tension is a necessary condition for well posedness while the low frequency main dynamics of interfacial waves are unaffected by it in order to derive a practical version of this criterion we work with a nondimensionalized version of the equations and allow for the possibility of various asymptotic regimes such as the shallow water limit this limit being singular we have to derive a new symbolic analysis of the dirichlet neumann operator that includes an infinitely smoothing tail accounting for the contribution of the bottom we then validate our criterion by comparison with experimental data in two important settings air water interfaces and internal waves the good agreement we observe allows us to discuss the scenario of wave breaking and the behavior of water brine interfaces and to propose a formula for the maximal amplitude of interfacial waves we also show how to rigorously justify two fluid asymptotic models used for applications and how to relate some of their properties to kelvin helmholtz instabilities

the korteweg de vries burgers and whitham limits for a spatially periodic boussinesq model we are interested in the korteweg de vries kdv burgers and whitham limits for a spatially periodic boussinesq model with non small contrast we prove estimates of the relations between the kdv burgers and whitham approximations and the true solutions of the original system that guarantee these amplitude equations make correct predictions about the dynamics of the spatially periodic boussinesq model over their natural timescales the proof is based on bloch wave analysis and energy estimates and is the first justification result of the kdv burgers and whitham approximations for a dispersive partial differential equation posed in a spatially periodic medium of non small contrast

justification of the nonlinear schrodinger equation for the evolution of gravity driven 2d surface water waves in a canal of finite depth in 1968 v e zakharov derived the nonlinear schrodinger equation for the two dimensional water wave problem in the absence of surface tension that is for the evolution of gravity driven surface water waves in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet in this paper we give a rigorous proof that the wave packets in the two dimensional water wave problem in a canal of finite depth can be approximated over a physically relevant timespan by solutions of the nonlinear schrodinger equation

a new nonlocal nonlinear diffusion of image processing abstract a novel nonlocal nonlinear diffusion is analyzed which has proven useful as a denoising tool in image processing the equation can be viewed as a new paradigm for the regularization of the well known perona malik equation the regularization is implemented via nonlinearity intensity reduction through fractional derivatives well posedness in the weak setting is established global existence and convergence to the average holds in the purely diffusive limit whereas an interesting dynamic behavior is engendered by the presence of nontrivial equilibria as the intensity of the nonlinearity is increased and comes close to the one of perona malik

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

the long wave limit for a boussinesq equation we consider the long wave limit for a boussinesq equation it is shown that for spatially localized initial conditions the dynamics are under a korteweg de vries kdv regime on a large time interval i e in this limit the solutions split up into two counterpropagating wave packets where each of the wave packets evolve independently and approximately as a solution of a kdv equation for the proof exact estimates between the long wave solutions of the boussinesq equation and the approximations obtained via the decoupled set of kdv equations are demonstrated we expect that this result holds for the water wave problem too