fully discrete entropy conservative schemes of arbitrary order we consider weak solutions of hyperbolic or hyperbolic elliptic systems of conservation laws in one space dimension and their approximation by finite difference schemes in conservative form the systems under consideration are endowed with an entropy entropy flux pair we introduce a general approach to construct second and third order accurate fully discrete in both space and time entropy conservative schemes in general these schemes are fully nonlinear implicit but in some important cases can be explicit or linear implicit furthermore semidiscrete entropy conservative schemes of arbitrary order are presented the entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization sensitive shock waves finally specific examples are investigated and tested numerically our approach extends the results and techniques by tadmor in numerical methods for compressible flows finite difference element and volume techniques asme new york 1986 pp 149 158 lefloch and rohde siam j numer anal 37 2000 pp 2023 2060

construction of non analytic solutions for the generalized kdv equation abstract in both the periodic and non periodic case we construct non analytic complex valued solutions for the generalized kdv equation with appropriate analytic initial data moreover for the kdv and mkdv we construct real valued non analytic solutions

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

lattices of jordan algebras commutative jordan algebras play a central part in orthogonal models the generations of these algebras is studied and applied in deriving lattices of such algebras these lattices constitute the natural framework for deriving new orthogonal models through factor aggregation and disaggregation

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

analyticity and smoothing effect for the korteweg de vries equation with a single point singularity we show that a solution of the cauchy problem for the kdv equation cases partial tv partial 3 x v partial x v 2 0 qquad t in t t x in bbb r v 0 x phi x endcases has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x 0 it is shown that for h s mathbb r s 3 4 data satisfying the condition sum k 0 infty frac a 0 k k x partial x k phi h s infty the solution is analytic in both space and time variable the above condition allows us to take as initial data the dirac delta measure or the cauchy principal value of 1 x the argument is based on the recent progress on the well posedness result by bourgain 2 and kenig ponce vega 20 and a systematic use of the dilation generator 3t partial t x partial x

binary operations and canonical forms for factorial and related models abstract binary operations on commutative jordan algebras are used to extend the grouping of treatments in blocks and the taking of fractional replicates to models where factors have arbitrary number of levels up to now these techniques had been restricted to models whose factors have a prime or a power of a prime number of levels moreover the binary operations will enable the use of simple models as building blocks for more complex designs

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

well posedness of the good boussinesq equation in analytic gevrey spaces and time regularity abstract the cauchy problem for the good boussinesq equation with data in analytic gevrey spaces on the line and the circle is considered and its local well posedness in these spaces is proved the proof is based on bilinear estimates in bourgain type spaces incorporating the symbol of the linear part of the equation and an exponential weight expressing the analytic gevrey regularity of the solution in the spatial variable also gevrey regularity of the solution in time variable is provided

entropy conservative and entropy stable schemes for nonconservative hyperbolic systems the vanishing viscosity limit of nonconservative hyperbolic systems depends heavily on the specific form of the viscosity numerical approximations such as the path consistent schemes of c pares siam j numer anal 41 2007 pp 169 185 may not converge to the physically relevant solutions of the system we construct entropy stable path consistent espc schemes to approximate nonconservative hyperbolic systems by combining entropy conservative discretizations with numerical diffusion operators that are based on the underlying viscous operator numerical experiments for the coupled burgers system and the two layer shallow water equations demonstrating the robustness of espc schemes are presented

binary operations on jordan algebras and orthogonal normal models jordan algebras are used to present normal orthogonal models in a canonical form binary operations are defined on these algebras which enable us to build complex models based on simpler ones properties of these operations and their relation with balanced models are studied the canonical model formulation is interesting because it leads to complete sufficient statistics these statistics are then used to obtain estimators in order to test hypothesis on the model parameters

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

jordan algebras and bayesian quadratic estimation of variance components the authors consider a linear model ey x cov y v where v is a quadratic subspace a jordan algebra using this property it is shown how multiplication computation of g inverses determination of a neutral element can be done more efficiently in v finally this technique is applied to obtain formulae for locally optimal estimators and bayesian estimators

analytic well posedness of periodic gkdv abstract in the periodic case it is proved that the cauchy problem for the generalized korteweg de vries equation gkdv is locally well posed in a class of analytic functions that can be extended holomorphically in a symmetric strip of the complex plane around the x axis thus the uniform analyticity radius of the solution does not change as time progresses also information about the regularity of the solution in the time variable is provided

gevrey regularity of the periodic gkdv equation abstract using the multilinear estimates which were derived for proving well posedness of the generalized korteweg de vries gkdv equation it is shown that if the initial data belongs to gevrey space g 1 in the space variable then the solution to the corresponding cauchy problem for gkdv belongs also to g in the space variable moreover the solution is not necessarily g in the time variable however it belongs to g 3 near 0 when 1 these are analytic regularity results for gkdv

entropy stable schemes for degenerate convection diffusion equations in this work we construct entropy stable schemes for nonlinear degenerate convection diffusion systems by extending the theoretical framework introduced by tadmor for conservation laws entropy stable finite difference schemes are developed for systems with viscous terms in both conservative and nonconservative formulations extended versions that include extra viscosity are proposed for both formulations in order to prevent the appearance of numerical spurious oscillations at discontinuities of the physical solution finally some numerical simulations are shown in order to illustrate the behavior and accuracy of the entropy stable methods for degenerate parabolic problems

the cauchy problem of a periodic higher order kdv equation in analytic gevrey spaces abstract this paper studies the periodic cauchy problem for a kdv equation whose dispersion is of order m 2 j 1 where j is a positive integer kdvm using bourgain gevrey type analytic spaces and appropriate bilinear estimates it is shown that local in time well posedness holds when the initial data belong to an analytic gevrey spaces of order this implies that in the space variable the regularity of the solution remains the same with that of the initial data it also implies that the size of the uniform radius of analyticity is preserved moreover the solution is not necessarily g in time however it belongs to g m r near zero for every x on the circle

new lower bounds on the radius of spatial analyticity for the kdv equation abstract the radius of spatial analyticity for solutions of the kdv equation is studied it is shown that the analyticity radius does not decay faster than t 1 4 as time t goes to infinity this improves the works of selberg and da silva 2017 30 and tesfahun 2017 34 our strategy mainly relies on a higher order almost conservation law in gevrey spaces which is inspired by the i method

white noise driven korteweg de vries equation abstract we consider a stochastic korteweg de vries equation on the real line the noise is additive we use function spaces similar to those introduced by bourgain to prove well posedness results for the korteweg de vries equation in l 2 r we are able to handle a noise which is locally white in space and time more precisely it is a space time white noise multiplied by an l 2 function of the space variable due to the lack of a priori estimates we can only get a local existence result in time however we obtain the global existence of l 2 r solutions when the covariance operator of the noise is hilbert schmidt in l 2 r

global solutions of the derivative schrodinger equation in a class of functions analytic in a strip lower bounds on the rate of decrease in time of a uniform radius of spatial analyticity for solutions of the derivative schrodinger equation are derived the bounds depend algebraically on time they are valid as long as the initial datum is of order one in l2 norm and satisfies suitable spatial decay requirements on the real axis

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation