nonlocal maximum principles for active scalars abstract active scalars appear in many problems of fluid dynamics the most common examples of active scalar equations are 2d euler burgers and 2d surface quasi geostrophic equations many questions about regularity and properties of solutions of these equations remain open we develop the idea of nonlocal maximum principle introduced in kiselev nazarov and volberg 2007 19 formulating a more general criterion and providing new applications the most interesting application is finite time regularization of weak solutions in the supercritical regime

blow up for the generalized surface quasi geostrophic equation with supercritical dissipation we prove the existence of singularities for the generalized surface quasi geostrophic gsqg equation with supercritical dissipation analogous results are obtained for the family of equations interpolating between gsqg and 2d euler

global existence of small data solutions for wave models with super exponential propagation speed abstract we consider semi linear classical damped wave models with time dependent speed of propagation and time dependent dissipation the right hand side is a power nonlinearity which can be interpreted only as a source term we are interested in the interplay of time dependent coefficients on the global existence of small data solutions the considerations are divided into two cases depending on the behavior of the propagation speed the super exponential and the sub exponential case in this paper we study the super exponential case one main tool of our approach is the use of matsumura type estimates for a family of cauchy problems depending on a parameter

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

on a multi dimensional transport equation with nonlocal velocity abstract we study a multi dimensional nonlocal active scalar equation of the form u t v u 0 in r r d where v 2 u with 1 2 we show that when 0 2 certain radial solutions develop gradient blowup in finite time in the case when 0 the equations are globally well posed with arbitrary initial data in suitable sobolev spaces

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

a weakly nonlinear analysis of elasto plastic microstructure models at certain critical values of the hardening modulus the governing equations of elasto plastic flow may lose their hyperbolicity and exhibit two modes of ill posedness shear band and flutter ill posedness these modes of ill posedness are characterized by the uncontrolled growth of modes at infinitely fine scales which ultimately violates the continuum assumption in previous work l an and a peirce siam j appl math 54 1994 pp 708 730 a continuum model accounting for microscale deformations was built linear analysis demonstrated the regularizing effect of the microstructure and provided a relationship between the width of the localized instabilities and the microlength scale in this paper a weakly nonlinear analysis is used to explore the immediate post critical behavior of the solutions for both one dimensional and anti plane shear models post critical deformations in the plastic regions are shown to be governed by the boussinesq equation one of the completely integrable pdes having so

simple expressions for a function occurring in diffraction theory the maliuzhinets integral function arises in connection with diffraction by a half plane two simple expressions are derived which when used in conjunction with known identities serve to approximate the function to a high degree of accuracy throughout the entire complex plane

formation of singularities for a transport equation with nonlocal velocity we study a 1d transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data by adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time

existence for the α patch model and the qg sharp front in sobolev spaces abstract we consider a family of contour dynamics equations depending on a parameter with 0 1 the vortex patch problem of the 2 d euler equation is obtained taking 0 and the case 1 corresponds to a sharp front of the qg equation we prove local in time existence for the family of equations in sobolev spaces

global attractors and their hausdorff dimensions for a class of kirchhoff models this paper studies the existence regularity and hausdorff dimensions of global attractors for a class of kirchhoff models arising in elastoplastic flow utt div u m 1 u ut 2u h ut g u f x it proves that under rather mild conditions the dynamical system associated with above mentioned models possesses in phase space x a global attractor which has further regularity in x 1 x and has finite hausdorff dimension for application the fact shows that for the concerned elastoplastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

on space time means and strong global solutions of nonlinear hyperbolic equations on demontre l existence de solutions globales appartenant a l loc r 2 l 2 2 r n pour des equations non lineaires de la forme t u u b x d u f u 0 t x r r n u 0 x 0 x t u 0 x 1 x x r n

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

analytic expressions for a function occurring in diffraction theory simple expressions are derived for the maliuzhinets integral function which arises in connection with diffraction by an impedance wedge the given expressions allow the computation of the function with remarkable accuracy throughout the entire complex plane for any exterior wedge angle

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

global existence and exponential stability of small solutions to nonlinear viscoelasticity the global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn dimensional euclidean space with smooth boundary dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions ast tending to is described time periodic solutions are also studied as an application of our main theorem nonlinear viscoelasticity strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated

decay estimates for fourth order wave equations may be thought of as a nonlinear beam equation in this paper we obtain both l l estimates and space time integrability estimates on solutions to the linear equation we also use these estimates to study the local existence and asymptotic behavior of solutions to the nonlinear equation for nonlinear terms which grow like a certain power of u the main l l estimate theorem 2 1 states that solutions of the linear equation with initial data u 0 ut 0 in w 2 q rn lq rn are bounded in l r w 2 q rn for 2 q 2 for all time and that their norm in this space decays at the optimal rate t n 2q 4 here and throughout 2 2n n 4 for n 5 for 1 n 4 1 2

global solutions for a supercritical drift diffusion equation abstract we study the global existence of solutions to a one dimensional drift diffusion equation with logistic term generalizing the classical parabolic elliptic keller segel aggregation equation arising in mathematical biology in particular we prove that there exists a global weak solution if the order of the fractional diffusion 1 c 1 2 where c 1 0 is an explicit constant depending on the physical parameters present in the problem chemosensitivity and strength of logistic damping furthermore in the range 1 c 2 2 with 0 c 2 c 1 the solution is globally smooth let us emphasize that when 1 the diffusion is in the supercritical regime

global attractor for a class of kirchhoff models the paper studies the longtime behavior of solutions to the initial boundary value problem ibvp for a class of kirchhoff models arising in elastoplastic flow utt div u m 1 u ut 2u h ut g u f x by combining the decomposition idea with the operate technique it proves that under rather mild conditions the dynamical system associated with above mentioned ibvp possesses in different phase spaces a global attractor which is connected respectively for application the fact shows that for the concerned elastoplastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space

blow up of solutions for a 1d transport equation with nonlocal velocity and supercritical dissipation we study a 1d transport equation with nonlocal velocity and supercritical dissipation we show that for a certain class of initial data the solution blows up in finite time

cauchy problem for a class of nonlinear dispersive wave equations arising in elasto plastic flow abstract the paper studies the existence both locally and globally in time stability decay estimates and blowup of solutions to the cauchy problem for a class of nonlinear dispersive wave equations arising in elasto plastic flow under the assumption that the nonlinear term of the equations is of polynomial growth order say it proves that when 1 the cauchy problem admits a unique local solution which is stable and can be continued to a global solution under rather mild conditions when 5 and the initial data is small enough the cauchy problem admits a unique global solution and its norm in l 1 p r decays at the rate 1 t p 2 2 p for 2 p 10 and if the initial energy is negative then under a suitable condition on the nonlinear term the local solutions of the cauchy problem blow up in finite time

global existence and global nonexistence of solutions of the cauchy problem for a nonlinearly damped wave equation abstract in this paper we consider the long time behavior of solutions of the initial value problem for semilinear wave equations of the form formula

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

well posedness for a transport equation with nonlocal velocity abstract we study a one dimensional transport equation with nonlocal velocity which was recently considered in the work of cordoba cordoba and fontelos a cordoba d cordoba m a fontelos formation of singularities for a transport equation with nonlocal velocity ann of math 2 162 3 2005 1377 1389 we show that in the subcritical and critical cases the problem is globally well posed with arbitrary initial data in h max 3 2 0 while in the supercritical case the problem is locally well posed with initial data in h 3 2 and is globally well posed under a smallness assumption some polynomial in time decay estimates are also discussed these results improve some previous results in a cordoba d cordoba m a fontelos formation of singularities for a transport equation with nonlocal velocity ann of math 2 162 3 2005 1377 1389

one dimensional model equations for hyperbolic fluid flow abstract in this paper we study the singularity formation for two nonlocal 1d active scalar equations focusing on the hyperbolic flow scenario those 1d equations can be regarded as simplified models of some 2d fluid equations

asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping we consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingvtt a vx x vt 0 x t r 0 subsequent to k nishihara j differential equations131 1996 171 188 in that article the system with dampingvt ux 0 ut p v x u p v 0 was treated and the convergence rates to the diffusion wave by l hsiao and t p liu comm math phys 143 1992 599 605 and l hsiao and t p liu chinese ann math ser b14 1993 465 480 were improved for initial data v0 u0 x satisfying v0 u0 v0 0 and v0 x v0 dx 0 this system is reduced to the quasilinear hyperbolic equation with linear dampingvtt p v0 vx p v0 x vt 0 and the decay rates v t l o t 1 2 vx t l o t 1 vt t l o t 3 2 were obtained provided that v vt t 0 h3 h2 is small and inl1 l1 from these decay rates vis expected to behave as a solution to the parabolic equation the aims of this paper are to seek the asymptotic profile x t satisfying t l o t 1 2 x t l o t 1 t t l o t 3 2 and to show that the solution of satisfies v t l o t 1 v x t l o t 3 2 v t t l o t 2 the proof is based on the elementary energy method and the green function method for the parabolic equation

global well posedness of the critical burgers equation in critical besov spaces abstract we make use of the method of modulus of continuity a kiselev f nazarov r shterenberg blow up and regularity for fractal burgers equation dyn partial differ equ 5 2008 211 240 and fourier localization technique h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 to prove the global well posedness of the critical burgers equation t u u x u u 0 in critical besov spaces b p 1 1 p r with p 1 where

global existence of small amplitude solutions to nonlinear klein gordon equations in four space time dimensions consider the following class of nonlinear wave equations square text u u text text f u u u n k g where t 2 1 2 2 2 3 2 is the d alembertian of the 4 dimensional minkowski space time and f a smooth function of u u t x and its first and second partial derivates vanishing together with its first derivatives at u u u 0 in other words f is at least quadratic in u u u in a neighborhood of the trivial solution u 0 we subject u to the pure initial value problem text u mkern 1mu varepsilon text f x u t varepsilon g x at t 0 i v p with f g c 0 3 and e 0 a small parameter

global existence asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term abstract the paper studies the global existence asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term it proves that under rather mild conditions on nonlinear terms and initial data the above mentioned problem admits a global weak solution and the solution decays exponentially to zero as t respectively in the states of large initial data and small initial energy in particular in the case of space dimension n 1 the weak solution is regularized to be a unique generalized solution and if the conditions guaranteeing the global existence of weak solutions are not valid then under the opposite conditions the solutions of above mentioned problem blow up in finite time and an example is given

potential well method for cauchy problem of generalized double dispersion equations abstract in this paper we study cauchy problem of generalized double dispersion equations u t t u x x u x x t t u x x x x f u x x where f u u p p 1 or u 2 k k 1 2 by introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions but also obtain the invariance of some sets and vacuum isolating of solutions in addition the global existence and finite time blow up of solutions for problem with critical initial conditions e 0 d i u 0 0 or i u 0 0 are proved

high frequency scattering from polygonal impedance cylinders and strips an analysis is presented for the high frequency diffraction by convex polygonal cylinders with arbitrary side impedances the authors extend their solution for the multiply diffracted fields associated with a double wedge see ibid vol 36 no 5 p 664 678 1988 the analysis thus utilizes the extended spectral ray method esrm that is applicable to nonray optical regions diffraction coefficients yielding a uniform total field are given for up to and including all fourth order mechanisms an important aspect of this analysis is the rigorous and uniform incorporation of the surface wave effects in the resulting diffraction coefficients a general polygon computer code was written that includes up to a third order mechanisms based on this code backscatter and bistatic patterns are given for impedance cylinders with triangular and square cross sections the results are found to be in remarkable agreement with corresponding moment method data as a matter of completeness uniform diffraction coefficients are presented for a strip which is a special case of a cylinder that can have unequal face impedances

finite dimensional attractors for the kirchhoff models with critical exponents the paper studies the existence of the finite dimensional global attractor and exponential attractor for the dynamical system associated with the kirchhoff models utt u m 1 u ut 2u h ut g u f x it proves that for the subcritical and critical cases 1 m n 2 n 2 with n 6 the above mentioned dynamical system possesses in phase spaces x1 a finite dimensional global attractor and a weak exponential attractor respectively

existence and regularity of rotating global solutions for the generalized surface quasi geostrophic equations motivated by the recent work of hassainia and hmidi z hassainia t hmidi on the vstates for the generalized quasi geostrophic equations arxiv preprint arxiv 1405 0858 we close the question of the existence of convex global rotating solutions for the generalized surface quasigeostrophic equation for 2 1 2 we also show c 1 regularity of their boundary for all 2 0 2

finite dimensional attractors for the kirchhoff models the paper studies the existence of the finite dimensional global attractor and exponential attractor for the dynamical system associated with the kirchhoff models arising in elasto plastic flow utt div u m 1 u ut 2u h ut g u f x by using the method of l trajectories and the operator technique it proves that under subcritical case 1 m n 2 n 2 the above mentioned dynamical system possesses in different phase spaces a finite dimensional weak global attractor and a weak exponential attractor respectively for application the fact shows that for the concerned elasto plastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the fractal dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

polynomial approximation of maliuzhinets function approximate formulas for the maliuzhinets functions which give the solutions for the wedge diffraction with face impedance are derived using the chebyshev polynomials for various wedge angles the coefficients of the polynomials are listed in a table although the formulas are derived for real variables it is found that they can be used for the restricted range of complex variables since the validity of the formulas depends on both real and imaginary part when applied to complex variables the equi error contour of the formulas is given for the special case where the exact solution is available an alternative but empirical formula for maliuzhinets functions is also obtained which can be applied to arbitrary wedge angles