stable self similar blowup in energy supercritical yang mills theory we consider the cauchy problem for an energy supercritical nonlinear wave equation that arises in 1 5 dimensional yang mills theory a certain self similar solution w 0 of this model is conjectured to act as an attractor for generic large data evolutions assuming mode stability of w 0 we prove a weak version of this conjecture namely that the self similar solution w 0 is nonlinearly stable phrased differently we prove that mode stability of w 0 implies its nonlinear stability the fact that this statement is not vacuous follows from careful numerical work by bizon and chmaj that verifies the mode stability of w 0 beyond reasonable doubt

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

representation of surfaces using spinor operators we use the concept of spinor operator to study some aspects of the classical theory of surfaces in three dimensions we show that a spinor operator representing a surface satisfies a pair of dirac like equations and that this spinor operator representation is a generalization of the weierstrass representation of minimal surfaces we also obtain spinor operator representations of a surface directly from its parameterization in terms of isothermal parameters without the need of solving the pair of dirac like equations we apply these results to minimal surfaces and their deformations and also show how they can be written in terms of quaternions and complex numbers we use the concept of spinor operator to study some aspects of the classical theory of surfaces in three dimensions we show that a spinor operator representing a surface satisfies a pair of dirac like equations and that this spinor operator representation is a generalization of the weierstrass representation of minimal surfaces we also obtain spinor operator representations of a surface directly from its parameterization in terms of isothermal parameters without the need of solving the pair of dirac like equations we apply these results to minimal surfaces and their deformations and also show how they can be written in terms of quaternions and complex numbers

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 regularity for supercritical nonlinear dissipative wave equations in 3d abstract the nonlinear wave equation u t t u u t p 1 u t 0 is shown to be globally well posed in the sobolev spaces of radially symmetric functions h rad k r 3 h rad k 1 r 3 for all p 3 and k 3 moreover global c solutions are obtained when the initial data are c 0 and exponent p is an odd integer the radial symmetry allows a reduction to the one dimensional case where an important observation of haraux 2009 can be applied i e dissipative nonlinear wave equations contract initial data in w k q r w k 1 q r for all k 1 2 and q 1

constant mean curvature surfaces via an integrable dynamical system it is shown that the equation which describes constant mean curvature surfaces via the generalized weierstrass enneper induction has hamiltonian form its simplest finite dimensional reduction is the integrable hamiltonian system with two degrees of freedom this finite dimensional system admits action and classes of equivalence of its trajectories are in one to one correspondence with different helicoidal constant mean curvature surfaces thus the interpretation of well known delaunay and do carmo dajczer surfaces via an integrable finite dimensional hamiltonian system is established

nonparametric and nonlinear reconstruction of surfaces from qualitative observations abstract the aim of this paper is to discuss the problem of reconstructing surfaces from qualitative observations the problem statement is motivated by examples from testing quality of materials the proposed estimator is nonlinear and nonparametric its consistency is discussed and the rate of convergence is given then we propose its smoothed and adaptive version and validate it using simulated data

on the blow up phenomenon of the critical nonlinear schrodinger equation abstract in this paper we consider the blow up phenomenon of critical nonlinear schrodinger equations in dimension 1d and 2d we define the minimal mass as the l 2 norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism asymptotic compactness properties and l 2 concentration are proved the proof relies on linear and nonlinear profile decompositions

the radial defocusing energy supercritical nls in dimension four abstract we consider the radial defocusing nonlinear schrodinger equations i u t u u p u with supercritical exponent p 4 in four space dimensions and prove that any radial solution that remains bounded in the critical sobolev space must be global and scatter

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

the defocusing energy supercritical nonlinear wave equation in three space dimensions we consider the defocusing nonlinear wave equation u tt u u p u 0 in the energy supercritical regime p 4 for even values of the power p we show that blowup or failure to scatter must be accompanied by blowup of the critical sobolev norm an equivalent formulation is that solutions with bounded critical sobolev norm are global and scatter the impetus to consider this problem comes from recent work of kenig and merle who treated the case of spherically symmetric solutions

global wellposedness of defocusing critical nonlinear schrodinger equation in the radial case is globally wellposed in time more precisely we obtain a unique solution u u ch1 0 such that for all time u t depends continuously on the data in fact the dependence is even real analytic here moreover there is scattering for t the same statement holds for radial data h s 1 and proves in particular global existence of classical solutions in the radially symmetric case also this issue was open thus this is the analogue for nls of the result for the wave equation with quintic nonlinearity obtained by m struwe str in the radial case and by m grillakis gr s s in general in the case of the wave equation the proof is based on the following two different facts

global existence for 2d nonlinear schrodinger equations via high low frequency decomposition method abstract we study global existence of solutions for the cauchy problem of the nonlinear schrodinger equation i u t u u 2 m u in the 2 dimension case where m is a positive integer m 2 using the high low frequency decomposition method we prove that if 10 m 6 10 m 5 s 1 then for any initial value h s r 2 the cauchy problem has a global solution in c r h s r 2 and it can be split into u t e i t y t with y c r h 1 r 2 satisfying y t h 1 1 t 2 1 s 10 m 5 s 10 m 6 where is an arbitrary sufficiently small positive number

global well posedness and scattering for the energy critical nonlinear schrodinger equation in r 3 we obtain global well posedness scattering and global l 10 spacetime bounds for energy class solutions to the quintic defocusing schrodinger equa tion in r 1 3 which is energy critical in particular this establishes global existence of classical solutions our work extends the results of bourgain 4 and grillakis 20 which handled the radial case the method is similar in spirit to the induction on energy strategy of bourgain 4 but we perform the induction analysis in both frequency space and physical space simultaneously and replace the morawetz inequality by an interaction variant first used in 12 13 the principal advantage of the interaction morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions in particular this interaction estimate together with an almost conservation argument controlling the movement of l 2 mass in fre quency space rules out the possibility of energy concentration

bilinear identities involving the k plane transform and fourier extension operators we prove certain l2 n bilinear estimates for fourier extension operators associated to spheres and hyperboloids under the action of the k plane transform as the estimates are l2 based they follow from bilinear identities in particular these are the analogues of a known identity for paraboloids and may be seen as higher dimensional versions of the classical l2 2 bilinear identity for fourier extension operators associated to curves in 2

global well posedness scattering and blow up for the energy critical focusing non linear wave equation we study the energy critical focusing non linear wave equation with data in the energy space in dimensions 3 4 and 5 we prove that for cauchy data of energy smaller than the one of the static solution w which gives the best constant in the sobolev embedding the following alternative holds if the initial data has smaller norm in the homogeneous sobolev space h1 than the one of w then we have global well posedness and scattering if the norm is larger than the one of w then we have break down in finite time

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

semilinear schrodinger flows on hyperbolic spaces scattering in h 1 we prove global well posedness and scattering in h1 for the defocusing nonlinear schrodinger equations left begin array ll i partial t delta g u u u 2 sigma u 0 phi end array right on the hyperbolic spaces mathbb h d d 2 for exponents sigma in 0 2 d 2 the main unexpected conclusion is scattering to linear solutions in the case of small exponents for comparison on euclidean spaces scattering in h1 is not known for any exponent sigma in 1 d 2 d and is known to fail for sigma in 0 1 d our main ingredients are certain noneuclidean global in time strichartz estimates and noneuclidean morawetz inequalities

some remarks on the schrodinger equation with a potential in lrtlsx we study the dispersive properties of the linear schrodinger equation with a time dependent potential v t x we show that an appropriate integrability condition in space and time on v i e the boundedness of a suitable lrtlsx norm is sufficient to prove the full set of strichartz estimates we also construct several counterexamples which show that our assumptions are optimal both for local and for global strichartz estimates in the class of large unsigned potentials v lrtlsx

linear systems identification from random threshold binary data a new identification problem of estimating parameters of linear dynamic systems from random threshold binary observations of its output and input is stated the only available data are collected as a result of checking whether a signal reached a randomly specified threshold at a randomly chosen instant of time the proposed estimation algorithm is based on the celebrated von neumann theorem which was earlier used mainly for generating random numbers strong consistency of parameters estimate from low cost output binary observations is proved assuming deterministic input signal of a finite duration possibilities of relaxing the assumption used in the theoretical part of the paper are considered by means of simulations

nonlinear profile decomposition for the h 12 h 12 rd energy subcritical wave equation abstract we consider the nonlinear wave equation u t t u u u 4 d 1 0 in dimensions d 2 this equation is critical in h 1 2 h 1 2 r d and subcritical with respect to the energy we prove the nonlinear profile decomposition the proof must address the lack of compactness due to the lorentz symmetry

quintic nls in the exterior of a strictly convex obstacle we consider the defocusing energy critical nonlinear schr odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions for the initial value problem with dirichlet boundary condition we prove global well posedness and scattering for all initial data in the energy space

the dynamics of the 3d radial nls with the combined terms in this paper we show the scattering and blow up result of the radial solution with the energy below the threshold for the nonlinear schrodinger equation nls with the combined terms iu t delta u u 4 u u 2 u qquad qquad qquad qquad rm cnls in the energy space h 1 mathbb r 3 the threshold is given by the ground state w for the energy critical nls iu t u u 4 u this problem was proposed by tao visan and zhang in comm pdes 32 1281 1343 2007 the main difficulty is lack of the scaling invariance illuminated by ibrahim et al in analysis pde 4 3 405 460 2011 we need to give the new radial profile decomposition with the scaling parameter then apply it to the scattering theory our result shows that the defocusing dot h 1 subcritical perturbation u 2 u does not affect the determination of the threshold of the scattering solution of cnls in the energy space

scattering theory for the radial h 12 critical wave equation with a cubic convolution abstract in this paper we study the global well posedness and scattering for the wave equation with a cubic convolution t 2 u u x 3 u 2 u in dimensions d 4 we prove that if the radial solution u with life span i obeys u u t l t i h x 1 2 r d h x 1 2 r d then u is global and scatters by the strategy derived from concentration compactness we show that the proof of the global well posedness and scattering is reduced to disprove the existence of two scenarios soliton like solution and high to low frequency cascade making use of the no waste duhamel formula and double duhamel trick we deduce that these two scenarios enjoy the additional regularity by the bootstrap argument of 7 this together with virial analysis implies the energy of such two scenarios is zero and so we get a contradiction

nonlinear scattering theory for a class of wave equations in hs abstract for the semi linear higher order wave equation and the nonlinear higher order schrodinger equation we show that the scattering operators map a band in h s into h s if the nonlinearities have sub critical powers in h s the smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing nls equation are also shown provided that the nonlinearities have subcritical growth in h 1 moreover the spatial decaying behavior of solutions in energy space for the defocusing nls equation are obtained

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

bounded solutions to an energy subcritical non linear wave equation on r3 abstract in this work we consider an energy subcritical semi linear wave equation 3 p 5 t 2 u u x u p 1 u x t r 3 r u t 0 u 0 h s p r 3 t u t 0 u 1 h s p 1 r 3 where s p 3 2 2 p 1 and the function r 3 1 1 is a radial continuous function with a limit at infinity we prove that unless the elliptic equation w x w p 1 w has a nonzero radial solution w c 2 r 3 h s p r 3 any radial solution u with a finite uniform upper bound on the critical sobolev norm u t t u t h s p h s p r 3 for all t in the maximal lifespan must be a global solution in time and scatter

conditional global existence and scattering for a semi linear skyrme equation with large data we study a generalization of energy super critical wave maps due to adkins and nappi that can also be viewed as a simplified version of the skyrme model these are maps from 1 3 dimensional minkowski space that take values in the 3 sphere and it follows that every finite energy adkins nappi wave map has a fixed topological degree which is an integer here we initiate the study of the large data dynamics for adkins nappi wave maps by proving that there is no type ii blow up in the class of maps with topological degree zero in particular any degree zero map whose critical norm stays bounded must be global in time and scatter to zero as t to pm infty

nondispersive radial solutions to energy supercritical non linear wave equations with applications in this paper we establish optimal pointwise decay estimates for non dispersive compact radial solutions to non linear wave equations in 3 dimensions in the energy supercritical range as an application we show for the full energy supercritical range in the defocusing case that if the scale invariant sobolev norm of a radial solution remains bounded in its maximal interval of existence then the solution must exist for all times and scatter

z2 gradings of clifford algebras and multivector structures let c v g be the real clifford algebra associated with the real vector space v e ndowed with a nondegenerate metric g i n this paper we study the class of z2 gradings of c v g which are somehow compatible with the multivector structure of the grassmann algebra over v a complete characterization for such z2 gradings is obtained by classifying all the even subalgebras coming from them an expression relating such subalgebras to the usual even part of c v g is also obtained finally we employ this framework to define spinor spaces and to parametrize all the possible signature changes on c v g by z2 gradings of this algebra

remarks on the inhomogeneous fractional nonlinear schrodinger equation using a sharp gagliardo nirenberg type inequality well posedness issues of the initial value problem for a fractional inhomogeneous schrodinger equation are investigated

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

the defocusing energy supercritical nls in four space dimensions abstract we consider a class of defocusing energy supercritical nonlinear schrodinger equations in four space dimensions following a concentration compactness approach we show that for 1 s c 3 2 any solution that remains bounded in the critical sobolev space h x s c r 4 must be global and scatter key ingredients in the proof include a long time strichartz estimate and a frequency localized interaction morawetz inequality

error bounds for reconstruction of a function f from a finite sequence operatorname sgn f t_i x_i consider reconstructing a function f t 0 leqq t leqq 1 from knowledge only of t i s i 1 leqq i leqq n where s i operatorname sgn f t i x i 1 leqq i leqq n and the x i are additive corruptions without the components x i f could not be reconstructed however for f continuous and for random uniform noise x masry and cambanis ieee trans inform theory it 26 1980 pp 50 58 it 27 1981 pp 84 96 show that f can be consistently estimated almost surely and in mean square as n to infty and establish rates of pointwise convergence through a somewhat different treatment in which the approximation of f t i is identified as a numerical integration problem rather than a statistical problem we obtain simple exact bounds on the error of estimation allow the noise x to be arbitrary random or deterministic and deal with the case of f having discontinuities the bounds yield substantially improved rates of convergence when the noise values x i

rough blowup solutions to the l 2 critical nls we study the singularity formation for the cubic focusing l 2 critical nonlinear schrodinger equation on mathbb r 2 in a series of recent works merle and raphael have completely described the so called log log blowup regime and proven its stability in the energy space h 1 our aim in this paper is to investigate the stability of this blowup regime under rough perturbations in the direction of developing a theory at the level of the critical space l 2 by blending the merle raphael techniques with the quantitative i method developed by colliander keel staffilani takaoka and tao for the study of the cauchy problem for rough data we obtain the stability of the log log regime in h s for all s 0

scattering of the energy critical nls with inverse square potential abstract we consider the initial value problem of the energy critical nonlinear schrodinger equation with an inverse square potential in dimension d 4 5 6 in the focusing case we achieve the global well posedness and scattering when the kinetic energy and energy of the solution are both below the threshold of the ground state solution in the defocusing case we prove that any h 1 bounded initial data leads to the global and scattering solution no radial assumption is made on the initial data

system identification from cheap qualitative output observations a new problem of system parameter estimation from qualitative data is stated and algorithms for its solution are proposed their convergence is proven

inward outward energy theory of non radial solutions to 3d semi linear wave equation abstract the topic of this paper is a semi linear energy sub critical defocusing wave equation t 2 u u u p 1 u in the 3 dimensional space with 3 p 5 we generalize inward outward energy theory and weighted morawetz estimates for radial solutions to the non radial case as an application we show that if 3 p 5 and 5 p 2 then the solution scatters as long as the initial data u 0 u 1 satisfy r 3 x 1 1 2 u 0 2 1 2 u 1 2 1 p 1 u 0 p 1 d x if p 3 we can also prove the scattering result if initial data u 0 u 1 are contained in the critical sobolev space and satisfy the inequality r 3 x 1 2 u 0 2 1 2 u 1 2 1 4 u 0 p 1 d x these assumptions on the decay rate of initial data as x are weaker than previously known scattering results in the non radial case

almost sure local well posedness and scattering for the 4d cubic nonlinear schrodinger equation abstract we consider the cauchy problem for the defocusing cubic nonlinear schrodinger equation in four space dimensions and establish almost sure local well posedness and conditional almost sure scattering for random initial data in h x s r 4 with 1 3 s 1 the main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear schrodinger equation which is inspired by certain function spaces used in the study of the schrodinger maps problem and is based on strichartz spaces as well as variants of local smoothing inhomogeneous local smoothing and maximal function spaces additionally we prove an almost sure scattering result for randomized radially symmetric initial data in h x s r 4 with 1 2 s 1

the solutions for nonlinear schrodinger equations abstract we consider the nonlinear schrodinger equation i u t u f u 0 f u u 2 m u n m n n x t r n r u 0 x u 0 x we give a formal solution of the cauchy problem by this formal solution we obtain that there exists a constant b independent of p n such that for any initial data u 0 h s b 0 s n 2 2 m n 1 4 n 2 s the schrodinger equation has a unique global solution u l r b p 2 s moreover when m 0 we give a unique weak solution of the cauchy problem in d distribution

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

stable blow up dynamics for energy supercritical wave equations we study the semilinear wave equation displaystyle partial t 2 psi delta psi vert psi vert p 1 psi for p 3 with radial data in three spatial dimensions there exists an explicit solution which blows up at t t 0 given by displaystyle psi t t x c p t t frac 2 p 1 where c p is a suitable constant we prove that the blow up described by psi t is stable in the sense that there exists an open set in a topology strictly stronger than the energy of radial initial data that leads to a solution which converges to psi t as t to t in the backward lightcone of the blow up point t r t 0

scattering theory for the defocusing energy supercritical nonlinear wave equation with a convolution abstract in this paper we study the global well posedness and scattering problem for the nonlinear wave equation with a convolution u t t u x u 2 u 0 in dimensions d 6 we show that if the solution u is apriorily bounded in the critical homogeneous sobolev space that is u l t h x s c h x s c 1 with s c 2 2 1 then u is global and it scatters the impetus to consider this problem stems from a series of recent works for the energy supercritical nonlinear wave equation and nonlinear schrodinger equation our analysis derived from the concentration compactness method to show that the proof of the global well posedness and scattering is reduced to disprove the existence of three scenarios the finite time blow up solution the soliton like solution and the low to high frequency cascade we note that authors preclude the finite time blow up solution to wave equation usually by the property of the finite speed of propagation in the previous literature however the finite speed of propagation is broken in the nonlocal nonlinear wave equation with a convolution nonlinearity for this we will initially establish the low regularity results for almost periodic solutions including the finite time blow up solutions and initially preclude the finite time blow up solutions by lowering the regularity

scattering for h 1 2 bounded solutions to the cubic defocusing nls in 3 dimensions we show that if a solution of the defocusing cubic nls in 3d remains bounded in the homogeneous sobolev norm of order 1 2 in its maximal interval of existence then the interval is infinite and the solution scatters no radial assumption is made

global well posedness and scattering for the defocusing mass critical nonlinear schrodinger equation for radial data in high dimensions we establish global well posedness and scattering for solutions to the defocusing mass critical pseudoconformal nonlinear schrodinger equation iut u u 4 nu for large spherically symmetric lx2 rn initial data in dimensions n 3 after using the concentration compactness reductions in 32 to reduce to eliminating blow up solutions that are almost periodic modulo scaling we obtain a frequency localized morawetz estimate and exclude a mass evacuation scenario somewhat analogously to 10 23 36 in order to conclude the argument

energy concentration of the focusing energy critical fnls abstract we consider the fractional nonlinear schrodinger equation fnls with non local dispersion and focusing energy critical hartree type nonlinearity x 2 u 2 u we consider the energy concentration phenomena of radial blowup solutions near the maximal existence time we use the concentration compactness approach of 19 for confined case and kinetic energy trapping approach of 21 for unconfined case

global well posedness and scattering for the defocusing energy supercritical cubic nonlinear wave equation abstract in this paper we consider the defocusing cubic nonlinear wave equation u t t u u 2 u 0 in the energy supercritical regime in dimensions d 6 with no radial assumption on the initial data we prove that if a solution satisfies an a priori bound in the critical homogeneous sobolev space throughout its maximal interval of existence that is u l t h x s c h x s c 1 then the solution is global and it scatters our analysis is based on the methods of the recent works of kenig and merle 2008 21 and killip and visan 2010 26 27 treating the energy supercritical nonlinear schrodinger and wave equations