are the incompressible 3d navier stokes equations locally ill posed in the natural energy space abstract an important open problem in the theory of the navier stokes equations is the uniqueness of the leray hopf weak solutions with l 2 initial data in this paper we give sufficient conditions for non uniqueness in terms of spectral properties of a natural linear operator associated to scale invariant solutions recently constructed in 8 if the spectral conditions are satisfied non uniqueness and ill posedness can appear for quite benign compactly supported data just at the borderline of applicability of the classical perturbation theory the verification of the spectral conditions seems to be approachable by numerical simulations which involve only smooth functions

on self similar singular solutions of the complex ginzburg landau equation we address the open problem of existence of singularities for the complex ginzburg landau equation using a combination of rigorous results and numerical computations we describe a countable family of self similar singularities our analysis includes the supercritical nonlinear schrodinger equation as a special case we also consider the problem of stability of these singularities 2001 john wiley sons inc

exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps abstract in this work we investigate stochastic partial differential equations with variable delays and jumps we derive by estimating the coefficients functions in the stochastic energy equality some sufficient conditions for exponential stability and almost sure exponential stability of energy solutions and generalize the results obtained by taniguchi t taniguchi the exponential stability for stochastic delay partial differential equations j math anal appl 331 2007 191 205 and wan and duan l wan j duan exponential stability of non autonomous stochastic partial differential equations with finite memory statist probab lett 78 5 2008 490 498 to cover a class of more general stochastic partial differential equations with jumps finally an illustrative example is established to demonstrate our established theory

integrability and superintegrability of the generalized n level many mode jaynes cummings and dicke models we analyze symmetries of the integrable generalizations of jaynes cummings and dicke models associated with simple lie algebras g and their reductive subalgebras gk t skrypnyk generalized n level jaynes cummings and dicke models classical rational r matrices and nested bethe ansatz j phys a math theor 41 475202 2008 we show that their symmetry algebras contain commutative subalgebras isomorphic to the cartan subalgebras of g which can be added to the commutative algebras of quantum integrals generated with the help of the quantum lax operators we diagonalize additional commuting integrals and constructed with their help the most general integrable quantum hamiltonian of the generalized n level many mode jaynes cummings and dicke type models using nested algebraic bethe ansatz

on uniqueness for the harmonic map heat flow in supercritical dimensions we examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d 3 it is shown that generically singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality we also discuss how uniqueness can be retrieved 2017 wiley periodicals inc

quantum integrable systems non skew symmetric r matrices and algebraic bethe ansatz we prove the integrability of the general quantum hamiltonian systems governed by an arbitrary non skew symmetric so 3 valued nondynamical classical r matrix with spectral parameters we consider the most interesting example of these quantum integrable systems namely the so 3 generalized gaudin systems in detail in the case of an arbitrary r matrix which is diagonal in the sl 2 basis we calculate the spectrum and the eigenvalues of the corresponding hamiltonians using the algebraic bethe ansatz technique

one dimensional stochastic burgers equation driven by levy processes abstract in this paper we proved the global existence and uniqueness of the strong weak and mild solutions for one dimensional burgers equation perturbed by a poisson form process a poisson form and q wiener process with the dirichlet bounded condition we also proved the existence of the invariant measure of these models

existence and uniqueness attraction for stochastic age structured population systems with diffusion and poisson jump in this paper the problems about the existence and uniqueness attraction for strong solution of stochastic age structured population systems with diffusion and poisson jump are considered under the non lipschitz condition with the lipschitz condition being considered as a special case the existence and uniqueness for such systems is firstly proved by using the burkholder davis gundy inequality b d g inequality and ito s formula and then by using a novel inequality technique some sufficient conditions ensuring the existence for the domain of attraction are established as another by product the exponential stability in mean square moment of strong solution for such systems can be also discussed

on global weak solutions to the cauchy problem for the navier stokes equations with large l3 initial data abstract the aim of the note is to discuss different definitions of solutions to the cauchy problem for the navier stokes equations with the initial data belonging to the lebesgue space l 3 r 3

on the volumorphism group the first conjugate point is always the hardest we find a simple local criterion for the existence of conjugate points on the group of volume preserving diffeomorphisms of a 3 manifold with the riemannian metric of ideal fluid mechanics in terms of an ordinary differential equation along each lagrangian path using this criterion we prove that the first conjugate point along a geodesic in this group is always pathological the differential of the exponential map always fails to be fredholm

the exponential stability for stochastic delay partial differential equations abstract in this paper we consider by the energy equality the exponential stability of energy solutions to a non linear stochastic functional partial differential equation with finite delay r 0 in separable hilbert spaces our method is very simple and is powerful for concrete stochastic delay partial differential equations

pth moment asymptotic stability for neutral stochastic functional differential equations with levy processes in this paper we will consider a class of neutral stochastic functional differential equations with levy processes levy processes contain a number of very important processes as special cases such as brownian motion the poisson process stable and self decomposable processes and subordinators and so on but its sample paths are discontinuity which makes the analysis more difficult in this paper we try to get over this difficulty the contributions of this paper are as follows a we will use lyapunov functional method to study the pth moment asymptotic stability and almost sure asymptotic stability of neutral stochastic functional differential equations with levy processes b under the result of a we will investigate two types of continuity of the solution continuous in the pth moment and continuous in probability finally we provide an example to illustrate the usefulness of the obtained results

classical yang baxter equations and quantum integrable systems quantum integrable models associated with nondegenerate solutions of classical yang baxter equations related to the simple lie algebras are investigated these models are diagonalized for rational and trigonometric solutions in the cases of sl n gl n o n and sp n algebras the analogy with the quantum inverse scattering method is demonstrated

weak and strong solutions of the complex ginzburg landau equation abstract the generalized complex ginzburg landau equation t a ra 1 iv a 1 i a 2 a has been proposed and studied as a model for turbulent dynamics in nonlinear partial differential equations it is a particularly interesting model in this respect because it is a dissipative version of the hamiltonian nonlinear schrodinger equation possessing solutions that form localized singularities in finite time in this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions appropriately defined weak solutions are established globally in time and unique strong solutions are found locally a new collection of a priori estimates are presented and we discuss the relationship of our results for the complex ginzburg landau equation to analogous issues for fluid turbulence described by the incompressible navier stokes equations

local structure of the self focusing singularity of the nonlinear schro dinger equation abstract for the cubic schrodinger equation in two dimensions we construct a family of singular solutions by perturbing slightly the dimension d 2 tod 2

z 2 graded classical r matrices and algebraic bethe ansatz applications to integrable models of quantum optics and nuclear physics we consider quantum integrable models based on the lie algebra gl n and non skew symmetric classical r matrices associated with z 2 gradings of gl n of the following type where among the considered models are gaudin type models with an external magnetic field used in nuclear physics to produce proton neutron bardeen cooper schrieer type models n level many mode jaynes cummings dicke type models of quantum optics matrix generalization of bose hubbard dimers etc we diagonalize the constructed models by means of the generalized nested bethe ansatz

the existence and asymptotic behaviour of energy solutions to stochastic age dependent population equations driven by levy processes in this paper we introduce a class of stochastic age dependent population equations driven by levy processes existence and uniqueness of energy solutions for stochastic age dependent population dynamic system are proved under lipschitz condition in hilbert space the moment boundedness of the approximate solution by the galerkin method is considered we discuss by using the energy equality the exponential stability theorems of the energy solution to stochastic age dependent population equations

about local continuity with respect to l_ 2 l2 initial data for energy solutions of the navier stokes equations in this paper we consider classes of initial data that ensure local in time hadamard well posedness of the associated weak leray hopf solutions of the three dimensional navier stokes equations in particular for any solenodial l 2 initial data u 0 belonging to certain subsets of vmo 1 mathbb r 3 we show that weak leray hopf solutions depend continuously with respect to small divergence free l 2 perturbations of the initial data u 0 on some finite time interval our main result is inspired by and improves upon previous work of the author barker in j math fluid mech 20 1 133 160 2018 and work of jean yves chemin commun pure appl math 64 12 1587 1598 2011 our method builds upon 4 9 in particular our method hinges on decomposition results for the initial data inspired by calderon trans am math soc 318 1 179 200 1990 together with use of persistence of regularity results the persistence of regularity statement presented may be of independent interest since it does not rely upon the solution or the initial data being in the perturbative regime

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

local in space estimates near initial time for weak solutions of the navier stokes equations and forward self similar solutions we show that the classical cauchy problem for the incompressible 3d navier stokes equations with 1 homogeneous initial data has a global scale invariant solution which is smooth for positive times our main technical tools are local in space regularity estimates near the initial time which are of independent interest

forward discretely self similar solutions of the navier stokes equations extending the work of jia and sverak on self similar solutions of the navier stokes equations we show the existence of large forward discretely self similar solutions

diagonalisation of gl n invariant transfer matrices and quantum n wave system lee model the algebraic bethe ansatz is constructed for the gl n invariant transfer matrices with arbitrary gl n spin for this purpose the notion of vacuum subspace is introduced it is shown that the gl n magnet can be interpreted as an integrable discrete approximation of vector or matrix nonlinear schrodinger models or of the quantum n wave system

generalized n level jaynes cummings and dicke models classical rational r matrices and algebraic bethe ansatz we construct integrable n level generalizations of jaynes cummings and dicke models corresponding to the simple reductive lie algebras of rank n we show that for each such lie algebra there exist many integrable jaynes cummings and dicke type models each of which is associated with the reductive subalgebras containing cartan subalgebra and is obtained via reduction from jaynes cummings or dicke type models with a maximal number of bosons we diagonalize the constructed jaynes cummings and dicke type hamiltonians in the physically most interesting case of using a nested algebraic bethe ansatz

general integrable n level many mode janes cummings dicke models and classical r matrices with spectral parameters using the technique of classical r matrices and quantum lax operators we construct the most general form of the quantum integrable n level many mode spin boson jaynes cummings dicke type hamiltonians describing an interaction of a molecule of n n level atoms with many modes of electromagnetic field and containing in general additional non linear interaction terms we explicitly obtain the corresponding quantum lax operators and spin boson analogs of the generalized gaudin hamiltonians and prove their quantum commutativity we investigate symmetries of the obtained models that are associated with the geometric symmetries of the classical r matrices and construct the corresponding algebra of quantum integrals we consider in detail three classes of non skew symmetric classical r matrices with spectral parameters and explicitly obtain the corresponding quantum lax operators and jaynes cummings dicke type hamiltonians depending on the considered r matrix

p euler equations and p navier stokes equations abstract we propose in this work new systems of equations which we call p euler equations and p navier stokes equations p euler equations are derived as the euler lagrange equations for the action represented by the benamou brenier characterization of wasserstein p distances with incompressibility constraint p euler equations have similar structures with the usual euler equations but the momentum is the signed p 1 th power of the velocity in the 2d case the p euler equations have streamfunction vorticity formulation where the vorticity is given by the p laplacian of the streamfunction by adding diffusion presented by laplacian of the velocity we obtain what we call p navier stokes equations if p the a priori energy estimates for the velocity and momentum have dual symmetries using these energy estimates and a time shift estimate we show the global existence of weak solutions for the p navier stokes equations in r d for p and p d 2 through a compactness criterion

global weak besov solutions of the navier stokes equations and applications we introduce a notion of global weak solution to the navier stokes equations in three dimensions with initial values in the critical homogeneous besov spaces dot b 1 frac 3 p p infty p 3 these solutions satisfy a certain stability property with respect to the weak ast convergence of initial conditions to illustrate this property we provide applications to blow up criteria minimal blow up initial data and forward self similar solutions our proof relies on a new splitting result in homogeneous besov spaces that may be of independent interest