global solutions and time decay of the non cutoff vlasov maxwell boltzmann system in the whole space the global in time classical solutions to the cauchy problem of the vlasov maxwell boltzmann system near maxwellians are obtained under the lower regularity index assumption and the weaker smallness condition on the initial perturbation in comparison with the work duan et al in kinet relat models 6 1 159 204 2013 in particular we show the relation between time decay rates and spatial derivatives of solutions our analysis relies on a refined energy estimates and the interpolation techniques between negative sobolev norms and positive sobolev norms without linear decay analysis

global smooth solutions and stabilization of nonlinear elastodynamic systems with locally distributed dissipation we consider the existence of global solutions of the nonlinear elastodynamic system with a locally distributed damping in a bounded domain we assume that the energy function satisfies the strong ellipticity condition at the zero equilibrium the local dissipation is in the form a x u where the nonnegative function a x is only positive on a small portion of the domain we show the existence of global smooth solutions when initial data are small in particular we obtain the exponential decay of the energy which implies the exponential stabilization of the system by internal feedback

energy method for boltzmann equation abstract a basic simple energy method for the boltzmann equation is presented here it is based on a new macro micro decomposition of the boltzmann equation as well as the h theorem this allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method as an illustration we apply the method for the study of the time asymptotic nonlinear stability of the global maxwellian states previous energy method starting with grad and finishing with ukai involves the spectral analysis and regularity of collision operator through sophisticated weighted norms

a generalized interpolation inequality and its application to the stabilization of damped equations in this paper we establish a generalized holder s or interpolation inequality for weighted spaces in which the weights are non necessarily homogeneous we apply it to the stabilization of some damped wave like evolution equations this allows obtaining explicit decay rates for smooth solutions for more general classes of damping operators in particular for 1 d models we can give an explicit decay estimate for pointwise damping mechanisms supported on any strategic point

an elementary proof for a compact imbedding result in generalized electromagnetic theory propriete de plongement compact et certaines de ses consequences on etudie la validite du plongement suivant r 9 g e q 1 d q g l 2 q loc g

on the linear damped wave equation abstract in this work we estimate the spectrum of the linear damped wave semigroup under homogeneous dirichlet boundary conditions by using the principal eigenvalue of an elliptic operator related to the equation our estimate is optimal for real eigenvalues then we analyze the behavior of the estimate as the damping amplitude grows to infinity when the damping changes of sign we extend a result of freitas 5 to show that the semigroup possesses at least two real eigenvalues greater than one if the amplitude is sufficiently large in particular the trivial state is unstable in strong contrast with the sign definited case finally we characterize the limiting behavior of the real eigenpairs which originate the inestability of the trivial state this analysis is based upon the behavior of the principal eigenpair of a singular perturbation problem at the singular limit our theory is of interest by itself and it has many applications to reaction diffusion systems c f for instance 6

regularizing effect and local existence for the non cutoff boltzmann equation the boltzmann equation without grad s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non integrable angular singularity of the cross section however even though this has been justified satisfactorily for the spatially homogeneous boltzmann equation it is still basically unsolved for the spatially inhomogeneous boltzmann equation in this paper by sharpening the coercivity and upper bound estimates for the collision operator establishing the hypo ellipticity of the boltzmann operator based on a generalized version of the uncertainty principle and analyzing the commutators between the collision operator and some weighted pseudo differential operators we prove the regularizing effect in all time space and velocity variables on the solutions when some mild regularity is imposed on these solutions for completeness we also show that when the initial data has this mild regularity and a maxwellian type decay in the velocity variable there exists a unique local solution with the same regularity so that this solution acquires the c regularity for any positive time

the damped string problem revisited abstract we revisit the damped string equation on a compact interval with a variety of boundary conditions and derive an infinite sequence of trace formulas associated with it employing methods familiar from supersymmetric quantum mechanics we also derive completeness and riesz basis results with parentheses for the associated root functions under less smoothness assumptions on the coefficients than usual using operator theoretic methods rather than detailed eigenvalue and root function asymptotics only

super golden gates for pu 2 abstract to each of the symmetry groups of the platonic solids we adjoin a carefully designed involution yielding topological generators of pu 2 which have optimal covering properties as well as efficient navigation these are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups the generators give super efficient 1 qubit quantum gates and are natural building blocks for the design of universal quantum gates

decay of solutions of the wave equation with arbitrary localized nonlinear damping abstract we study the problem of decay rate for the solutions of the initial boundary value problem to the wave equation governed by localized nonlinear dissipation and without any assumption on the dynamics i e the control geometric condition is not satisfied we treat separately the autonomous and the non autonomous cases providing regular initial data without any assumption on an observation subdomain we prove that the energy decays at last as fast as the logarithm of time our result is a generalization of lebeau in a boutet de monvel v marchenko eds algebraic and geometric methods in mathematical physics kluwer academic publishers dordrecht the netherlands 1996 pp 73 result in the autonomous case and nakao adv math sci appl 7 1 1997 317 work in the non autonomous case in order to prove that result we use a new method based on the fourier bross iaglintzer fbi transform

weighted l2 estimates for dissipative wave equations with variable coefficients abstract we establish weighted l 2 estimates for the wave equation with variable damping u t t u a u t 0 in r n where a x a 0 1 x with a 0 0 and 0 1 in particular we show that the energy of solutions decays at a polynomial rate t n 2 1 if a x a 0 x for large x we derive these results by strengthening significantly the multiplier method this approach can be adapted to other hyperbolic equations with damping

sharp energy decay estimates for the wave equation with a local degenerate dissipation we derive a fast decay estimate for the wave equation with a local degenerate dissipation of the type a x u t in a bounded domain w the dissipative coefficient a x is a nonnegative function only on a neighborhood of some part of the boundary w and may vanish somewhere in w the results obtained extend and improve on earlier results of nakao as well as those of tcheugoue tebou the method of proof is based on multipliers technique and some modified interpolation and difference inequalities

well posedness and stability of a hinged plate equation with a localized nonlinear structural damping abstract we consider an n dimensional plate equation in a bounded domain with a locally distributed nonlinear dissipation involving the laplacian the dissipation is effective in a neighborhood of a suitable portion of the boundary when the space dimension equals two the associated linear equation corresponds to the plate equation with a localized viscoelastic or structural damping first we prove existence uniqueness and smoothness results then using an appropriate perturbed energy coupled with multiplier technique we directly prove exponential and polynomial decay estimates for the underlying energy to the author s best knowledge the perturbed energy approach is new in the framework of stabilization of second order evolution equations with locally distributed damping

exponential decay of solutions of maxwell s equations coupled with a first order ordinary differential equation for the polarization this paper is concerned with maxwell s equations involving a first order ordinary differential equation for the dielectric polarization the main goal is the exponential convergence of the electromagnetic field to an asymptotic state which is determined by the initial data this is achieved by the construction of a suitable energy functional involving a vector potential which provides an additional dissipative term for the magnetic field

strong decay for wave equations with nonlinear nonmonotone damping thus the energy is nonincreasing and the term p represents damping here and always the ymbols i ii denote the norm on the sobolev space wk p o 1 the classical example p d u d a positive constant shows that under reasonable coercivity hypotheses concerning f any solution u u decays to zero exponentially with respect to the trong topology on the energy space hd o 1 x l o 1 this property is preserved even if tl d x with d 0 on a certain nonvoid subinterval i e 0 l see l 4 moreover the same is true if p d g u d as above and g is globally lipschitz and satisfies the growth condition g w w 2 const w2 wer 1 2 cf 3 in the case f 0 3 d x g ut g monotone the strong decay to zero follows from a variant of the lasalle invariance principle see 5 91 in some particular cases the speed of convergence is obtained lo

decay estimates for some semilinear damped hyperbolic problems let be a bounded open domain in rn g r r a non decreasing continuous function such that g 0 0 and h e lloc1 r l2 under suitable assumptions on g and h the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u lu g u h t x as t the results obtained by use of differential inequalities can be applied to the case of the semilinear wave equation u u delta u g text u t text h text in r times omega text u text 0 on r times partial omega in r u 0 onr for instance if g s c left s right p 1 s d left s right q 1 s with c d 0 and 1 p q n 2 q n 2 then if h in l infty r l 2 omega all solutions are bounded in the energy space for t 0 and if u v are two such solutions the energy norm of u t v t decays like t 1 p 1 as t

total energy decay for the wave equation in exterior domains with a dissipation near infinity we show that the energy of solutions to the initial boundary value problem for the wave equation in exterior domains with a dissipation which is localized only near infinity tends to zero as the time goes to infinity we do not make any geometrical condition like star shapedness on the boundary

oscillatory phenomena associated to semilinear wave equations in one spatial dimension let g be a nonincreasing odd c function and i 0 we establish that for any solution u s c r hq 0 of the equation utt uxx g u 0 and any io 0 l the function t u t xo satisfies the following alternative either u t x0 0 vt g r or vo r there exist t and t2 in a a 21 such that u ti xo 0 and u t2 xo 0 we study the structure of the set of points satisf ing the first possibility we give analogous results for ux and for some other homogeneous boundary conditions

the vlasov maxwell boltzmann system near maxwellians perhaps the most fundamental model for dynamics of dilute charged particles is described by the vlasov maxwell boltzmann system in which particles interact with themselves through collisions and with their self consistent electromagnetic field despite its importance no global in time solutions weak or strong have been constructed so far it is shown in this article that any initially smooth periodic small perturbation of a given global maxwellian which preserves the same mass total momentum and reduced total energy 22 leads to a unique global in time classical solution for such a master system the construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation the linear collision operator l not its time integration is positive definite for any solutionf t x v with small amplitude to the vlasov maxwell boltzmann system 8 and 12 as a by product such an estimate also yields an exponential decay for the simpler vlasov poisson boltzmann system 24

negative sobolev spaces and the two species vlasov maxwell landau system in the whole space abstract a global solvability result of the cauchy problem of the two species vlasov maxwell landau system near a given global maxwellian is established by employing an approach different than that of 2 compared with that of 2 the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker our analysis does not rely on the decay of the corresponding linearized system and the duhamel principle and thus it can be used to treat the one species vlasov maxwell landau system for the case of 3 and the one species vlasov maxwell boltzmann system for the case of 1 1 to deduce the global existence results together with the corresponding temporal decay estimates

on a first order hyperbolic system including bean s model for superconductors with displacement current abstract this paper deals with a class of first order systems with generally multivalued damping terms including bean s model for superconductors with displacement current the main goal is to prove the convergence of solutions to stationary states as t

stability of solutions for some classes of nonlinear damped wave equations we consider two classes of semilinear wave equations with nonnegative damping which may be of type on off or integrally positive in both cases we give a sufficient condition for the asymptotic stability of the solutions in the case of integrally positive damping we show that such a condition is also necessary

the rate at which energy decays in a viscously damped hinged euler bernoulli beam we study the best decay rate of the solutions of a damped euler bernoulli beam equation with a homogeneous dirichlet boundary conditions we show that the fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup if the damping coefficient is in l infty 0 1

semiclassical analysis of general second order elliptic operators on bounded domains in this work we ascertain the semiclassical behavior of the fundamental energy and the ground state of an arbitrary second order elliptic operator not necessarily selfadjoint on a bounded domain our analysis provides us with substantial improvements of many previous results found in the context of quantum mechanics for c perturbations of the laplacian

optimal large time behavior of the vlasov maxwell boltzmann system in the whole space in this paper we study the large time behavior of classical solutions to the two species vlasov maxwell boltzmann system in the whole space input amssym bbb r 3 the existence of global in time nearby maxwellian solutions is known from strain in 2006 however the asymptotic behavior of these solutions has been a challenging open problem building on our previous work on time decay for the simpler vlasov poisson boltzmann system we prove that these solutions converge to the global maxwellian with the optimal decay rate of o t 3 2 3 2r in the l l norm for any 2 r if initial perturbation is smooth enough and decays in space velocity fast enough at infinity moreover some explicit rates for the electromagnetic field tending to 0 are also provided 2011 wiley periodicals inc

stability analysis of dissipative systems subject to nonlinear damping via lyapunov techniques in this paper we provide a general strategy based on lyapunov functionals to analyze global asymptotic stability of linear infinite dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymptotically stable with a linear damping to do so we use the fact that for any linear infinite dimensional system that is globally exponentially stable there exists a lyapunov functional then we derive a lyapunov functional for the nonlinear system which is the sum of a lyapunov functional coming from the linear system and another term that compensates the nonlinearity our results are then applied to the linearized korteweg de vries equation and some wave equations

algorithmic enumeration of ideal classes for quaternion orders we provide algorithms to count and enumerate representatives of the right ideal classes of an eichler order in a quaternion algebra defined over a number field we analyze the run time of these algorithms and consider several related problems including the computation of two sided ideal classes isomorphism classes of orders connecting ideals for orders and ideal principalization we conclude by giving the complete list of definite eichler orders with class number at most 2

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

asymptotic behavior of electromagnetic waves in nonlinear dielectric media this paper is concerned with maxwell s equations including a nonlinear dielectric polarization and a generally nonlinear law for the electric current the large time behavior of the electromagnetic field is investigated in the case of a bounded spatial domain as well as in the exterior domain case

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

weak asymptotic decay via a relaxed invariance principle for a wave equation with nonlinear non monotone damping this paper considers the problem of asymptotic decay as t of solutions of the wave equation u tt u a x u t u t x a bounded open connected set in n n 1 with smooth boundary u 0 on the nonlinear function is not assumed to be globally lipschitz continuous 0 y 2 y n 1 0 y 1 y 1 y n 1 0 for all y n 1 is not assumed to be monotone in y 1 under additional restrictions on the kernel of conditions are given which imply that u u t converges to 0 0 weakly in h h 1 0 l 2 as t the work generalises earlier results of dafermos and haraux where strong decay in h as t was obtained in the case y 1 y n 1 q y 1 q monotone on

global existence and full regularity of the boltzmann equation without angular cutoff we prove the global existence and uniqueness of classical solutions around an equilibrium to the boltzmann equation without angular cutoff in some sobolev spaces in addition the solutions thus obtained are shown to be non negative and c in all variables for any positive time in this paper we study the maxwellian molecule type collision operator with mild singularity one of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator this norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated h theorem

stabilization of trajectories for some weakly damped hyperbolic equations in this paper we study the phenomenon of stabilization of trajectories for a wave equation in a bounded open domain endowed with a weak dissipative mechanism by stabilization we mean that the self oscillations induced by the wave equation are damped out asymptotically and so we are left when time tends to infinity either with an equilibrium if the system is autonomous or with a forced oscillation if the system was submitted to an exterior periodic or almost periodic excitation the common point between the two situations is the extinction of selfoscillations in the presence of a weak damping and it is extremely satisfactory to discover that the difficulty is overcome in both cases by the use of one simple and elegant idea due to c m dafermos and which appeared in s precisely in the study of some partially damped hyperbolic systems the paper is divided in four sections in section 1 we recall the idea of s which will be used in sections 2 and 3 in section 2 we generalize the result of s proposition 2 31 in a framework which allows the existence of nontrivial equilibria a simple example from continuum mechanics is given in section 3 we apply the same device to establish total damping in the sense of s for a wave equation with single valued damping term and almost periodic forcing the result which we obtain is not covered by the general theorems of 7 or 8 a simple example is studied in section 4 we give two counterexamples which show that the result of section 3 cannot be extended to the framework of weak solutions or to other boundary conditions in each specific case a careful analysis will be necessary the notation which is used throughout the text is the standard one with 145 0022 0396185 3 00

the boltzmann equation without angular cutoff in the whole space i global existence for soft potential it is known that the singularity in the non cutoff cross section of the boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable by defining and analyzing a non isotropy norm which precisely captures the dissipation in the linearized collision operator we first give a new and precise coercivity estimate for the non cutoff boltzmann equation for general physical cross sections then the cauchy problem for the boltzmann equation is considered in the framework of small perturbation of an equilibrium state in this part for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator we derive some new functional estimates on the nonlinear collision operator together with the coercivity estimates we prove the global existence of classical solutions for the boltzmann equation in weighted sobolev spaces

singular internal stabilization of the wave equation abstract we consider an initial and boundary value problem for the one and two dimensional wave equation with nonlinear damping concentrated on an interior point and respectively on an interior curve in the two dimensional case our main result asserts that generically i e for almost all interior curves the solutions decay to zero in the energy space when the domain is strictly convex we show that whatever the interior curve is the decay is not uniform we generalize in this way results known in one space dimension our main improvement of existing one dimensional results consists in giving sharp decay rates provided that the initial data are regular and the damping term is linear a crucial intermediate step is the proof of a generalization of ingham s inequality on nonharmonic fourier series

optimal design of the damping set for the stabilization of the wave equation abstract we consider the problem of optimizing the shape and position of the damping set for the internal stabilization of the linear wave equation in r n n 1 2 in a first theoretical part we reformulate the problem into an equivalent non convex vector variational one using a characterization of divergence free vector fields then by means of gradient young measures we obtain a relaxed formulation of the problem in which the original cost density is replaced by its constrained quasi convexification this implies that the new relaxed problem is well posed in the sense that there exists a minimizer and in addition the infimum of the original problem coincides with the minimum of the relaxed one in a second numerical part we address the resolution of the relaxed problem using a first order gradient descent method we present some numerical experiments which highlight the influence of the over damping phenomena and show that for large values of the damping potential the original problem has no minimizer we then propose a penalization technique to recover the minimizing sequences of the original problem from the optimal solution of the relaxed one

on a uniqueness theorem of l amerio and g prouse we give two generalizations of a theorem of l amerio and g prouse concerning uniqueness of the almost periodic solution of the wave equation with a local multivalued damping term and almost periodic forcing counterexamples are given which show that these results may fail if some hypotheses of the theorems are dropped they also show that the two generalizations are relatively independent

stabilization of the wave equation with localized nonlinear damping abstract we prove some decay estimates of the energy of the wave equation in a bounded domain the damping is nonlinear and is effective only in a neighborhood of a suitable subset of the boundary the method of proof is direct and is based on the multipliers technique on some integral inequalities due to haraux and komornik and on a judicious idea of conrad and rao

stabilization of the elastodynamic equations with a degenerate locally distributed dissipation we consider the dynamic elasticity equations with a locally distributed damping in a bounded domain the local dissipation of the form a x yt involves coefficients a that vanish on a negligible portion of the subset where the damping is effective using multiplier techniques interpolation inequalities and a judicious application of holder inequality we prove sharp energy decay estimates the results obtained generalize and improve on earlier works by nakao and the author in the framework of the ordinary wave equation

existence of global classical solutions of the initial boundary value problem for some nonlinear wave equations in fact we shall assume a x u is strictly increasing in u and also impose a certain smallness condition on the initial data zq u and the forcing term f but it is weaker compared with earlier works see theorem 1 1 for a moment we assume further g x u u 3 0 for simplicity when n 1 the global existence of the smooth solutions for o l 0 2 is easily proved by a standard energy method cf strauss 121 nakao 7 when iv 2 we can also prove a similar result if b x o satisfies the conditions 8 x u du b 0

stabilization of the euler bernoulli plate with variable coefficients by nonlinear internal feedback we consider the energy decay for solutions of the euler bernoulli plate equation with variable coefficients where a nonlinear internal feedback acts in a suitable subregion of the domain the riemannian geometric method is used to deal with variable coefficient problems when the feedback region has a structure similar to that for the wave equation with constant coefficients we establish the stabilization of the system in the case of fixed boundary conditions several energy decay rates are established according to various growth restrictions on the nonlinear feedback near the origin and at infinity we further show that unlike for the case of constant coefficients choices of such feedback regions depend not only on the type of boundary conditions but also on the curvature of a riemannian metric based on the coefficients of the system