quenching of flames by fluid advection we consider a simple scalar reaction advection diffusion equation with ignitiontype nonlinearity and discuss the following question what kinds of velocity profiles are capable of quenching any given flame provided the velocity s amplitude is adequately large even for shear flows the answer turns out to be surprisingly subtle if the velocity profile changes in space so that it is nowhere identically constant or if it is identically constant only in a region of small measure then the flow can quench any initial data but if the velocity profile is identically constant in a sizable region then the ensuing flow is incapable of quenching large enough flames no matter how much larger the amplitude of this velocity is the constancy region must be wider across than a couple of laminar propagating front widths the proof uses a linear pde associated to the nonlinear problem and quenching follows when the pde is hypoelliptic the techniques used allow the derivation of new nearly optimal bounds on the speed of traveling wave solutions c 2001 john wiley sons inc

enlargeability and index theory let m be a closed enlargeable spin manifold we show nontriviality of the universal index obstruction in the k theory of the maximal c algebra of the fundamental group of m our proof is independent of the injectivity of the baum connes assembly map for 1 m and relies on the construction of a certain infinite dimensional flat vector bundle out of a sequence of finite dimensional vector bundles on m whose curvatures tend to zero besides the well known fact that m does not carry a metric with positive scalar curvature our results imply that the classifying map m b 1 m sends the fundamental class of m to a nontrivial homology class in h b 1 m q this answers a question of burghelea 1983

model flames in the boussinesq limit the effects of feedback we have studied the fully nonlinear behavior of premixed flames in a gravitationally stratified medium subject to the boussinesq approximation the key results include the establishment of criteria for when such flames propagate as simple planar flames elucidation of scaling laws for the effective flame speed and a study of the stability properties of these flames the simplicity of some of our scaling results suggests that analytical work may further advance our understandings of buoyant flames

the sharp hardy uncertainty principle for schrodinger evolutions we give a new proof of hardy s uncertainty principle up to the end point case which is only based on calculus the method allows us to ex tend hardy s uncertainty principle to schrodinger equations with non constant coefficients we also deduce optimal gaussian decay bounds for solutions to these schrodinger equations

geometric optics approach to reaction diffusion equations the speed of propagation of traveling waves of reaction diffusion equations is calculated by an asymptotic method similar to geometrical optics this also works for reaction diffusion waves in random media

existence of traveling waves in the stokes boussinesq system for reactive flows abstract we consider the stokes boussinesq equations in a slanted that is not aligned with gravity s direction cylinder of any dimension and with an arbitrary rayleigh number we prove the existence of a non planar traveling wave solution propagating at a constant speed and satisfying the dirichlet boundary condition in the velocity and the neumann condition in the temperature

uniqueness properties of solutions of schrodinger equations under suitable assumptions on the potentials v and a we prove that if u c 0 1 h1 is a solution of the linear schrodinger equation i t x u vu a xu on rd 0 1 and if u 0 in x r 0 1 for some r 0 then u 0 in rd 0 1 as a consequence we obtain uniqueness properties of solutions of nonlinear schrodinger equations of the form i t x u g x t u u xu xu on rd 0 1 where g is a suitable nonlinear term the main ingredient in our proof is a carleman inequality of the form e x1 v lx2lt2 e x1 xv bx 2lt2 c e x1 i t x v bx1 2lt2 for any v c r h1 with v t 0 for t 0 1 in this inequality bx 2 and bx1 2 are banach spaces of functions on rd and e x1 is a suitable weight

hardy uncertainty principle and unique continuation properties of covariant schrodinger flows abstract we prove a logarithmic convexity result for exponentially weighted l 2 norms of solutions to electromagnetic schrodinger equation without needing to assume smallness of the magnetic potential as a consequence we can prove a unique continuation result in the style of the hardy uncertainty principle which generalizes the analogous theorems which have been recently proved by escauriaza kenig ponce and vega

existence of planar flame fronts in convective diffusive periodic media we prove the existence of planar travelling wave solutions in a reaction diffusion convection equation with combustion nonlinearity and self adjoint linear part in rn n 1 the linear part involves diffusion convection terms and periodic coefficients these travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media we use the method of continuation spectral theory and the maximum principle uniqueness and monotonicity properties of solutions follow from a previous paper these properties are essential to overcoming the lack of compactness and the degeneracy in the problem

on unique continuation for the generalized schrodinger equations abstract in this paper we study the unique continuation properties of solutions of the generalized schrodinger equations we prove the carleman inequalities as well as the exponential decay weighted estimates as a consequence we obtain sufficient conditions on the behavior of the solution u at two different times t 0 0 and t 1 1 which guarantee that u 0 is the unique solution

unique continuation properties of the higher order nonlinear schrodinger equations in one dimension abstract in this paper we study the unique continuation properties of the higher order nonlinear schrodinger equations in one dimension and show exponential decay weighted estimates as well as a l p type carleman estimate based on the littlewood paley theory as a consequence we obtain that if u is a solution of the schrodinger equation such that there exists x 0 with supp u 0 supp u 1 x 0 or x 0 then u 0 in r 0 1

existence and nonexistence of traveling waves and reaction diffusion front propagation in periodic media we review the existence results of traveling wave solutions to the reaction diffusion equations with periodic diffusion convection coefficients and combustion bistable nonlinearities we prove that whenever traveling waves exist the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions in the case of bistable nonlinearity and one space dimension we give an example of nonexistence of traveling waves which causes quenching localization of wavefront propagation quenching localization only occurs when the variations of the media from their constant mean values are large enough our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example finally we comment on the role of the effective wave speeds in determining the effective wavefront equation hamilton jacobi equation of the reactiondiffusion equations under the small diffusion fast reaction limit with a formal geometric optics expansion

gaussian decay of harmonic oscillators and related models abstract we prove that the decay of the eigenfunctions of harmonic oscillators uniform electric or magnetic fields is not stable under 0 order complex perturbations even if bounded of these hamiltonians in the sense that we can produce solutions to the evolutionary schrodinger flows associated to the hamiltonians with a stronger gaussian decay at two distinct times we then characterize in a quantitative way the sharpest possible gaussian decay of solutions as a function of the oscillation frequency or the strength of the field depending on the hamiltonian which is considered this is connected to the hardy s uncertainty principle for free schrodinger evolutions

the ivp for a higher dimensional version of the benjamin ono equation in weighted sobolev spaces abstract we study the initial value problem associated to a higher dimensional version of the benjamin ono equation our purpose is to establish local well posedness results in weighted sobolev spaces and to determinate according to them some sharp unique continuation properties of the solution flow in consequence optimal decay rate for this model is determined a key ingredient is the deduction of a new commutator estimate involving riesz transforms

homology of finite k area we use gromov s k area to define homology groups for compact smooth manifolds in fact this theory collects obstructions to the enlargeability of the manifold and its nontrivial submanifolds moreover using the k area homology we can rephrase some classic results about positive scalar curvature we also show that the functor of k area homology determines the functor of singular homology on the category of compact smooth manifolds

uniqueness properties for discrete equations and carleman estimates abstract using carleman estimates we give a lower bound for solutions to the discrete schrodinger equation in both dynamic and stationary settings that allows us to prove uniqueness results under some assumptions on the decay of the solutions

unique continuation for schrodinger evolutions with applications to profiles of concentration and traveling waves we prove unique continuation properties for solutions of the evolution schrodinger equation with time dependent potentials as an application of our method we also obtain results concerning the possible concentration profiles of blow up solutions and the possible profiles of the traveling waves solutions of semi linear schrodinger equations

on the existence of traveling waves in the 3d boussinesq system we extend earlier work on traveling waves in premixed flames in a gravitationally stratified medium subject to the boussinesq approximation for three dimensional channels not aligned with the gravity direction and under the dirichlet boundary conditions in the fluid velocity it is shown that a non planar traveling wave corresponding to a non zero reaction exists under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the prandtl and rayleigh numbers or when the advection term in the flow equations is neglected

uniqueness properties of solutions to schrodinger equations nx j 1 2 u x 2 x 0 harmonic functions in the unit ball x r n x 2 since harmonic functions are still real analytic in x r n x 1 in fact it is well known that if p x d is a linear elliptic differential operator with real analytic coefficients andp x d u 0 in a open set r n then u is real analytic in hence the s u c p also holds for such solutions through the work of hadamard 28 on the uniqueness of the cauchy problem which is closely related to the strong unique continuation property discussed earlier it became clear for applications in nonlinear problems that it would be desirable to establish the strong unique continuation property for operators whose coefficients are not necessarily real analytic oreven c the first results in this direction were found in the pioneering work of carleman 9 when

hardy uncertainty principle convexity and parabolic evolutions we give a new proof of the l2 version of hardy s uncertainty principle based on calculus and on its dynamical version for the heat equation the reasonings rely on new log convexity properties and the derivation of optimal gaussian decay bounds for solutions to the heat equation with gaussian decay at a future time we extend the result to heat equations with lower order variable coefficient

systoles des surfaces plates singulieres de genre deux the extremal metrics for the isosystolic problem on surfaces of genus two are studied in this paper we show that contrary to e calabi s conjecture in genus three no flat metric with conical singularities is extremal for this problem in genus two

stability of planar traveling waves in a keller segel equation on an infinite strip domain abstract we consider a simplified model of tumor angiogenesis described by a keller segel equation on the two dimensional domain x y r s where s is the circle of perimeter it is known that the system allows planar traveling wave solutions of an invading type in case that is sufficiently small we establish the nonlinear stability of traveling wave solutions in the absence of chemical diffusion if the initial perturbation is sufficiently small in some weighted sobolev space when chemical diffusion is present it can be shown that the system is linearly stable lastly we prove that any solution with our front condition eventually becomes planar under certain regularity conditions

traveling fronts in a reactive boussinesq system bounds and stability this paper considers a simplified model of active combustion in a fluid flow with the reaction influencing the flow the model consists of a reaction diffusion advection equation coupled with an incompressible navier stokes system under the boussinesq approximation in an infinite vertical strip we prove that for certain ignition nonlinearities including all that are c2 and for any domain width planar traveling front solutions are nonlinearly and exponentially stable within certain weighted h2 spaces provided that the rayleigh number is small enough the same result holds for bistable nonlinearities in unweighted h2 spaces we also obtain uniform bounds on the nusselt number the bulk burning rate and the average maximum vertical velocity for chemistries that include bistable and ignition nonlinearities

positive scalar curvature and 10 8 type inequalities on 4 manifolds with periodic ends we show 10 8 type inequalities for some end periodic 4 manifolds which have positive scalar curvature metrics on the ends as an application we construct a new family of closed 4 manifolds which do not admit positive scalar curvature metrics

the ivp for the benjamin ono equation in weighted sobolev spaces abstract in this work we continue our study initiated in fonseca and ponce 2011 11 on the uniqueness properties of real solutions to the ivp associated to the benjamin ono bo equation in particular we shall show that the uniqueness results established in fonseca and ponce 2011 11 do not extend to any pair of non vanishing solutions of the bo equation also we shall prove that the uniqueness result established in fonseca and ponce 2011 11 under a hypothesis involving information of the solution at three different times cannot be relaxed to two different times

fronts in reactive convection bounds stability and instability this paper examines a simplified active combustion model in which the reaction influences the flow we consider front propagation in a reactive boussinesq system in an infinite vertical strip nonlinear stability of planar fronts is established for narrow domains when the rayleigh number is not too large planar fronts are shown to be linearly unstable with respect to long wavelength perturbations if the rayleigh number is sufficiently large we also prove uniform bounds on the bulk burning rate and the nusselt number in the kpp reaction case 2003 wiley periodicals inc

on three manifolds dominated by circle bundles we determine which three manifolds are dominated by products the result is that a closed oriented connected three manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of s2 s1 this characterization can also be formulated in terms of thurston geometries or in terms of purely algebraic properties of the fundamental group we also determine which three manifolds are dominated by non trivial circle bundles and which three manifold groups are presentable by products