In this paper we consider time dependent Schrodinger equations on the one-dimensional torus $T := R /(2 pi Z)$ of the form $partial_t u = ii {cal V}(t)[u]$ where ${cal V}(t)$ is a time dependent, self-adjoint pseudo-differential operator of the form ${cal V}(t) = V(t, x) |D|^M + {cal W}(t)$, $M > 1$, $|D| := sqrt{- partial_{xx}}$, $V$ is a smooth function uniformly bounded from below and ${cal W}$ is a time-dependent pseudo-differential operator of order strictly smaller than $M$. We prove that the solutions of the Schrodinger equation $partial_t u = ii {cal V}(t)[u]$ grow at most as $t^e$, $t to + infty$ for any $e > 0$. The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field $ii {cal V}(t)$ which uses Egorov type theorems and pseudo-differential calculus.
本文考虑一维环面上的时变薛定谔方程 $T := R /(2 pi Z)$ 形式的 $partial_t u = ii {cal V}(t)[u]$ 在哪里 ${cal V}(t)$ 一个时间相关的，自伴随的伪微分算子是这样的形式吗 ${cal V}(t) = V(t, x) |D|^M + {cal W}(t)$， $M > 1$， $|D| := sqrt{- partial_{xx}}$， $V$ 光滑函数从下到上有界吗 ${cal W}$ 一个时间相关的伪微分算子的阶是否严格小于 $M$． 我们证明了薛定谔方程的解 $partial_t u = ii {cal V}(t)[u]$ 最多成长为 $t^e$， $t to + infty$ 对于任何 $e > 0$． 证明是基于对常数系数的简化，直到平滑向量场的余数 $ii {cal V}(t)$ 它使用了Egorov型定理和伪微分学。
{"title":"On the growth of Sobolev norms for a class of linear Schrödinger equations on the torus with superlinear dispersion","authors":"Riccardo Montalto","doi":"10.3233/ASY-181470","DOIUrl":"https://doi.org/10.3233/ASY-181470","url":null,"abstract":"In this paper we consider time dependent Schrodinger equations on the one-dimensional torus $T := R /(2 pi Z)$ of the form $partial_t u = ii {cal V}(t)[u]$ where ${cal V}(t)$ is a time dependent, self-adjoint pseudo-differential operator of the form ${cal V}(t) = V(t, x) |D|^M + {cal W}(t)$, $M > 1$, $|D| := sqrt{- partial_{xx}}$, $V$ is a smooth function uniformly bounded from below and ${cal W}$ is a time-dependent pseudo-differential operator of order strictly smaller than $M$. We prove that the solutions of the Schrodinger equation $partial_t u = ii {cal V}(t)[u]$ grow at most as $t^e$, $t to + infty$ for any $e > 0$. The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field $ii {cal V}(t)$ which uses Egorov type theorems and pseudo-differential calculus.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"64 1","pages":"85-114"},"PeriodicalIF":0.0,"publicationDate":"2017-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90571939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator $-Delta$ in the perforated domain $Omegasetminus bigcup_{ iin 2varepsilonmathbb Z^d }B_{a_varepsilon}(i),$ $a_varepsilonllvarepsilon,$ to the limit operator $-Delta+mu_{iota}$ on $L^2(Omega)$, where $mu_iotainmathbb C$ is a constant depending on the choice of boundary conditions. This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problems on Domains with Many Tiny Holes. Pro c. Japan Acad., 61, Ser. A (1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.
{"title":"Norm-resolvent convergence in perforated domains","authors":"K. Cherednichenko, P. Dondl, F. Rösler","doi":"10.3233/ASY-181481","DOIUrl":"https://doi.org/10.3233/ASY-181481","url":null,"abstract":"For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator $-Delta$ in the perforated domain $Omegasetminus bigcup_{ iin 2varepsilonmathbb Z^d }B_{a_varepsilon}(i),$ $a_varepsilonllvarepsilon,$ to the limit operator $-Delta+mu_{iota}$ on $L^2(Omega)$, where $mu_iotainmathbb C$ is a constant depending on the choice of boundary conditions. \u0000This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problems on Domains with Many Tiny Holes. Pro c. Japan Acad., 61, Ser. A (1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"19 1","pages":"163-184"},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84590385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose several continuous data assimilation (downscaling) algorithms based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We show that for sufficiently large choices of the control parameter and resolution and assuming that the observed data is error-free, the solution of the controlled system converges exponentially (in $L^2$ and $H^1$ norms) to the reference solution independently of the initial data chosen for the controlled system. Furthermore, we show that a similar result holds when controls are placed only on the horizontal (or vertical) variables, or on a single Els"asser variable, under more restrictive conditions on the control parameter and resolution. Finally, using the data assimilation system, we show the existence of abridged determining modes, nodes and volume elements.
{"title":"Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields","authors":"A. Biswas, Joshua Hudson, Adam Larios, Yuan Pei","doi":"10.3233/ASY-171454","DOIUrl":"https://doi.org/10.3233/ASY-171454","url":null,"abstract":"We propose several continuous data assimilation (downscaling) algorithms based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We show that for sufficiently large choices of the control parameter and resolution and assuming that the observed data is error-free, the solution of the controlled system converges exponentially (in $L^2$ and $H^1$ norms) to the reference solution independently of the initial data chosen for the controlled system. Furthermore, we show that a similar result holds when controls are placed only on the horizontal (or vertical) variables, or on a single Els\"asser variable, under more restrictive conditions on the control parameter and resolution. Finally, using the data assimilation system, we show the existence of abridged determining modes, nodes and volume elements.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"1 1","pages":"1-43"},"PeriodicalIF":0.0,"publicationDate":"2017-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91295778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we establish compactness and existence results to a Branson-Paneitz type problem on a bounded domain of R^n with Navier boundary condition.
{"title":"Q-curvature type problem on bounded domains of R n","authors":"W. Abdelhedi, H. Chtioui, H. Hajaiej","doi":"10.3233/ASY-181473","DOIUrl":"https://doi.org/10.3233/ASY-181473","url":null,"abstract":"In this paper, we establish compactness and existence results to a Branson-Paneitz type problem on a bounded domain of R^n with Navier boundary condition.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"37 1","pages":"143-170"},"PeriodicalIF":0.0,"publicationDate":"2017-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89448904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Bchatnia, Sabrine Chebbi, M. Hamouda, A. Soufyane
In this paper, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We first investigate the strong stability of this system, then we devote our efforts to obtain the strong lower energy estimates using Alabau--Boussouira's energy comparison principle introduced in cite{2} (see also cite{alabau}). One of the main advantages of these results is that they allows us to prove the optimality of the asymptotic results (as $trightarrow infty$) obtained in cite{ali}. We also extend to our model the nice results achieved in cite{alabau} for the case of nonlinearly damped Timoshenko system with thermoelasticity. The optimality of our results is also investigated through some explicit examples of the nonlinear damping term. The proof of our results relies on the approach in cite{AB1, AB2}.
{"title":"Lower bound and optimality for a nonlinearly damped Timoshenko system with thermoelasticity","authors":"A. Bchatnia, Sabrine Chebbi, M. Hamouda, A. Soufyane","doi":"10.3233/ASY-191519","DOIUrl":"https://doi.org/10.3233/ASY-191519","url":null,"abstract":"In this paper, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We first investigate the strong stability of this system, then we devote our efforts to obtain the strong lower energy estimates using Alabau--Boussouira's energy comparison principle introduced in cite{2} (see also cite{alabau}). One of the main advantages of these results is that they allows us to prove the optimality of the asymptotic results (as $trightarrow infty$) obtained in cite{ali}. We also extend to our model the nice results achieved in cite{alabau} for the case of nonlinearly damped Timoshenko system with thermoelasticity. The optimality of our results is also investigated through some explicit examples of the nonlinear damping term. The proof of our results relies on the approach in cite{AB1, AB2}.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"19 1","pages":"73-91"},"PeriodicalIF":0.0,"publicationDate":"2017-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91116039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider fourth order ordinary differential operator with compactly supported coefficients on the line. We define resonances as zeros of the Fredholm determinant which is analytic on a four sheeted Riemann surface. We determine asymptotics of the number of resonances in complex discs at large radius. We consider resonances of an Euler-Bernoulli operator on the real line with the positive coefficients which are constants outside some finite interval. We show that the Euler-Bernoulli operator has no eigenvalues and resonances iff the positive coefficients are constants on the whole axis.
{"title":"Resonances of 4th order differential operators","authors":"A. Badanin, E. Korotyaev","doi":"10.3233/ASY-181489","DOIUrl":"https://doi.org/10.3233/ASY-181489","url":null,"abstract":"We consider fourth order ordinary differential operator with compactly supported coefficients on the line. We define resonances as zeros of the Fredholm determinant which is analytic on a four sheeted Riemann surface. We determine asymptotics of the number of resonances in complex discs at large radius. We consider resonances of an Euler-Bernoulli operator on the real line with the positive coefficients which are constants outside some finite interval. We show that the Euler-Bernoulli operator has no eigenvalues and resonances iff the positive coefficients are constants on the whole axis.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"26 1","pages":"137-177"},"PeriodicalIF":0.0,"publicationDate":"2017-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85513535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we use the Feynman-Kac formula to derive a moment representation for the 2D parabolic Anderson model in small time, which is related to the intersection local time of planar Brownian motions.
{"title":"Moments of 2D parabolic Anderson model","authors":"Yu Gu, Weijun Xu","doi":"10.3233/ASY-171460","DOIUrl":"https://doi.org/10.3233/ASY-171460","url":null,"abstract":"In this note, we use the Feynman-Kac formula to derive a moment representation for the 2D parabolic Anderson model in small time, which is related to the intersection local time of planar Brownian motions.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"30 1","pages":"151-161"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77277706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. Alessandrini, M. Hoop, Romina Gaburro, E. Sincich
We consider the inverse boundary value problem of determining the potential $q$ in the equation $Delta u + qu = 0$ in $Omegasubsetmathbb{R}^n$, from local Cauchy data. A result of global Lipschitz stability is obtained in dimension $ngeq 3$ for potentials that are piecewise linear on a given partition of $Omega$. No sign, nor spectrum condition on $q$ is assumed, hence our treatment encompasses the reduced wave equation $Delta u + k^2c^{-2}u=0$ at fixed frequency $k$.
我们考虑了用局部柯西数据确定$Omegasubsetmathbb{R}^n$方程$Delta u + qu = 0$中势$q$的反边值问题。对于在$Omega$的给定分区上分段线性的势，在$ngeq 3$维上得到了全局Lipschitz稳定性的结果。在$q$上没有符号，也没有频谱条件，因此我们的处理包含固定频率$k$的简化波动方程$Delta u + k^2c^{-2}u=0$。
{"title":"Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data","authors":"G. Alessandrini, M. Hoop, Romina Gaburro, E. Sincich","doi":"10.3233/ASY-171457","DOIUrl":"https://doi.org/10.3233/ASY-171457","url":null,"abstract":"We consider the inverse boundary value problem of determining the potential $q$ in the equation $Delta u + qu = 0$ in $Omegasubsetmathbb{R}^n$, from local Cauchy data. A result of global Lipschitz stability is obtained in dimension $ngeq 3$ for potentials that are piecewise linear on a given partition of $Omega$. No sign, nor spectrum condition on $q$ is assumed, hence our treatment encompasses the reduced wave equation $Delta u + k^2c^{-2}u=0$ at fixed frequency $k$.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"6 1","pages":"115-149"},"PeriodicalIF":0.0,"publicationDate":"2017-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86525938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze a general class of difference operators He = Te + Ve on � 2 ((eZ) d ), where Ve is a one-well potential and e is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of He. These are obtained from eigenfunctions or quasimodes for the operator He, acting on L 2 (R d ), via restriction to the lattice (eZ) d .
我们分析了一类一般的差分算子He = Te + Ve on 2 ((eZ) d)，其中Ve是单井势，e是一个小参数。构造了与He的低特征值相关的特征函数的wkb型渐近展开式。这些是通过对晶格(eZ) d的限制，从作用于l2 (rd)的算子He的本征函数或准模中得到的。
{"title":"Asymptotic eigenfunctions for a class of difference operators","authors":"M. Klein, Elke Rosenberger","doi":"10.3233/ASY-2010-1025","DOIUrl":"https://doi.org/10.3233/ASY-2010-1025","url":null,"abstract":"We analyze a general class of difference operators He = Te + Ve on � 2 ((eZ) d ), where Ve is a one-well potential and e is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of He. These are obtained from eigenfunctions or quasimodes for the operator He, acting on L 2 (R d ), via restriction to the lattice (eZ) d .","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"18 1","pages":"1-36"},"PeriodicalIF":0.0,"publicationDate":"2017-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75505015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the location of the transmission eigenvalues in the isotropic case when the restrictions of the refraction indices on the boundary coincide. Under some natural conditions we show that there exist parabolic transmission eigenvalue-free regions.
{"title":"Parabolic transmission eigenvalue-free regions in the degenerate isotropic case","authors":"G. Vodev","doi":"10.3233/ASY-171443","DOIUrl":"https://doi.org/10.3233/ASY-171443","url":null,"abstract":"We study the location of the transmission eigenvalues in the isotropic case when the restrictions of the refraction indices on the boundary coincide. Under some natural conditions we show that there exist parabolic transmission eigenvalue-free regions.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"1 1","pages":"147-168"},"PeriodicalIF":0.0,"publicationDate":"2017-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87860971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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