SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4712-4741, August 2024. Abstract. In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient [math] in the Schrödinger equation [math], from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain types of geometric optics solutions can reach, and a stability estimate based on the unique continuation property for the linear equation.
{"title":"Partial Data Inverse Problems for the Nonlinear Time-Dependent Schrödinger Equation","authors":"Ru-Yu Lai, Xuezhu Lu, Ting Zhou","doi":"10.1137/23m1587993","DOIUrl":"https://doi.org/10.1137/23m1587993","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4712-4741, August 2024. <br/> Abstract. In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient [math] in the Schrödinger equation [math], from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain types of geometric optics solutions can reach, and a stability estimate based on the unique continuation property for the linear equation.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4742-4758, August 2024. Abstract. We prove a sharp global [math] estimate for potentials of optimal transport maps that take a certain class of nonconvex planar domains to convex ones.
{"title":"Sobolev Regularity for Optimal Transport Maps of Nonconvex Planar Domains","authors":"Connor Mooney, Arghya Rakshit","doi":"10.1137/23m1582436","DOIUrl":"https://doi.org/10.1137/23m1582436","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4742-4758, August 2024. <br/> Abstract. We prove a sharp global [math] estimate for potentials of optimal transport maps that take a certain class of nonconvex planar domains to convex ones.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4662-4686, August 2024. Abstract. We investigate the distributional extension of the [math]-plane transform in [math] and of related operators. We parameterize the [math]-plane domain as the Cartesian product of the Stiefel manifold of orthonormal [math]-frames in [math] with [math]. This parameterization imposes an isotropy condition on the range of the [math]-plane transform which is analogous to the even condition on the range of the Radon transform. We use our distributional formalism to investigate the invertibility of the dual [math]-plane transform (the “backprojection” operator). We provide a systematic construction (via a completion process) to identify Banach spaces in which the backprojection operator is invertible and present some prototypical examples. These include the space of isotropic finite Radon measures and isotropic [math]-functions for [math]. Finally, we apply our results to study a new form of regularization for inverse problems.
{"title":"Distributional Extension and Invertibility of the [math]-Plane Transform and Its Dual","authors":"Rahul Parhi, Michael Unser","doi":"10.1137/23m1556721","DOIUrl":"https://doi.org/10.1137/23m1556721","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4662-4686, August 2024. <br/> Abstract. We investigate the distributional extension of the [math]-plane transform in [math] and of related operators. We parameterize the [math]-plane domain as the Cartesian product of the Stiefel manifold of orthonormal [math]-frames in [math] with [math]. This parameterization imposes an isotropy condition on the range of the [math]-plane transform which is analogous to the even condition on the range of the Radon transform. We use our distributional formalism to investigate the invertibility of the dual [math]-plane transform (the “backprojection” operator). We provide a systematic construction (via a completion process) to identify Banach spaces in which the backprojection operator is invertible and present some prototypical examples. These include the space of isotropic finite Radon measures and isotropic [math]-functions for [math]. Finally, we apply our results to study a new form of regularization for inverse problems.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4356-4374, August 2024. Abstract. The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier–Stokes equations. In [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945–1950], Li and Sinai have proposed a renormalization approach to the problem of the existence of finite-time blow-up solutions of this equation. In this paper, we revisit the renormalization problem for the quasi-geostrophic blow-ups, prove the existence of a family of renormalization fixed points, and deduce the existence of real [math] solutions to the quasi-geostrophic equation whose energy and enstrophy become unbounded in finite time, different from those found in [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945–1950].
SIAM 数学分析期刊》，第 56 卷第 4 期，第 4356-4374 页，2024 年 8 月。 摘要。一维准地转方程是著名的纳维-斯托克斯方程的一维傅里叶空间类似方程。在 [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp.在本文中，我们重新审视了准地转吹胀的重正化问题，证明了重正化定点族的存在，并推导出了与[D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp.
{"title":"Renormalization and Existence of Finite-Time Blow-Up Solutions for a One-Dimensional Analogue of the Navier–Stokes Equations","authors":"Denis Gaidashev, Alejandro Luque","doi":"10.1137/23m1551481","DOIUrl":"https://doi.org/10.1137/23m1551481","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4356-4374, August 2024. <br/> Abstract. The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier–Stokes equations. In [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945–1950], Li and Sinai have proposed a renormalization approach to the problem of the existence of finite-time blow-up solutions of this equation. In this paper, we revisit the renormalization problem for the quasi-geostrophic blow-ups, prove the existence of a family of renormalization fixed points, and deduce the existence of real [math] solutions to the quasi-geostrophic equation whose energy and enstrophy become unbounded in finite time, different from those found in [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945–1950].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4466-4493, August 2024. Abstract. Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schrödinger equation. Explicit formulas for the transmission coefficient and [math]-matrix of the classical one-dimensional Schrödinger operator with arbitrary compactly supported potential are obtained as a consequence. The formulas involve operator theoretic analogues of the standard hyperbolic functions and provide new tools with which to analyze acoustic and quantum scattering in one dimension.
{"title":"Explicit Solution of the 1D Schrödinger Equation","authors":"Peter C. Gibson","doi":"10.1137/22m1514441","DOIUrl":"https://doi.org/10.1137/22m1514441","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4466-4493, August 2024. <br/> Abstract. Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schrödinger equation. Explicit formulas for the transmission coefficient and [math]-matrix of the classical one-dimensional Schrödinger operator with arbitrary compactly supported potential are obtained as a consequence. The formulas involve operator theoretic analogues of the standard hyperbolic functions and provide new tools with which to analyze acoustic and quantum scattering in one dimension.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jean-Claude Saut, Shihan Sun, Yuexun Wang, Yi Zhang
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4440-4465, August 2024. Abstract. This paper aims to show that the Cauchy problem of the Burgers equation with a weakly dispersive perturbation involving the Bessel potential (generalization of the Fornberg–Whitham equation) can exhibit wave breaking for initial data with large slope. We also comment on the dispersive properties of the equation.
{"title":"Wave Breaking for the Generalized Fornberg–Whitham Equation","authors":"Jean-Claude Saut, Shihan Sun, Yuexun Wang, Yi Zhang","doi":"10.1137/23m1603431","DOIUrl":"https://doi.org/10.1137/23m1603431","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4440-4465, August 2024. <br/> Abstract. This paper aims to show that the Cauchy problem of the Burgers equation with a weakly dispersive perturbation involving the Bessel potential (generalization of the Fornberg–Whitham equation) can exhibit wave breaking for initial data with large slope. We also comment on the dispersive properties of the equation.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4324-4355, August 2024. Abstract. In this paper, we study a mathematical model describing the movement of a colloidal particle in a fixed, bounded three dimensional container filled with a nematic liquid crystal fluid. The motion of the fluid is governed by the Beris–Edwards model for nematohydrodynamics equations, which couples the incompressible Navier–Stokes equations with a parabolic system. The dynamics of colloidal particle within the nematic liquid crystal is described by the conservation laws of linear and angular momentum. We prove the existence of global weak solutions for the coupled system.
{"title":"Global Existence of Weak Solutions for a Model of Nematic Liquid Crystal-Colloidal Interactions","authors":"Zhiyuan Geng, Arnab Roy, Arghir Zarnescu","doi":"10.1137/23m161149x","DOIUrl":"https://doi.org/10.1137/23m161149x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4324-4355, August 2024. <br/> Abstract. In this paper, we study a mathematical model describing the movement of a colloidal particle in a fixed, bounded three dimensional container filled with a nematic liquid crystal fluid. The motion of the fluid is governed by the Beris–Edwards model for nematohydrodynamics equations, which couples the incompressible Navier–Stokes equations with a parabolic system. The dynamics of colloidal particle within the nematic liquid crystal is described by the conservation laws of linear and angular momentum. We prove the existence of global weak solutions for the coupled system.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4594-4622, August 2024. Abstract. We establish a global uniqueness result for an inverse boundary problem with partial data for the magnetic Schrödinger operator with a magnetic potential of class [math], and an electric potential of class [math]. Our result is an extension, in terms of the regularity of the potentials, of the results [D. Dos Santos Ferreira et al., Comm. Math. Phys., 271 (2007), pp. 467–488] and [K. Knudsen and M. Salo, Inverse Probl. Imaging, 1 (2007), pp. 349–369]. As a consequence, we also show global uniqueness for a partial data inverse boundary problem for the advection-diffusion operator with the advection term of class [math].
{"title":"Partial Data Inverse Problems for Magnetic Schrödinger Operators with Potentials of Low Regularity","authors":"Salem Selim","doi":"10.1137/22m1530707","DOIUrl":"https://doi.org/10.1137/22m1530707","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4594-4622, August 2024. <br/> Abstract. We establish a global uniqueness result for an inverse boundary problem with partial data for the magnetic Schrödinger operator with a magnetic potential of class [math], and an electric potential of class [math]. Our result is an extension, in terms of the regularity of the potentials, of the results [D. Dos Santos Ferreira et al., Comm. Math. Phys., 271 (2007), pp. 467–488] and [K. Knudsen and M. Salo, Inverse Probl. Imaging, 1 (2007), pp. 349–369]. As a consequence, we also show global uniqueness for a partial data inverse boundary problem for the advection-diffusion operator with the advection term of class [math].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4415-4439, August 2024. Abstract. We study the focusing mass-subcritical biharmonic nonlinear Schrödinger equation (BNLS) on the product space [math]. Following the crucial scaling arguments introduced in [Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp. 73–96] we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number [math] that sharply determines the [math]-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer scale-invariant and the arguments from [Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp. 73–96] for determining the sharp [math]-dependence of the ground states fail. The main novelty of the present paper is to address this difficult and interesting issue: Using a different scaling argument, we show that [math]-independence of ground states with small mass still holds in the case [math] and [math]. Additionally, we also prove that ground states with sufficiently large mass must possess nontrivial [math]-dependence by appealing to some novel construction of test functions. The latter particularly holds for all parameters lying in the full mass-subcritical regime.
{"title":"On Existence and Stability Results for Normalized Ground States of Mass-Subcritical Biharmonic Nonlinear Schrödinger Equation on [math]","authors":"Hichem Hajaiej, Yongming Luo, Lingjie Song","doi":"10.1137/22m1543707","DOIUrl":"https://doi.org/10.1137/22m1543707","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4415-4439, August 2024. <br/> Abstract. We study the focusing mass-subcritical biharmonic nonlinear Schrödinger equation (BNLS) on the product space [math]. Following the crucial scaling arguments introduced in [Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp. 73–96] we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number [math] that sharply determines the [math]-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer scale-invariant and the arguments from [Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp. 73–96] for determining the sharp [math]-dependence of the ground states fail. The main novelty of the present paper is to address this difficult and interesting issue: Using a different scaling argument, we show that [math]-independence of ground states with small mass still holds in the case [math] and [math]. Additionally, we also prove that ground states with sufficiently large mass must possess nontrivial [math]-dependence by appealing to some novel construction of test functions. The latter particularly holds for all parameters lying in the full mass-subcritical regime.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4253-4295, August 2024. Abstract. Laplace learning is a popular machine learning algorithm for finding missing labels from a small number of labeled feature vectors using the geometry of a graph. More precisely, Laplace learning is based on minimizing a graph-Dirichlet energy, equivalently a discrete Sobolev [math] seminorm, constrained to taking the values of known labels on a given subset. The variational problem is asymptotically ill-posed as the number of unlabeled feature vectors goes to infinity for finite given labels due to a lack of regularity in minimizers of the continuum Dirichlet energy in any dimension higher than one. In particular, continuum minimizers are not continuous. One solution is to consider higher-order regularization, which is the analogue of minimizing Sobolev [math] seminorms. In this paper we consider the asymptotics of minimizing a graph variant of the Sobolev [math] seminorm with pointwise constraints. We show that, as expected, one needs [math], where [math] is the dimension of the data manifold. We also show that there must be an upper bound on the connectivity of the graph; that is, highly connected graphs lead to degenerate behavior of the minimizer even when [math].
{"title":"Consistency of Fractional Graph-Laplacian Regularization in Semisupervised Learning with Finite Labels","authors":"Adrien Weihs, Matthew Thorpe","doi":"10.1137/23m1559087","DOIUrl":"https://doi.org/10.1137/23m1559087","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4253-4295, August 2024. <br/> Abstract. Laplace learning is a popular machine learning algorithm for finding missing labels from a small number of labeled feature vectors using the geometry of a graph. More precisely, Laplace learning is based on minimizing a graph-Dirichlet energy, equivalently a discrete Sobolev [math] seminorm, constrained to taking the values of known labels on a given subset. The variational problem is asymptotically ill-posed as the number of unlabeled feature vectors goes to infinity for finite given labels due to a lack of regularity in minimizers of the continuum Dirichlet energy in any dimension higher than one. In particular, continuum minimizers are not continuous. One solution is to consider higher-order regularization, which is the analogue of minimizing Sobolev [math] seminorms. In this paper we consider the asymptotics of minimizing a graph variant of the Sobolev [math] seminorm with pointwise constraints. We show that, as expected, one needs [math], where [math] is the dimension of the data manifold. We also show that there must be an upper bound on the connectivity of the graph; that is, highly connected graphs lead to degenerate behavior of the minimizer even when [math].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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