Pub Date : 2021-05-03DOI: 10.1512/iumj.2022.71.9753
Mayuresh Londhe
. This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincar´e recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance properties of the supports of the measures mentioned.
{"title":"Recurrence in the dynamics of meromorphic correspondences and holomorphic semigroups","authors":"Mayuresh Londhe","doi":"10.1512/iumj.2022.71.9753","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9753","url":null,"abstract":". This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincar´e recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance properties of the supports of the measures mentioned.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43302890","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}
Pub Date : 2021-04-20DOI: 10.1512/iumj.2023.72.9425
Wenhui Chen, R. Ikehata, A. Palmieri
We study a fundamental model in nonlinear acoustics, precisely, the general Blackstock's model (that is, without Becker's assumption) in the whole space $mathbb{R}^n$. This model describes nonlinear acoustics in perfect gases under the irrotational flow. By means of the Fourier analysis we will derive $L^2$ estimates for the solution of the linear homogeneous problem and its derivatives. Then, we will apply these estimates to study three different topics: the optimality of the decay estimates in the case $ngeqslant 5$ and the optimal growth rate for the $L^2$-norm of the solution for $n=3,4$; the singular limit problem in determining the first- and second-order profiles for the solution of the linear Blackstock's model with respect to the small thermal diffusivity; the proof of the existence of global (in time) small data Sobolev solutions with suitable regularity for a nonlinear Blackstock's model.
{"title":"Asymptotic behaviors for Blackstock's model of thermoviscous flow","authors":"Wenhui Chen, R. Ikehata, A. Palmieri","doi":"10.1512/iumj.2023.72.9425","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9425","url":null,"abstract":"We study a fundamental model in nonlinear acoustics, precisely, the general Blackstock's model (that is, without Becker's assumption) in the whole space $mathbb{R}^n$. This model describes nonlinear acoustics in perfect gases under the irrotational flow. By means of the Fourier analysis we will derive $L^2$ estimates for the solution of the linear homogeneous problem and its derivatives. Then, we will apply these estimates to study three different topics: the optimality of the decay estimates in the case $ngeqslant 5$ and the optimal growth rate for the $L^2$-norm of the solution for $n=3,4$; the singular limit problem in determining the first- and second-order profiles for the solution of the linear Blackstock's model with respect to the small thermal diffusivity; the proof of the existence of global (in time) small data Sobolev solutions with suitable regularity for a nonlinear Blackstock's model.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49531081","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}
Pub Date : 2021-04-03DOI: 10.1512/iumj.2023.72.9394
A. Lorent, G. Peng
The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose $Gamma$-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in the variational analysis of this problem. Establishing fine structures of entropy solutions of the Eikonal equation, e.g. concentration of entropy measures on $mathcal{H}^1$-rectifiable sets in $2$D, is arguably the key missing part for a proof of the full $Gamma$-convergence of the Aviles-Giga functional. In the first part of this work, for $pin left(1,frac{4}{3}right]$ we establish an $L^p$ version of the main theorem of Ghiraldin and Lamy [Comm. Pure Appl. Math. 73 (2020), no. 2, 317-349]. Specifically we show that if $m$ is a solution to the Eikonal equation, then $min B^{frac{1}{3}}_{3p,infty,loc}$ is equivalent to all entropy productions of $m$ being in $L^p_{loc}$. This result also shows that as a consequence of a weak form of the Aviles-Giga conjecture (namely the conjecture that all solutions to the Eikonal equation whose entropy productions are in $L^p_{loc}$ are rigid) - the rigidity/flexibility threshold of the Eikonal equation is exactly the space $ B^{frac{1}{3}}_{3,infty,loc}$. In the second part of this paper, under the assumption that all entropy productions are in $L^p_{loc}$, we establish a factorization formula for entropy productions of solutions of the Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this formula is control of all entropy productions by the Jin-Kohn entropies in the $L^p$ setting - this is a strong extension of an earlier result of the authors [Annales de l'Institut Henri Poincar'{e}. Analyse Non Lin'{e}aire 35 (2018), no. 2, 481-516].
Eikonal方程自然产生于二阶Aviles-Giga-泛函的极限，其$Gamma$-收敛是一个长期存在的具有挑战性的问题。Eikonal方程的熵解理论在该问题的变分分析中起着核心作用。建立Eikonal方程的熵解的精细结构，例如在$2$D中$mathcal｛H｝^1$-可直集上的熵测度的集中，可以说是证明Aviles Giga泛函的完全$Gamma$-收敛性的关键缺失部分。在这项工作的第一部分中，对于$pInleft（1，frac｛4｝｛3｝right]$，我们建立了Ghiraldin和Lamy主要定理的$L^p$版本[Comm.Pure Appl.Math.73（2020），no.2317-349]。具体地说，我们证明了如果$m$是Eikonal方程的解，那么B^｛frac｛1｝｛3｝｝_｛3p，fty，loc｝$中的$m等价于$L^ p_｛loc｝$中$m$的所有熵产生。这一结果还表明，作为Aviles-Giga猜想的一种弱形式（即熵产生在$L^p_{loc}$中的Eikonal方程的所有解都是刚性的猜想）的结果，Eikonal方程式的刚性/柔性阈值正是空间$B^｛frac｛1｝｛3｝｝_｛3，infty，loc｝$。在本文的第二部分中，在所有熵产生都在$L^p_{loc}$的假设下，我们用两个Jin-Kohn熵建立了Eikonal方程解的熵产生的因子分解公式。这个公式的一个结果是在$L^p$设置下，金-科恩熵对所有熵产生的控制——这是作者[Annales de L’Institut Henri Poincar'｛e｝.Analysis Non-Lin{e}aire35（2018），第2号，481-516]。
{"title":"Factorization for entropy production of the Eikonal equation and regularity","authors":"A. Lorent, G. Peng","doi":"10.1512/iumj.2023.72.9394","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9394","url":null,"abstract":"The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose $Gamma$-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in the variational analysis of this problem. Establishing fine structures of entropy solutions of the Eikonal equation, e.g. concentration of entropy measures on $mathcal{H}^1$-rectifiable sets in $2$D, is arguably the key missing part for a proof of the full $Gamma$-convergence of the Aviles-Giga functional. In the first part of this work, for $pin left(1,frac{4}{3}right]$ we establish an $L^p$ version of the main theorem of Ghiraldin and Lamy [Comm. Pure Appl. Math. 73 (2020), no. 2, 317-349]. Specifically we show that if $m$ is a solution to the Eikonal equation, then $min B^{frac{1}{3}}_{3p,infty,loc}$ is equivalent to all entropy productions of $m$ being in $L^p_{loc}$. This result also shows that as a consequence of a weak form of the Aviles-Giga conjecture (namely the conjecture that all solutions to the Eikonal equation whose entropy productions are in $L^p_{loc}$ are rigid) - the rigidity/flexibility threshold of the Eikonal equation is exactly the space $ B^{frac{1}{3}}_{3,infty,loc}$. In the second part of this paper, under the assumption that all entropy productions are in $L^p_{loc}$, we establish a factorization formula for entropy productions of solutions of the Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this formula is control of all entropy productions by the Jin-Kohn entropies in the $L^p$ setting - this is a strong extension of an earlier result of the authors [Annales de l'Institut Henri Poincar'{e}. Analyse Non Lin'{e}aire 35 (2018), no. 2, 481-516].","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41670199","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}
Pub Date : 2021-03-24DOI: 10.1512/iumj.2022.71.9369
F. Bayart, Fernando Costa, Q. Menet
We investigate the existence of a common hypercyclic vector for a family (Tλ)λ∈Λ of hypercyclic operators acting on the same Banach space X. We give positive and negative results involving the dimension of Λ and the regularity of each map λ ∈ Λ 7→ T λ x, x ∈ X, n ∈ N.
{"title":"Common hypercyclic vectors and dimension of the parameter set","authors":"F. Bayart, Fernando Costa, Q. Menet","doi":"10.1512/iumj.2022.71.9369","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9369","url":null,"abstract":"We investigate the existence of a common hypercyclic vector for a family (Tλ)λ∈Λ of hypercyclic operators acting on the same Banach space X. We give positive and negative results involving the dimension of Λ and the regularity of each map λ ∈ Λ 7→ T λ x, x ∈ X, n ∈ N.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49591063","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}
Pub Date : 2021-03-08DOI: 10.1512/iumj.2021.70.9364
M. Renzi
We construct 3-dimensional once-Extended Topological Quantum Field Theories (ETQFTs for short) out of (possibly non-semisimple) modular categories, and we explicitly identify linear categories and functors in their image. The circle category of an ETQFT produced by our construction is equivalent to the full subcategory of projective objects of the underlying modular category. In particular, it need not be semisimple.
{"title":"Extended TQFTs from non-semisimple modular categories","authors":"M. Renzi","doi":"10.1512/iumj.2021.70.9364","DOIUrl":"https://doi.org/10.1512/iumj.2021.70.9364","url":null,"abstract":"We construct 3-dimensional once-Extended Topological Quantum Field Theories (ETQFTs for short) out of (possibly non-semisimple) modular categories, and we explicitly identify linear categories and functors in their image. The circle category of an ETQFT produced by our construction is equivalent to the full subcategory of projective objects of the underlying modular category. In particular, it need not be semisimple.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47996667","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}
Pub Date : 2021-02-25DOI: 10.1512/iumj.2021.70.9435
Panayotis Smyrnelis
. We establish a comparison principle providing accurate upper bounds for the modulus of vector valued minimizers of an energy functional, associated when the potential is smooth, to elliptic gradient systems. Our assumptions are very mild: we assume that the potential is lower semicontinuous, and satisfies a monotonicity condition in a neighbourhood of its minimum. As a consequence, we give a sufficient condition for the existence of dead core regions, where the minimizer is equal to one of the minima of the potential.
{"title":"A comparison principle for vector-valued minimzers of semilinear elliptic energy, with application to dead cores","authors":"Panayotis Smyrnelis","doi":"10.1512/iumj.2021.70.9435","DOIUrl":"https://doi.org/10.1512/iumj.2021.70.9435","url":null,"abstract":". We establish a comparison principle providing accurate upper bounds for the modulus of vector valued minimizers of an energy functional, associated when the potential is smooth, to elliptic gradient systems. Our assumptions are very mild: we assume that the potential is lower semicontinuous, and satisfies a monotonicity condition in a neighbourhood of its minimum. As a consequence, we give a sufficient condition for the existence of dead core regions, where the minimizer is equal to one of the minima of the potential.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45670810","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}
Pub Date : 2021-02-14DOI: 10.1512/iumj.2023.72.9404
C. Miao, Jason Murphy, Jiqiang Zheng
We adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the $3d$ focusing cubic NLS with a repulsive potential. We treat both the case of short-range potentials as previously considered in the work of Hong, as well as the inverse-square potential, previously considered in the work of the authors.
{"title":"Threshold scattering for the focusing NLS with a repulsive potential","authors":"C. Miao, Jason Murphy, Jiqiang Zheng","doi":"10.1512/iumj.2023.72.9404","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9404","url":null,"abstract":"We adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the $3d$ focusing cubic NLS with a repulsive potential. We treat both the case of short-range potentials as previously considered in the work of Hong, as well as the inverse-square potential, previously considered in the work of the authors.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66765773","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}
Pub Date : 2021-02-09DOI: 10.1512/iumj.2023.72.9363
F. Golse, T. Paul
We present a time dependent quantum perturbation result, uniform in the Planck constant for potential whose gradient is bounded a.e..We show also that the classical limit of the perturbed quantum dynamics remains in a tubular neighborhood of the classical unperturbed one, the size of this neighborhood being of the order of the square root of the size of the perturbation. We treat both Schr"odinger and von Neumann-Heisenberg equations.
{"title":"Time dependent quantum perturbations uniform in the semiclassical regime","authors":"F. Golse, T. Paul","doi":"10.1512/iumj.2023.72.9363","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9363","url":null,"abstract":"We present a time dependent quantum perturbation result, uniform in the Planck constant for potential whose gradient is bounded a.e..We show also that the classical limit of the perturbed quantum dynamics remains in a tubular neighborhood of the classical unperturbed one, the size of this neighborhood being of the order of the square root of the size of the perturbation. We treat both Schr\"odinger and von Neumann-Heisenberg equations.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42269671","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}
Pub Date : 2021-01-20DOI: 10.1512/iumj.2023.72.9304
Chen-Chih Lai, Juncheng Wei, Yifu Zhou
We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in $mathbb{R}^2$ that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and transported by ambient fluid velocity, and the fluid flow is forced by the friction induced by the cells. The main result of the paper is to show the global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass.
{"title":"Global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass","authors":"Chen-Chih Lai, Juncheng Wei, Yifu Zhou","doi":"10.1512/iumj.2023.72.9304","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9304","url":null,"abstract":"We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in $mathbb{R}^2$ that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and transported by ambient fluid velocity, and the fluid flow is forced by the friction induced by the cells. The main result of the paper is to show the global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44845471","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}
Pub Date : 2021-01-14DOI: 10.1512/iumj.2023.72.9307
Vyacheslav Krushkal, Paul Wedrich
The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. We formulate a stable homotopy refinement of the Blanchet theory, based on a comparison of the Blanchet and Khovanov chain complexes associated to link diagrams. The construction of the stable homotopy type relies on the signed Burnside category approach of Sarkar-Scaduto-Stoffregen.
{"title":"mathfrak{gl}_2 foams and the Khovanov homotopy type","authors":"Vyacheslav Krushkal, Paul Wedrich","doi":"10.1512/iumj.2023.72.9307","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9307","url":null,"abstract":"The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. We formulate a stable homotopy refinement of the Blanchet theory, based on a comparison of the Blanchet and Khovanov chain complexes associated to link diagrams. The construction of the stable homotopy type relies on the signed Burnside category approach of Sarkar-Scaduto-Stoffregen.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48948441","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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