Pub Date : 2025-09-01Epub Date: 2025-04-25DOI: 10.1016/j.matpur.2025.103719
Isabelle Catto , Long Meng
Existence of minimizers for the Dirac–Fock model for crystals was recently proved by Paturel and Séré and the authors [9]. In this paper, inspired by Ghimenti and Lewin's result [13] for the periodic Hartree–Fock model, we prove that the Fermi level of any periodic Dirac–Fock minimizer is either empty or totally filled when and . Here c is the speed of light, α is the fine structure constant, and is a constant only depending on the number of electrons and on the charge of nuclei per cell. More importantly, we provide an explicit upper bound for .
Our result implies that any minimizer of the periodic Dirac–Fock model is a projector when and . In particular, the non-relativistic regime (i.e., ) and the weak coupling regime (i.e., ) are covered.
The proof is based on a delicate study of a second-order expansion of the periodic Dirac–Fock functional composed with a retraction that was introduced by Séré in [24] for atoms and molecules and later extended to the case of crystals in [9].
{"title":"Properties of periodic Dirac–Fock functional and minimizers","authors":"Isabelle Catto , Long Meng","doi":"10.1016/j.matpur.2025.103719","DOIUrl":"10.1016/j.matpur.2025.103719","url":null,"abstract":"<div><div>Existence of minimizers for the Dirac–Fock model for crystals was recently proved by Paturel and Séré and the authors <span><span>[9]</span></span>. In this paper, inspired by Ghimenti and Lewin's result <span><span>[13]</span></span> for the periodic Hartree–Fock model, we prove that the Fermi level of any periodic Dirac–Fock minimizer is either empty or totally filled when <span><math><mfrac><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></mfrac><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>cri</mi></mrow></msub></math></span> and <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>. Here <em>c</em> is the speed of light, <em>α</em> is the fine structure constant, and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>cri</mi></mrow></msub></math></span> is a constant only depending on the number of electrons and on the charge of nuclei per cell. More importantly, we provide an explicit upper bound for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>cri</mi></mrow></msub></math></span>.</div><div>Our result implies that any minimizer of the periodic Dirac–Fock model is a projector when <span><math><mfrac><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></mfrac><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>cri</mi></mrow></msub></math></span> and <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>. In particular, the non-relativistic regime (i.e., <span><math><mi>c</mi><mo>≫</mo><mn>1</mn></math></span>) and the weak coupling regime (i.e., <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo>≪</mo><mn>1</mn></math></span>) are covered.</div><div>The proof is based on a delicate study of a second-order expansion of the periodic Dirac–Fock functional composed with a retraction that was introduced by Séré in <span><span>[24]</span></span> for atoms and molecules and later extended to the case of crystals in <span><span>[9]</span></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"201 ","pages":"Article 103719"},"PeriodicalIF":2.1,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144067943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-04-25DOI: 10.1016/j.matpur.2025.103724
Hui Li , Weiren Zhao
In this paper, we study the instability effect of viscous dissipation in a domain without boundaries. We construct a shear flow that is initially spectrally stable but evolves into a spectrally unstable state under the influence of viscous dissipation. To the best of our knowledge, this is the first result of viscosity driven instability that is not caused by boundaries.
{"title":"Viscosity driven instability of shear flows without boundaries","authors":"Hui Li , Weiren Zhao","doi":"10.1016/j.matpur.2025.103724","DOIUrl":"10.1016/j.matpur.2025.103724","url":null,"abstract":"<div><div>In this paper, we study the instability effect of viscous dissipation in a domain without boundaries. We construct a shear flow that is initially spectrally stable but evolves into a spectrally unstable state under the influence of viscous dissipation. To the best of our knowledge, this is the first result of viscosity driven instability that is not caused by boundaries.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"200 ","pages":"Article 103724"},"PeriodicalIF":2.1,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-04-28DOI: 10.1016/j.matpur.2025.103723
Lucio Galeati
We show that, for any Leray solution u to the 3D Navier–Stokes equations with , the associated deterministic and stochastic Lagrangian trajectories are unique for Lebesgue a.e. initial condition. Additionally, if , then pathwise uniqueness is established for the stochastic Lagrangian trajectories starting from every initial condition. The result sharpens and extends the original one by Robinson and Sadowski [1] and is based on rather different techniques. A key role is played by a newly established asymmetric Lusin–Lipschitz property of Leray solutions u, in the framework of (random) Regular Lagrangian flows.
{"title":"Almost-everywhere uniqueness of Lagrangian trajectories for 3D Navier–Stokes revisited","authors":"Lucio Galeati","doi":"10.1016/j.matpur.2025.103723","DOIUrl":"10.1016/j.matpur.2025.103723","url":null,"abstract":"<div><div>We show that, for any Leray solution <em>u</em> to the 3D Navier–Stokes equations with <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, the associated deterministic and stochastic Lagrangian trajectories are unique for <em>Lebesgue a.e.</em> initial condition. Additionally, if <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>, then pathwise uniqueness is established for the stochastic Lagrangian trajectories starting from <em>every</em> initial condition. The result sharpens and extends the original one by Robinson and Sadowski <span><span>[1]</span></span> and is based on rather different techniques. A key role is played by a newly established asymmetric Lusin–Lipschitz property of Leray solutions <em>u</em>, in the framework of (random) Regular Lagrangian flows.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"200 ","pages":"Article 103723"},"PeriodicalIF":2.1,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we identify the Fokker-Planck equation for (reflected) Sticky Brownian Motion as a Wasserstein gradient flow in the space of probability measures. The driving functional is the relative entropy with respect to a non-standard reference measure, the sum of an absolutely continuous interior part plus a singular part supported on the boundary. Taking the small time-step limit in a minimizing movement (JKO scheme) we prove existence of weak solutions for the coupled system of PDEs satisfying in addition an Energy Dissipation Inequality.
{"title":"Sticky-reflecting diffusion as a Wasserstein gradient flow","authors":"Jean-Baptiste Casteras , Léonard Monsaingeon , Filippo Santambrogio","doi":"10.1016/j.matpur.2025.103721","DOIUrl":"10.1016/j.matpur.2025.103721","url":null,"abstract":"<div><div>In this paper we identify the Fokker-Planck equation for (reflected) Sticky Brownian Motion as a Wasserstein gradient flow in the space of probability measures. The driving functional is the relative entropy with respect to a non-standard reference measure, the sum of an absolutely continuous interior part plus a singular part supported on the boundary. Taking the small time-step limit in a minimizing movement (JKO scheme) we prove existence of weak solutions for the coupled system of PDEs satisfying in addition an Energy Dissipation Inequality.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"199 ","pages":"Article 103721"},"PeriodicalIF":2.1,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143906987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-01Epub Date: 2025-04-25DOI: 10.1016/j.matpur.2025.103722
Tommaso Cortopassi
We prove a novel stability estimate in between the regular Lagrangian flow of a Sobolev vector field and a piecewise affine approximation of such flow. This approximation of the flow is obtained by a (sort of) explicit Euler method, and it is the crucial tool to prove approximation results for the solution of the continuity equation by using the representation of the solution as the push-forward via the regular Lagrangian flow of the initial datum. We approximate the solution in two ways, using different approximations for both the flow and the initial datum. In the first case we give an estimate, which however holds only in probability, of the Wasserstein distance between the solution of the continuity equation and a discrete approximation of such solution. The approximate solution is defined as the push-forward of weighted Dirac deltas (whose centers are chosen in a probabilistic way). In the second case we give a deterministic estimate of the Wasserstein distance using a slightly different approximation of the regular Lagrangian flow and requiring more regularity on the velocity field u than in the previous case. An advantage of both approximations is that they provide an algorithm which is easily parallelizable and does not rely on any particular structure of the mesh with which we discretize (only in space) the domain. We also compare our estimates to similar ones previously obtained in [27], and we show how under certain hypotheses our method provides better convergence rates.
{"title":"An explicit Euler method for Sobolev vector fields with applications to the continuity equation on non Cartesian grids","authors":"Tommaso Cortopassi","doi":"10.1016/j.matpur.2025.103722","DOIUrl":"10.1016/j.matpur.2025.103722","url":null,"abstract":"<div><div>We prove a novel stability estimate in <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>)</mo></math></span> between the regular Lagrangian flow of a Sobolev vector field and a piecewise affine approximation of such flow. This approximation of the flow is obtained by a (sort of) explicit Euler method, and it is the crucial tool to prove approximation results for the solution of the continuity equation by using the representation of the solution as the push-forward via the regular Lagrangian flow of the initial datum. We approximate the solution in two ways, using different approximations for both the flow and the initial datum. In the first case we give an estimate, which however holds only in probability, of the Wasserstein distance between the solution of the continuity equation and a discrete approximation of such solution. The approximate solution is defined as the push-forward of weighted Dirac deltas (whose centers are chosen in a probabilistic way). In the second case we give a deterministic estimate of the Wasserstein distance using a slightly different approximation of the regular Lagrangian flow and requiring more regularity on the velocity field <em>u</em> than in the previous case. An advantage of both approximations is that they provide an algorithm which is easily parallelizable and does not rely on any particular structure of the mesh with which we discretize (only in space) the domain. We also compare our estimates to similar ones previously obtained in <span><span>[27]</span></span>, and we show how under certain hypotheses our method provides better convergence rates.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"199 ","pages":"Article 103722"},"PeriodicalIF":2.1,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143906988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-01Epub Date: 2025-04-28DOI: 10.1016/j.matpur.2025.103725
The Anh Bui
Let L be the Dunkl Laplacian on the Euclidean space associated with a normalized root R and a multiplicity function . In this paper, we first prove that the Besov and Triebel-Lizorkin spaces associated with the Dunkl Laplacian L are identical to the Besov and Triebel-Lizorkin spaces defined in the space of homogeneous type , where . Next, consider the Dunkl transform denoted by . We introduce the multiplier operator , defined as , where m is a bounded function defined on . Our second aim is to prove multiplier theorems, including the Hörmander multiplier theorem, for on the Besov and Tribel-Lizorkin spaces in the space of homogeneous type . Importantly, our findings present novel results, even in the specific case of the Hardy spaces.
{"title":"Harmonic analysis in Dunkl settings","authors":"The Anh Bui","doi":"10.1016/j.matpur.2025.103725","DOIUrl":"10.1016/j.matpur.2025.103725","url":null,"abstract":"<div><div>Let <em>L</em> be the Dunkl Laplacian on the Euclidean space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> associated with a normalized root <em>R</em> and a multiplicity function <span><math><mi>k</mi><mo>(</mo><mi>ν</mi><mo>)</mo><mo>≥</mo><mn>0</mn><mo>,</mo><mi>ν</mi><mo>∈</mo><mi>R</mi></math></span>. In this paper, we first prove that the Besov and Triebel-Lizorkin spaces associated with the Dunkl Laplacian <em>L</em> are identical to the Besov and Triebel-Lizorkin spaces defined in the space of homogeneous type <span><math><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo><mo>‖</mo><mo>⋅</mo><mo>‖</mo><mo>,</mo><mi>d</mi><mi>w</mi><mo>)</mo></math></span>, where <span><math><mi>d</mi><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∏</mo></mrow><mrow><mi>ν</mi><mo>∈</mo><mi>R</mi></mrow></msub><msup><mrow><mo>〈</mo><mi>ν</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow><mrow><mi>k</mi><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></msup><mi>d</mi><mi>x</mi></math></span>. Next, consider the Dunkl transform denoted by <span><math><mi>F</mi></math></span>. We introduce the multiplier operator <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>, defined as <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>f</mi><mo>=</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>m</mi><mi>F</mi><mi>f</mi><mo>)</mo></math></span>, where <em>m</em> is a bounded function defined on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. Our second aim is to prove multiplier theorems, including the Hörmander multiplier theorem, for <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> on the Besov and Tribel-Lizorkin spaces in the space of homogeneous type <span><math><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo><mo>‖</mo><mo>⋅</mo><mo>‖</mo><mo>,</mo><mi>d</mi><mi>w</mi><mo>)</mo></math></span>. Importantly, our findings present novel results, even in the specific case of the Hardy spaces.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"199 ","pages":"Article 103725"},"PeriodicalIF":2.1,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-01Epub Date: 2025-02-25DOI: 10.1016/j.matpur.2025.103687
Marco Cirant , Fanze Kong , Juncheng Wei , Xiaoyu Zeng
This paper is devoted to the study of Mean-field Games (MFG) systems in the mass-critical exponent case. We first derive the optimal Gagliardo-Nirenberg type inequality associated with the potential-free MFG system. Then, under some mild assumptions on the potential function, we show that there exists a critical mass such that the MFG system admits a least-energy solution if and only if the total mass of population density M satisfies . Moreover, the blow-up behavior of energy minimizers is characterized as . In particular, by considering the precise asymptotic expansions of the potential, we establish the refined blow-up behavior of ground states as . While studying the existence of least-energy solutions, we establish new local estimates for solutions to Hamilton-Jacobi equations with superlinear gradient terms.
{"title":"Critical mass phenomena and blow-up behaviors of ground states in stationary second order mean-field games systems with decreasing cost","authors":"Marco Cirant , Fanze Kong , Juncheng Wei , Xiaoyu Zeng","doi":"10.1016/j.matpur.2025.103687","DOIUrl":"10.1016/j.matpur.2025.103687","url":null,"abstract":"<div><div>This paper is devoted to the study of Mean-field Games (MFG) systems in the mass-critical exponent case. We first derive the optimal Gagliardo-Nirenberg type inequality associated with the potential-free MFG system. Then, under some mild assumptions on the potential function, we show that there exists a critical mass <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> such that the MFG system admits a least-energy solution if and only if the total mass of population density <em>M</em> satisfies <span><math><mi>M</mi><mo><</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Moreover, the blow-up behavior of energy minimizers is characterized as <span><math><mi>M</mi><mo>↗</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. In particular, by considering the precise asymptotic expansions of the potential, we establish the refined blow-up behavior of ground states as <span><math><mi>M</mi><mo>↗</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. While studying the existence of least-energy solutions, we establish new local <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msup></math></span> estimates for solutions to Hamilton-Jacobi equations with superlinear gradient terms.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"198 ","pages":"Article 103687"},"PeriodicalIF":2.1,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-01Epub Date: 2025-02-21DOI: 10.1016/j.matpur.2025.103685
L. Huysmans , Edriss S. Titi
We study the vanishing viscosity/diffusivity limit for the transport of a passive scalar by a bounded, divergence-free vector field . This is described by the Cauchy problem to the PDE , or with viscosity , to the PDE . In the first part of this work, we construct a bounded, divergence-free vector field for which, for any non-constant initial datum, the viscous solutions along different subsequences of the vanishing viscosity limit converge to different solutions to the inviscid problem. In the second part, we construct another bounded, divergence-free vector field for which, for every initial datum, the vanishing viscosity limit of solutions exists, is unique, and converges to an inviscid solution; however, when the initial datum is not constant, this inviscid limit is physically inadmissible due to increasing energy/entropy.
{"title":"Non-uniqueness & inadmissibility of the vanishing viscosity limit of the passive scalar transport equation","authors":"L. Huysmans , Edriss S. Titi","doi":"10.1016/j.matpur.2025.103685","DOIUrl":"10.1016/j.matpur.2025.103685","url":null,"abstract":"<div><div>We study the vanishing viscosity/diffusivity limit for the transport of a passive scalar <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi>R</mi></math></span> by a bounded, divergence-free vector field <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. This is described by the Cauchy problem to the PDE <span><math><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>f</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, or with viscosity <span><math><mi>ν</mi><mo>></mo><mn>0</mn></math></span>, to the PDE <span><math><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>f</mi><mo>)</mo><mo>−</mo><mi>ν</mi><mi>Δ</mi><mi>f</mi><mo>=</mo><mn>0</mn></math></span>. In the first part of this work, we construct a bounded, divergence-free vector field <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> for which, for any non-constant initial datum, the viscous solutions along different subsequences of the vanishing viscosity limit converge to different solutions to the inviscid problem. In the second part, we construct another bounded, divergence-free vector field <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> for which, for every initial datum, the vanishing viscosity limit of solutions exists, is unique, and converges to an inviscid solution; however, when the initial datum is not constant, this inviscid limit is physically inadmissible due to increasing energy/entropy.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"198 ","pages":"Article 103685"},"PeriodicalIF":2.1,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-01Epub Date: 2025-02-21DOI: 10.1016/j.matpur.2025.103689
Nuno Costa Dias , Franz Luef , João Nuno Prata
We revisit the uncertainty principle from the point of view suggested by A. Wigderson and Y. Wigderson. This approach is based on a primary uncertainty principle from which one can derive several inequalities expressing the impossibility of a simultaneous sharp localization in time and frequency. Moreover, it requires no specific properties of the Fourier transform and can therefore be easily applied to all operators satisfying the primary uncertainty principle. A. Wigderson and Y. Wigderson also suggested many generalizations to higher dimensions and stated several conjectures which we address in the present paper. We argue that we have to consider a more general primary uncertainty principle to prove the results suggested by the authors. As a by-product we obtain some new inequalities akin to the Cowling-Price uncertainty principle, a generalization of the Heisenberg uncertainty principle, and derive the entropic uncertainty principle from the primary uncertainty principles.
{"title":"On Wigdersons' approach to the uncertainty principle","authors":"Nuno Costa Dias , Franz Luef , João Nuno Prata","doi":"10.1016/j.matpur.2025.103689","DOIUrl":"10.1016/j.matpur.2025.103689","url":null,"abstract":"<div><div>We revisit the uncertainty principle from the point of view suggested by A. Wigderson and Y. Wigderson. This approach is based on a primary uncertainty principle from which one can derive several inequalities expressing the impossibility of a simultaneous sharp localization in time and frequency. Moreover, it requires no specific properties of the Fourier transform and can therefore be easily applied to all operators satisfying the primary uncertainty principle. A. Wigderson and Y. Wigderson also suggested many generalizations to higher dimensions and stated several conjectures which we address in the present paper. We argue that we have to consider a more general primary uncertainty principle to prove the results suggested by the authors. As a by-product we obtain some new inequalities akin to the Cowling-Price uncertainty principle, a generalization of the Heisenberg uncertainty principle, and derive the entropic uncertainty principle from the primary uncertainty principles.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"198 ","pages":"Article 103689"},"PeriodicalIF":2.1,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-01Epub Date: 2025-02-24DOI: 10.1016/j.matpur.2025.103691
Elia Fusi , Federico Giusti
Given a compact Chern-Ricci flat balanced orbifold, we show that its blow-up at a finite family of smooth points admits constant Chern scalar curvature balanced metrics, extending Arezzo-Pacard's construction to the balanced setting. Moreover, if the orbifold has isolated singularities and admits crepant resolutions, we show that they always carry Chern-Ricci flat balanced metrics, without any further hypothesis. Along the way, we study two Lichnerowicz-type operators originating from complex connections and investigate the relation between their kernel and holomorphic vector fields, with the aim of discussing the general constant Chern scalar curvature balanced case. Ultimately, we provide a variation of the main Theorem assuming the existence of a special -form and we present several classes of examples in which all our results can be applied.
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