Pub Date : 2025-06-16DOI: 10.1016/j.matpur.2025.103755
Giulio Ciraolo, Matteo Cozzi, Michele Gatti
A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations in must be radial and radially decreasing. In this paper, we consider both energy solutions in and non-energy local weak solutions to small perturbations of these equations, and study its quantitative stability counterpart.
To the best of our knowledge, the present work provides the first quantitative stability result for non-energy solutions to semilinear equations involving the Laplacian, even for the critical nonlinearity.
Gidas, Ni &；Nirenberg断言，在无穷远处衰减的半线性方程Δu+f(u)=0的正经典解在Rn中必须是径向和径向递减的。本文考虑了这些方程D1,2（Rn）的能量解和小扰动的非能量局部弱解，并研究了它们的定量稳定性对应项。据我们所知，目前的工作提供了第一个涉及拉普拉斯方程的半线性方程的非能量解的定量稳定性结果，甚至对于临界非线性也是如此。
{"title":"A quantitative study of radial symmetry for solutions to semilinear equations in Rn","authors":"Giulio Ciraolo, Matteo Cozzi, Michele Gatti","doi":"10.1016/j.matpur.2025.103755","DOIUrl":"10.1016/j.matpur.2025.103755","url":null,"abstract":"<div><div>A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations <span><math><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> must be radial and radially decreasing. In this paper, we consider both energy solutions in <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> and non-energy local weak solutions to small perturbations of these equations, and study its quantitative stability counterpart.</div><div>To the best of our knowledge, the present work provides the first quantitative stability result for non-energy solutions to semilinear equations involving the Laplacian, even for the critical nonlinearity.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103755"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321252","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 : 2025-06-16DOI: 10.1016/j.matpur.2025.103756
Zhaosheng Feng , Junjie Zhang , Shenzhou Zheng
We consider a nonlinear elliptic equation of the form , where the principle part depends on the solution itself and the right-hand data μ is a signed Radon measure. The associated nonlinearity is assumed to satisfy the -BMO condition in x and the Lipschitz continuity condition in u, and its growth in Du is like the -Laplacian, while the boundary of underlying domain is assumed to be Reifenberg flat. We establish an optimal global Calderón-Zygmund type estimate in weighted Lorentz spaces for the gradients of very weak solutions to such a measure data problem. This is achieved by developing the perturbation method and modifying the weighted Vitali type covering argument.
{"title":"Weighted gradient estimates to nonlinear elliptic equations of p(x)-growth with measure data","authors":"Zhaosheng Feng , Junjie Zhang , Shenzhou Zheng","doi":"10.1016/j.matpur.2025.103756","DOIUrl":"10.1016/j.matpur.2025.103756","url":null,"abstract":"<div><div>We consider a nonlinear elliptic equation of the form <span><math><mo>−</mo><mtext>div</mtext><mi>A</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo><mo>=</mo><mi>μ</mi></math></span>, where the principle part depends on the solution itself and the right-hand data <em>μ</em> is a signed Radon measure. The associated nonlinearity is assumed to satisfy the <span><math><mo>(</mo><mi>δ</mi><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span>-BMO condition in <em>x</em> and the Lipschitz continuity condition in <em>u</em>, and its growth in <em>Du</em> is like the <span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>-Laplacian, while the boundary of underlying domain is assumed to be Reifenberg flat. We establish an optimal global Calderón-Zygmund type estimate in weighted Lorentz spaces for the gradients of very weak solutions to such a measure data problem. This is achieved by developing the perturbation method and modifying the weighted Vitali type covering argument.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103756"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320783","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}
We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of inhomogeneous hypocoercivity. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must require an additional regularity assumption on the initial data to ensure the decay.
Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memory.
{"title":"Large-time asymptotics for hyperbolic systems with non-symmetric relaxation: An algorithmic approach","authors":"Timothée Crin-Barat , Lorenzo Liverani , Ling-Yun Shou , Enrique Zuazua","doi":"10.1016/j.matpur.2025.103757","DOIUrl":"10.1016/j.matpur.2025.103757","url":null,"abstract":"<div><div>We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of <em>inhomogeneous hypocoercivity</em>. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must require an additional regularity assumption on the initial data to ensure the decay.</div><div>Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memory.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103757"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290769","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 : 2025-06-16DOI: 10.1016/j.matpur.2025.103753
Purba Das , Donghan Kim
We study the concept of (generalized) p-th variation of a real-valued continuous function along a general class of refining sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of -norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to -norm of the Schauder coefficients. This result provides an isomorphism between the space of α-Hölder continuous functions with finite (generalized) p-th variation along a given partition sequence and a subclass of infinite-dimensional matrices equipped with an appropriate norm, in the spirit of Ciesielski.
{"title":"On isomorphism of the space of continuous functions with finite p-th variation along a partition sequence","authors":"Purba Das , Donghan Kim","doi":"10.1016/j.matpur.2025.103753","DOIUrl":"10.1016/j.matpur.2025.103753","url":null,"abstract":"<div><div>We study the concept of (generalized) <em>p</em>-th variation of a real-valued continuous function along a general class of refining sequence of partitions. We show that the finiteness of the <em>p</em>-th variation of a given function is closely related to the finiteness of <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-norm of the Schauder coefficients. This result provides an isomorphism between the space of <em>α</em>-Hölder continuous functions with finite (generalized) <em>p</em>-th variation along a given partition sequence and a subclass of infinite-dimensional matrices equipped with an appropriate norm, in the spirit of Ciesielski.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103753"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320784","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 : 2025-06-13DOI: 10.1016/j.matpur.2025.103763
Vincenzo Antonelli , Francesco Malaspina , Simone Marchesi , Joan Pons-Llopis
In this work we study the moduli spaces of instanton bundles on the flag twistor space . We stratify them in terms of the minimal twist supporting global sections and we introduce the notion of (special) ‘t Hooft bundle on F. In particular we prove that there exist μ-stable ‘t Hooft bundles for each admissible charge k. We completely describe the geometric structure of the moduli space of (special) ‘t Hooft bundles for arbitrary charge k. Along the way to reach these goals, we describe the possible structures of multiple curves supported on some rational curves in F as well as the family of del Pezzo surfaces realized as hyperplane sections of F. Finally we investigate the splitting behavior of ‘t Hooft bundles when restricted to conics.
{"title":"‘t Hooft bundles on the complete flag threefold and moduli spaces of instantons","authors":"Vincenzo Antonelli , Francesco Malaspina , Simone Marchesi , Joan Pons-Llopis","doi":"10.1016/j.matpur.2025.103763","DOIUrl":"10.1016/j.matpur.2025.103763","url":null,"abstract":"<div><div>In this work we study the moduli spaces of instanton bundles on the flag twistor space <span><math><mi>F</mi><mo>:</mo><mo>=</mo><mi>F</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. We stratify them in terms of the minimal twist supporting global sections and we introduce the notion of (special) ‘t Hooft bundle on <em>F</em>. In particular we prove that there exist <em>μ</em>-stable ‘t Hooft bundles for each admissible charge <em>k</em>. We completely describe the geometric structure of the moduli space of (special) ‘t Hooft bundles for arbitrary charge <em>k</em>. Along the way to reach these goals, we describe the possible structures of multiple curves supported on some rational curves in <em>F</em> as well as the family of del Pezzo surfaces realized as hyperplane sections of <em>F</em>. Finally we investigate the splitting behavior of ‘t Hooft bundles when restricted to conics.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103763"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313662","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 : 2025-06-13DOI: 10.1016/j.matpur.2025.103761
Jean-Michel Coron , Shengquan Xiang
In this paper, we study the controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important properties, such as small-time local controllability, local quantitative rapid stabilization, obstruction to semi-global asymptotic stabilization, and global controllability to geodesics. Surprisingly, due to the geometric feature of the equation we can also prove the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding open problem for nonlinear heat equations.
{"title":"Global controllability to harmonic maps of the heat flow from a circle to a sphere","authors":"Jean-Michel Coron , Shengquan Xiang","doi":"10.1016/j.matpur.2025.103761","DOIUrl":"10.1016/j.matpur.2025.103761","url":null,"abstract":"<div><div>In this paper, we study the controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important properties, such as small-time local controllability, local quantitative rapid stabilization, obstruction to semi-global asymptotic stabilization, and global controllability to geodesics. Surprisingly, due to the geometric feature of the equation we can also prove the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding open problem for nonlinear heat equations.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103761"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491518","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 : 2025-06-13DOI: 10.1016/j.matpur.2025.103759
Lei Zhang , Xiao-Qiang Zhao
In this paper, we study the propagation dynamics for a large class of time and space heterogeneous reaction-diffusion equations , where represents the shifting distance, and the nonlinearity is asymptotically of KPP type as and is negative as . Let be the spreading speed of the limiting equation . Under the assumption that the shifting speed admits a uniform mean c, we show that the solutions with compactly supported initial data go to zero eventually when , the leftward spreading speed is when , and the rightward spreading speed is c and when and , respectively. We also establish the existence, uniqueness and nonexistence of the forced traveling wave in terms of the sign of .
{"title":"Spreading properties and forced traveling waves of reaction-diffusion equations in a time-heterogeneous shifting environment","authors":"Lei Zhang , Xiao-Qiang Zhao","doi":"10.1016/j.matpur.2025.103759","DOIUrl":"10.1016/j.matpur.2025.103759","url":null,"abstract":"<div><div>In this paper, we study the propagation dynamics for a large class of time and space heterogeneous reaction-diffusion equations <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>u</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo>−</mo><mi>ω</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span>, where <span><math><mi>ω</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> represents the shifting distance, and the nonlinearity <span><math><mi>u</mi><mi>g</mi><mo>(</mo><mi>ξ</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span> is asymptotically of KPP type as <span><math><mi>ξ</mi><mo>→</mo><mo>−</mo><mo>∞</mo></math></span> and is negative as <span><math><mi>ξ</mi><mo>→</mo><mo>+</mo><mo>∞</mo></math></span>. Let <span><math><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> be the spreading speed of the limiting equation <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>u</mi><mi>g</mi><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span>. Under the assumption that the shifting speed <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></math></span> admits a uniform mean <em>c</em>, we show that the solutions with compactly supported initial data go to zero eventually when <span><math><mi>c</mi><mo>≤</mo><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the leftward spreading speed is <span><math><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> when <span><math><mi>c</mi><mo>></mo><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, and the rightward spreading speed is <em>c</em> and <span><math><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> when <span><math><mi>c</mi><mo>∈</mo><mo>(</mo><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>,</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span> and <span><math><mi>c</mi><mo>≥</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, respectively. We also establish the existence, uniqueness and nonexistence of the forced traveling wave in terms of the sign of <span><math><mi>c</mi><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103759"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322088","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 : 2025-06-13DOI: 10.1016/j.matpur.2025.103760
Yaxiong Liu , Zhuo Liu , Hui Yang , Xiangyu Zhou
In this paper, we obtain a Le Potier-type isomorphism theorem twisted with multiplier submodule sheaves, which relates a holomorphic vector bundle endowed with a strongly Nakano semi-positive singular Hermitian metric to the tautological line bundle with the induced metric. As applications, we obtain a Kollár-type injectivity theorem, a Nadel-type vanishing theorem, and a singular holomorphic Morse inequality for holomorphic vector bundles and so on.
{"title":"A Le Potier-type isomorphism twisted with multiplier submodule sheaves","authors":"Yaxiong Liu , Zhuo Liu , Hui Yang , Xiangyu Zhou","doi":"10.1016/j.matpur.2025.103760","DOIUrl":"10.1016/j.matpur.2025.103760","url":null,"abstract":"<div><div>In this paper, we obtain a Le Potier-type isomorphism theorem twisted with multiplier submodule sheaves, which relates a holomorphic vector bundle endowed with a strongly Nakano semi-positive singular Hermitian metric to the tautological line bundle with the induced metric. As applications, we obtain a Kollár-type injectivity theorem, a Nadel-type vanishing theorem, and a singular holomorphic Morse inequality for holomorphic vector bundles and so on.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103760"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313732","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 : 2025-06-13DOI: 10.1016/j.matpur.2025.103762
Yu-Chu Lin , Haitao Wang , Kung-Chien Wu
Consider the Boltzmann equation in the perturbation regime. Since the macroscopic quantities in the background global Maxwellian are obtained through measurements, there are typically some errors involved. This paper investigates the effect of background variations on the solution for a given initial perturbation. Our findings demonstrate that the solution changes continuously with variations in the background and provide a sharp time decay estimate of the associated errors. The proof relies on refined estimates for the linearized solution operator and a proper decomposition of the nonlinear solution.
{"title":"Stability of background perturbation for Boltzmann equation","authors":"Yu-Chu Lin , Haitao Wang , Kung-Chien Wu","doi":"10.1016/j.matpur.2025.103762","DOIUrl":"10.1016/j.matpur.2025.103762","url":null,"abstract":"<div><div>Consider the Boltzmann equation in the perturbation regime. Since the macroscopic quantities in the background global Maxwellian are obtained through measurements, there are typically some errors involved. This paper investigates the effect of background variations on the solution for a given initial perturbation. Our findings demonstrate that the solution changes continuously with variations in the background and provide a sharp time decay estimate of the associated errors. The proof relies on refined estimates for the linearized solution operator and a proper decomposition of the nonlinear solution.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103762"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338507","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 : 2025-06-13DOI: 10.1016/j.matpur.2025.103764
Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring
We consider local weak solutions to the fractional p-Poisson equation of order s, i.e. . In the range and we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any that The qualitative result is accompanied by a local quantitative estimate.
{"title":"Gradient estimates for the fractional p-Poisson equation","authors":"Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring","doi":"10.1016/j.matpur.2025.103764","DOIUrl":"10.1016/j.matpur.2025.103764","url":null,"abstract":"<div><div>We consider local weak solutions to the fractional <em>p</em>-Poisson equation of order <em>s</em>, i.e. <span><math><msup><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi></math></span>. In the range <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></math></span> we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any <span><math><mi>q</mi><mo>></mo><mn>1</mn></math></span> that<span><span><span><math><mi>f</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mfrac><mrow><mi>q</mi><mi>p</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msubsup><mspace></mspace><mo>⟹</mo><mspace></mspace><mi>∇</mi><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mi>q</mi><mi>p</mi></mrow></msubsup><mo>.</mo></math></span></span></span> The qualitative result is accompanied by a local quantitative estimate.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103764"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338508","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}
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