Pub Date : 2024-10-14DOI: 10.1016/j.jde.2024.10.001
Thialita M. Nascimento, Eduardo V. Teixeira
We prove that if satisfies in , in the viscosity sense, for some fully nonlinear -elliptic operator, then , with appropriate estimates, for a sharp exponent verifying The upper bound is conjectured to be the optimal one. Thus, the main new information proven in this paper is that the sharp Hessian integrability exponent for viscosity supersolutions in the plane remains at least 81.45% of its upper bound. This greatly improves previous known estimates.
{"title":"On the sharp Hessian integrability conjecture in the plane","authors":"Thialita M. Nascimento, Eduardo V. Teixeira","doi":"10.1016/j.jde.2024.10.001","DOIUrl":"10.1016/j.jde.2024.10.001","url":null,"abstract":"<div><div>We prove that if <span><math><mi>u</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> satisfies <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>≤</mo><mn>0</mn></math></span> in <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, in the viscosity sense, for some fully nonlinear <span><math><mo>(</mo><mi>λ</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></math></span>-elliptic operator, then <span><math><mi>u</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>ε</mi></mrow></msup><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>)</mo></math></span>, with appropriate estimates, for a sharp exponent <span><math><mi>ε</mi><mo>=</mo><mi>ε</mi><mo>(</mo><mi>λ</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></math></span> verifying<span><span><span><math><mfrac><mrow><mn>1.629</mn></mrow><mrow><mfrac><mrow><mi>Λ</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfrac><mo><</mo><mi>ε</mi><mo>(</mo><mi>λ</mi><mo>,</mo><mi>Λ</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mi>Λ</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfrac><mo>.</mo></math></span></span></span> The upper bound is conjectured to be the optimal one. Thus, the main new information proven in this paper is that the sharp Hessian integrability exponent for viscosity supersolutions in the plane remains <em>at least</em> 81.45% of its upper bound. This greatly improves previous known estimates.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"414 ","pages":"Pages 890-903"},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432984","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 : 2024-10-14DOI: 10.1016/j.jde.2024.09.030
Zhenyu Wan , Ying Wang , Min Zhu
This paper aims to understand a blow-up mechanism on a family of shallow-water models with linear dispersion, which are linked with the modified Camassa-Holm equation and the Novikov equation. We first demonstrate the local well-posedness of the model equation in Besov spaces. Our blow-up analysis begins with two cases where the first case is and then we deduce the results on the curvature blow-up in finite time. To overcome the lack of conservation in the functional due to weak linear dispersion, we can determine a suitable alternative via a slight modification to conserved quantity (see Lemma 4.1). Furthermore, we explore the formation of singularities in another case when nonlocal terms are absent. Lastly, we investigate the Gevrey regularity and analyticity of solutions for Cauchy problem within a specified range of Gevrey-Sobolev spaces by employing the generalized Ovsyannikov theorem and study the continuity of the data-to-solution mapping.
{"title":"On the Cauchy problem for a combined mCH-Novikov integrable equation with linear dispersion","authors":"Zhenyu Wan , Ying Wang , Min Zhu","doi":"10.1016/j.jde.2024.09.030","DOIUrl":"10.1016/j.jde.2024.09.030","url":null,"abstract":"<div><div>This paper aims to understand a blow-up mechanism on a family of shallow-water models with linear dispersion, which are linked with the modified Camassa-Holm equation and the Novikov equation. We first demonstrate the local well-posedness of the model equation in Besov spaces. Our blow-up analysis begins with two cases where the first case is <span><math><mn>2</mn><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>3</mn><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≠</mo><mn>0</mn></math></span> and then we deduce the results on the curvature blow-up in finite time. To overcome the lack of conservation in the functional due to weak linear dispersion, we can determine a suitable alternative via a slight modification to conserved quantity <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><mi>u</mi><mo>]</mo></math></span> (see <span><span>Lemma 4.1</span></span>). Furthermore, we explore the formation of singularities in another case when nonlocal terms are absent. Lastly, we investigate the Gevrey regularity and analyticity of solutions for Cauchy problem within a specified range of Gevrey-Sobolev spaces by employing the generalized Ovsyannikov theorem and study the continuity of the data-to-solution mapping.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 727-767"},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432940","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}
This article is concerned with “up to -regularity results” about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators.
First of all, an estimate on the norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities.
We then prove the interior -regularity and the -regularity up to the boundary of weak solutions, which extends previous results by the authors (Su et al., 2022, [20]), where the nonlinearities considered were of subcritical type.
In addition, we establish the interior -regularity of solutions for all and the -regularity up to the boundary for all , with sharp regularity exponents.
For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue.
{"title":"On some regularity properties of mixed local and nonlocal elliptic equations","authors":"Xifeng Su , Enrico Valdinoci , Yuanhong Wei , Jiwen Zhang","doi":"10.1016/j.jde.2024.10.003","DOIUrl":"10.1016/j.jde.2024.10.003","url":null,"abstract":"<div><div>This article is concerned with “up to <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>-regularity results” about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators.</div><div>First of all, an estimate on the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities.</div><div>We then prove the interior <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>-regularity and the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>-regularity up to the boundary of weak solutions, which extends previous results by the authors (Su et al., 2022, <span><span>[20]</span></span>), where the nonlinearities considered were of subcritical type.</div><div>In addition, we establish the interior <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>-regularity of solutions for all <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>-regularity up to the boundary for all <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span>, with sharp regularity exponents.</div><div>For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 576-613"},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432942","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 : 2024-10-10DOI: 10.1016/j.jde.2024.09.058
T.J. Doumatè , J. Kotounou , L.A. Leadi , R.B. Salako
We study the asymptotic dynamics of solutions to a diffusive epidemic model with varying population dynamics. The large-time behavior of solutions is completely described in spatially homogeneous environments. When the environment is spatially heterogeneous, it is shown that there exist two critical numbers such that if the ratio of the infected population diffusion rate and the susceptible population rate either exceeds or is less than , then the epidemic model has an endemic equilibrium (EE) solution if and only if the basic reproduction number (BRN) exceeds one. The unique EE is non-degenerate if . Furthermore, results on the global dynamics of solutions are established when . Our results shed some light on the differences on disease predictions for constant total population size models versus varying population size models. Results on the asymptotic profiles of the EEs for small population diffusion rates are also established.
我们研究了具有不同人口动态的扩散性流行病模型解的渐近动态。在空间均质环境中，解的大时间行为被完全描述。当环境在空间上是异质的时，研究表明存在两个临界数 1≤σ⁎≤σ⁎<∞ ，即如果受感染种群扩散速率与易感种群速率之比 dIdS 超过 σ⁎，或小于 σ⁎，那么当且仅当基本繁殖数（BRN）超过 1 时，该流行病模型才有流行均衡（EE）解。如果 dIdS≥σ⁎ ，则唯一的 EE 是非退化的。此外，当 σ⁎=1 时，还建立了关于解的全局动力学的结果。我们的结果揭示了总种群数量恒定模型与种群数量变化模型在疾病预测方面的差异。此外，我们还得出了小种群扩散率 EE 的渐近曲线结果。
{"title":"Dynamics of classical solutions to a diffusive epidemic model with varying population demographics","authors":"T.J. Doumatè , J. Kotounou , L.A. Leadi , R.B. Salako","doi":"10.1016/j.jde.2024.09.058","DOIUrl":"10.1016/j.jde.2024.09.058","url":null,"abstract":"<div><div>We study the asymptotic dynamics of solutions to a diffusive epidemic model with varying population dynamics. The large-time behavior of solutions is completely described in spatially homogeneous environments. When the environment is spatially heterogeneous, it is shown that there exist two critical numbers <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>σ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>≤</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo><</mo><mo>∞</mo></math></span> such that if the ratio <span><math><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>I</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfrac></math></span> of the infected population diffusion rate and the susceptible population rate either exceeds <span><math><msup><mrow><mi>σ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> or is less than <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span>, then the epidemic model has an endemic equilibrium (EE) solution if and only if the basic reproduction number (BRN) exceeds one. The unique EE is non-degenerate if <span><math><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>I</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfrac><mo>≥</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Furthermore, results on the global dynamics of solutions are established when <span><math><msup><mrow><mi>σ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mn>1</mn></math></span>. Our results shed some light on the differences on disease predictions for constant total population size models versus varying population size models. Results on the asymptotic profiles of the EEs for small population diffusion rates are also established.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 491-530"},"PeriodicalIF":2.4,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425004","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 : 2024-10-09DOI: 10.1016/j.jde.2024.10.005
William R. Green , Connor Lane , Benjamin Lyons , Shyam Ravishankar , Aden Shaw
We investigate dispersive estimates for the massless three dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a decay rate as an operator from to regardless of the existence of zero energy eigenfunctions. We also show this decay rate may be improved to for any at the cost of spatial weights. This estimate, along with the conservation law allows one to deduce a family of Strichartz estimates in the case of a threshold eigenvalue. We classify the structure of threshold obstructions as being composed of zero energy eigenfunctions. Finally, we show the Dirac evolution is bounded for all time with minimal requirements on the decay of the potential and smoothness of initial data.
{"title":"The massless Dirac equation in three dimensions: Dispersive estimates and zero energy obstructions","authors":"William R. Green , Connor Lane , Benjamin Lyons , Shyam Ravishankar , Aden Shaw","doi":"10.1016/j.jde.2024.10.005","DOIUrl":"10.1016/j.jde.2024.10.005","url":null,"abstract":"<div><div>We investigate dispersive estimates for the massless three dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a <span><math><msup><mrow><mo>〈</mo><mi>t</mi><mo>〉</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> decay rate as an operator from <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> to <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> regardless of the existence of zero energy eigenfunctions. We also show this decay rate may be improved to <span><math><msup><mrow><mo>〈</mo><mi>t</mi><mo>〉</mo></mrow><mrow><mo>−</mo><mn>1</mn><mo>−</mo><mi>γ</mi></mrow></msup></math></span> for any <span><math><mn>0</mn><mo>≤</mo><mi>γ</mi><mo><</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span> at the cost of spatial weights. This estimate, along with the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> conservation law allows one to deduce a family of Strichartz estimates in the case of a threshold eigenvalue. We classify the structure of threshold obstructions as being composed of zero energy eigenfunctions. Finally, we show the Dirac evolution is bounded for all time with minimal requirements on the decay of the potential and smoothness of initial data.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 449-490"},"PeriodicalIF":2.4,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425003","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 : 2024-10-08DOI: 10.1016/j.jde.2024.09.057
Rong Zhou, Shi-Liang Wu
In this paper, we investigate the spreading properties for a predator-prey system with nonlocal dispersal and climate change. We are concerned with the case when the prey grow relatively rapidly at one side of the habitat and grow relatively slowly at another side of the habitat. We are interested in the effect of the climate change on the spreading speed of the predator and prey. In the case where the predator is faster than the prey, we show that the predator and the prey have the same leftward spreading speed and the same rightward spreading speed, respectively, which depend on c, the climate change speed, and , the maximum and minimum speeds of the prey without predator. While in the case where the prey is faster than the predator, we find that the solution can form a multi-layer wave and the two species have different leftward spreading speeds and different rightward spreading speeds, which depend on c, and , the maximum and minimum speeds of the predator when the density of the prey attains its maximum and minimum capacity.
在本文中，我们研究了具有非局部扩散和气候变化的捕食者-猎物系统的扩散特性。我们关注的是猎物在栖息地一侧生长相对较快而在栖息地另一侧生长相对较慢的情况。我们感兴趣的是气候变化对捕食者和猎物扩散速度的影响。在捕食者速度快于猎物的情况下，我们发现捕食者和猎物的向左扩散速度和向右扩散速度分别相同，这取决于气候变化速度 c 和 c1⁎（±∞），即猎物在没有捕食者的情况下的最大和最小速度。而在猎物的速度快于捕食者的情况下，我们发现解可以形成多层波浪，两种物种的左向展向速度和右向展向速度不同，分别取决于 c、c1⁎(±∞)和 c2⁎(±∞)，即当猎物密度达到最大和最小容量时捕食者的最大和最小速度。
{"title":"Spreading properties for a predator-prey system with nonlocal dispersal and climate change","authors":"Rong Zhou, Shi-Liang Wu","doi":"10.1016/j.jde.2024.09.057","DOIUrl":"10.1016/j.jde.2024.09.057","url":null,"abstract":"<div><div>In this paper, we investigate the spreading properties for a predator-prey system with nonlocal dispersal and climate change. We are concerned with the case when the prey grow relatively rapidly at one side of the habitat and grow relatively slowly at another side of the habitat. We are interested in the effect of the climate change on the spreading speed of the predator and prey. In the case where the predator is faster than the prey, we show that the predator and the prey have the same leftward spreading speed and the same rightward spreading speed, respectively, which depend on <em>c</em>, the climate change speed, and <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mo>±</mo><mo>∞</mo><mo>)</mo></math></span>, the maximum and minimum speeds of the prey without predator. While in the case where the prey is faster than the predator, we find that the solution can form a multi-layer wave and the two species have different leftward spreading speeds and different rightward spreading speeds, which depend on <em>c</em>, <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mo>±</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mo>±</mo><mo>∞</mo><mo>)</mo></math></span>, the maximum and minimum speeds of the predator when the density of the prey attains its maximum and minimum capacity.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"415 ","pages":"Pages 791-828"},"PeriodicalIF":2.4,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417322","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 : 2024-10-04DOI: 10.1016/j.jde.2024.09.054
Roberto de A. Capistrano–Filho , Thiago Yukio Tanaka
The main goal of this manuscript is to prove the existence of insensitizing controls for the fourth-order dispersive nonlinear Schrödinger equation with cubic nonlinearity. To obtain the main result we prove a null controllability property for a coupled fourth-order Schrödinger cascade type system with zero-order coupling which is equivalent to the insensitizing control problem. Precisely, employing a new Carleman estimates, we first obtain a null controllability result for the linearized system around zero, and then the null controllability for the nonlinear case is extended using an inverse mapping theorem.
{"title":"Controls insensitizing the norm of solution of a Schrödinger type system with mixed dispersion","authors":"Roberto de A. Capistrano–Filho , Thiago Yukio Tanaka","doi":"10.1016/j.jde.2024.09.054","DOIUrl":"10.1016/j.jde.2024.09.054","url":null,"abstract":"<div><div>The main goal of this manuscript is to prove the existence of insensitizing controls for the fourth-order dispersive nonlinear Schrödinger equation with cubic nonlinearity. To obtain the main result we prove a null controllability property for a coupled fourth-order Schrödinger cascade type system with zero-order coupling which is equivalent to the insensitizing control problem. Precisely, employing a new Carleman estimates, we first obtain a null controllability result for the linearized system around zero, and then the null controllability for the nonlinear case is extended using an inverse mapping theorem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 357-395"},"PeriodicalIF":2.4,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425001","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 : 2024-10-04DOI: 10.1016/j.jde.2024.09.056
Xue Wang, Xiaojing Xu
This paper concerns the Cauchy problem to the compressible magnetohydrodynamic equations in with the constant state of density at far field being vacuum or nonvacuum. Under the conditions that the adiabatic constant , the shear viscosity coefficient μ is a positive constant, and the bulk one with , we establish the global existence and uniqueness of strong solutions. In particular, the initial data can be arbitrarily large and the density is allowed to vanish initially. These results generalize and improve previous ones by Huang-Li (2022) and Jiu-Wang-Xin (2018) for compressible Navier-Stokes equations. This paper introduces some key weighted estimates on H and presents some delicate analysis to exploit the decay properties of solutions due to the strong coupling and interplay interaction.
本文涉及远场密度恒定状态为真空或非真空的 R2 中可压缩磁流体动力学方程的 Cauchy 问题。在绝热常数γ>1、剪切粘度系数μ为正常数、体积系数λ(ρ)=ρβ（β>4/3）的条件下，我们建立了强解的全局存在性和唯一性。特别是，初始数据可以任意大，而且允许密度在初始时消失。这些结果概括并改进了黄立（2022）和裘旺新（2018）之前针对可压缩纳维-斯托克斯方程的结果。本文介绍了对 H 的一些关键加权估计，并提出了一些精细分析，以利用强耦合和相互作用引起的解的衰减特性。
{"title":"Global existence of strong solutions to the compressible magnetohydrodynamic equations with large initial data and vacuum in R2","authors":"Xue Wang, Xiaojing Xu","doi":"10.1016/j.jde.2024.09.056","DOIUrl":"10.1016/j.jde.2024.09.056","url":null,"abstract":"<div><div>This paper concerns the Cauchy problem to the compressible magnetohydrodynamic equations in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with the constant state of density at far field being vacuum or nonvacuum. Under the conditions that the adiabatic constant <span><math><mi>γ</mi><mo>></mo><mn>1</mn></math></span>, the shear viscosity coefficient <em>μ</em> is a positive constant, and the bulk one <span><math><mi>λ</mi><mo>(</mo><mi>ρ</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mi>β</mi></mrow></msup></math></span> with <span><math><mi>β</mi><mo>></mo><mn>4</mn><mo>/</mo><mn>3</mn></math></span>, we establish the global existence and uniqueness of strong solutions. In particular, the initial data can be arbitrarily large and the density is allowed to vanish initially. These results generalize and improve previous ones by Huang-Li (2022) and Jiu-Wang-Xin (2018) for compressible Navier-Stokes equations. This paper introduces some key weighted estimates on <em>H</em> and presents some delicate analysis to exploit the decay properties of solutions due to the strong coupling and interplay interaction.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"415 ","pages":"Pages 722-763"},"PeriodicalIF":2.4,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417303","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 : 2024-10-04DOI: 10.1016/j.jde.2024.09.023
Qiuyi Dai, Xing Yi
<div><div>Let <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> be a bounded convex domain with boundary ∂Ω and <span><math><mi>ν</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> be the unit outer vector normal to ∂Ω at <em>x</em>. Let <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> be the unit sphere in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. Then, the Gauss mapping <span><math><mi>g</mi><mo>:</mo><mo>∂</mo><mi>Ω</mi><mo>→</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, defined almost everywhere with respect to surface measure <em>σ</em>, is given by <span><math><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>ν</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. For <span><math><mn>0</mn><mo><</mo><mi>β</mi><mo><</mo><mn>1</mn></math></span>, it is well known that the following problem of sub-linear elliptic equation<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>φ</mi><mo>=</mo><msup><mrow><mi>φ</mi></mrow><mrow><mi>β</mi></mrow></msup><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>φ</mi><mo>></mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>φ</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span> has a unique solution. Moreover, it is easy to prove that each component of <span><math><mi>∇</mi><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is well-defined almost everywhere on ∂Ω with respect to <em>σ</em>. Therefore, we can assign a measure <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>Ω</mi></mrow></msub></math></span> on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> such that <span><math><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mo>=</mo><msub><mrow><mi>g</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mo>|</mo><mi>∇</mi><mi>φ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>σ</mi><mo>)</mo></math></span>. That is<span><span><span><math><munder><mo>∫</mo><mrow><msup><mrow><mi>S</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></munder><mi>f</mi><mo>(</mo><mi>ξ</mi><mo>)</mo><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mo>(</mo><mi>ξ</mi><mo>)</mo><mo>=</mo><munder><mo>∫</mo><mrow><mo>∂</mo><mi>Ω</mi></mrow></munder><mi>f</mi><mo>(</mo><mi>ν</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>|</mo><mi>∇</mi><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>σ</mi></math></span></span></span> for every <span><math><mi>f</mi><mo>∈</mo><mi>C</mi><mo>(</mo><msup><mrow><mi>S</mi></mrow>
{"title":"Minkowski problems arise from sub-linear elliptic equations","authors":"Qiuyi Dai, Xing Yi","doi":"10.1016/j.jde.2024.09.023","DOIUrl":"10.1016/j.jde.2024.09.023","url":null,"abstract":"<div><div>Let <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> be a bounded convex domain with boundary ∂Ω and <span><math><mi>ν</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> be the unit outer vector normal to ∂Ω at <em>x</em>. Let <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> be the unit sphere in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. Then, the Gauss mapping <span><math><mi>g</mi><mo>:</mo><mo>∂</mo><mi>Ω</mi><mo>→</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, defined almost everywhere with respect to surface measure <em>σ</em>, is given by <span><math><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>ν</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. For <span><math><mn>0</mn><mo><</mo><mi>β</mi><mo><</mo><mn>1</mn></math></span>, it is well known that the following problem of sub-linear elliptic equation<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>φ</mi><mo>=</mo><msup><mrow><mi>φ</mi></mrow><mrow><mi>β</mi></mrow></msup><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>φ</mi><mo>></mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>φ</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span> has a unique solution. Moreover, it is easy to prove that each component of <span><math><mi>∇</mi><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is well-defined almost everywhere on ∂Ω with respect to <em>σ</em>. Therefore, we can assign a measure <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>Ω</mi></mrow></msub></math></span> on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> such that <span><math><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mo>=</mo><msub><mrow><mi>g</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mo>|</mo><mi>∇</mi><mi>φ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>σ</mi><mo>)</mo></math></span>. That is<span><span><span><math><munder><mo>∫</mo><mrow><msup><mrow><mi>S</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></munder><mi>f</mi><mo>(</mo><mi>ξ</mi><mo>)</mo><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mo>(</mo><mi>ξ</mi><mo>)</mo><mo>=</mo><munder><mo>∫</mo><mrow><mo>∂</mo><mi>Ω</mi></mrow></munder><mi>f</mi><mo>(</mo><mi>ν</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>|</mo><mi>∇</mi><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>σ</mi></math></span></span></span> for every <span><math><mi>f</mi><mo>∈</mo><mi>C</mi><mo>(</mo><msup><mrow><mi>S</mi></mrow>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"415 ","pages":"Pages 764-790"},"PeriodicalIF":2.4,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417321","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 : 2024-10-04DOI: 10.1016/j.jde.2024.09.059
Wei Hong, Wei Liu, Luhan Yang
This paper is devoted to proving the small noise asymptotic behavior, particularly large deviation principle, for multi-scale stochastic dynamical systems with fully local monotone coefficients driven by multiplicative noise. The main techniques rely on the weak convergence approach, the theory of pseudo-monotone operators and the time discretization scheme. The main results derived in this paper have broad applications to various multi-scale models, where the slow component could be such as stochastic porous medium equations, stochastic Cahn-Hilliard equations and stochastic 2D Liquid crystal equations.
{"title":"Large deviation principle for multi-scale fully local monotone stochastic dynamical systems with multiplicative noise","authors":"Wei Hong, Wei Liu, Luhan Yang","doi":"10.1016/j.jde.2024.09.059","DOIUrl":"10.1016/j.jde.2024.09.059","url":null,"abstract":"<div><div>This paper is devoted to proving the small noise asymptotic behavior, particularly large deviation principle, for multi-scale stochastic dynamical systems with fully local monotone coefficients driven by multiplicative noise. The main techniques rely on the weak convergence approach, the theory of pseudo-monotone operators and the time discretization scheme. The main results derived in this paper have broad applications to various multi-scale models, where the slow component could be such as stochastic porous medium equations, stochastic Cahn-Hilliard equations and stochastic 2D Liquid crystal equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 396-448"},"PeriodicalIF":2.4,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425002","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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