Journal of Differential Equations最新文献

{"title":"Large time behavior of the full compressible Navier-Stokes-Maxwell system with a nonconstant background density","authors":"Xin Li","doi":"10.1016/j.jde.2024.10.010","DOIUrl":"10.1016/j.jde.2024.10.010","url":null,"abstract":"<div><div>We study the Cauchy problem for the full compressible Navier-Stokes-Maxwell system with a nonconstant background density in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. By means of suitable choosing of symmetrizers and weighted energy estimates with some new developments, we establish the global existence and uniqueness of the classical solution provided that the initial data are near this equilibrium. Furthermore, by using the spectrum analysis on the linearized homogeneous system of the full compressible Navier-Stokes-Maxwell equations and refining the convergence property, we obtain the time-algebraic convergence rates of the perturbed solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442175","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}

{"title":"Characterising blenders via covering relations and cone conditions","authors":"Maciej J. Capiński ,&nbsp;Bernd Krauskopf ,&nbsp;Hinke M. Osinga ,&nbsp;Piotr Zgliczyński","doi":"10.1016/j.jde.2024.10.004","DOIUrl":"10.1016/j.jde.2024.10.004","url":null,"abstract":"<div><div>We present a characterisation of a blender based on the topological alignment of certain sets in phase space in combination with cone conditions. Importantly, the required conditions can be verified by checking properties of a single iterate of the diffeomorphism, which is achieved by finding finite series of sets that form suitable sequences of alignments. This characterisation is applicable in arbitrary dimension. Moreover, the approach naturally extends to establishing <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-persistent heterodimensional cycles. Our setup is flexible and allows for a rigorous, computer-assisted validation based on interval arithmetic.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432937","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}

{"title":"Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in R3","authors":"Dengjun Guo,&nbsp;Lifeng Zhao","doi":"10.1016/j.jde.2024.10.008","DOIUrl":"10.1016/j.jde.2024.10.008","url":null,"abstract":"<div><div>We consider the three-dimensional incompressible Euler equation<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>Ω</mi><mo>+</mo><mi>U</mi><mo>⋅</mo><mi>∇</mi><mi>Ω</mi><mo>−</mo><mi>Ω</mi><mo>⋅</mo><mi>∇</mi><mi>U</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mi>Ω</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mrow></math></span></span></span> in the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Under the assumption that <span><math><msup><mrow><mi>Ω</mi></mrow><mrow><mi>z</mi></mrow></msup></math></span> is helical and in the absence of vorticity stretching, we prove the global well-posedness of weak solutions in <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mo>⋂</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>. Moreover, the vortex transport formula and the conservation of the energy and the second momentum are also obtained in our article, which will serve as valuable tools in our subsequent exploration of the dynamics of helical vortex filaments.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442589","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}

{"title":"Miura transformations and large-time behaviors of the Hirota-Satsuma equation","authors":"Deng-Shan Wang,&nbsp;Cheng Zhu,&nbsp;Xiaodong Zhu","doi":"10.1016/j.jde.2024.10.006","DOIUrl":"10.1016/j.jde.2024.10.006","url":null,"abstract":"<div><div>The good Boussinesq equation has several modified versions, such as the modified Boussinesq equation, Mikhailov-Lenells equation and Hirota-Satsuma equation. This work builds the full relations among these equations by Miura transformation and invertible linear transformations and draws a pyramid diagram to demonstrate such relations. The direct and inverse spectral analysis shows that the solution of Riemann-Hilbert problem for Hirota-Satsuma equation has a simple pole at origin, the solution of Riemann-Hilbert problem for the good Boussinesq equation has double pole at origin, while the solution of Riemann-Hilbert problem for the modified Boussinesq equation and Mikhailov-Lenells equation doesn't have singularity at origin. Further, the large-time asymptotic behaviors of the Hirota-Satsuma equation with Schwartz class initial value are studied by Deift-Zhou nonlinear steepest descent analysis. In such initial conditions, the asymptotic expressions away from the origin are derived and it is shown that the leading term of asymptotic formulas matches well with the direct numerical simulations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432843","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}

{"title":"The convergence problem of the generalized Korteweg-de Vries equation in Fourier-Lebesgue space","authors":"Qiaoqiao Zhang ,&nbsp;Wei Yan ,&nbsp;Jinqiao Duan ,&nbsp;Meihua Yang","doi":"10.1016/j.jde.2024.10.007","DOIUrl":"10.1016/j.jde.2024.10.007","url":null,"abstract":"<div><div>In this paper, we investigate the pointwise convergence problem of the generalized Korteweg-de Vries (gKdV) equation with data in the Fourier-Lebesgue space. Firstly, for the Airy equation, we show the almost everywhere pointwise convergence with data in <span><math><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>s</mi><mo>,</mo><mfrac><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>s</mi><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mn>5</mn><mo>≤</mo><mi>α</mi><mo>&lt;</mo><mo>∞</mo><mo>)</mo></math></span>, furthermore, we show that the maximal function estimate related to the Airy equation can fail with data in <span><math><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>s</mi><mo>,</mo><mfrac><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>(</mo><mi>s</mi><mo>&lt;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>)</mo></math></span>. Then, for the gKdV equation, we establish the pointwise convergence results with the data in <span><math><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>(</mo><mn>5</mn><mo>≤</mo><mi>α</mi><mo>&lt;</mo><mfrac><mrow><mn>23</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></math></span>, in particular, we establish the pointwise convergence results with small data in <span><math><msup><mrow><mover><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, which implies that the pointwise convergence of generalized KdV equation is closely related to the pointwise convergence of linear KdV equation in the Fourier-Lebesgue spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432842","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}

{"title":"The heat flow in nonlinear Hodge theory under general growth","authors":"Christoph Hamburger","doi":"10.1016/j.jde.2024.09.043","DOIUrl":"10.1016/j.jde.2024.09.043","url":null,"abstract":"<div><div>We study the <em>nonlinear Hodge system</em> <span><math><mi>d</mi><mi>ω</mi><mo>=</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ω</mi><mo>=</mo><mn>0</mn></math></span> for an exterior form <em>ω</em> on a compact oriented Riemannian manifold <em>M</em>. Its solutions are called <em>ρ-harmonic forms</em>. Here the <em>ρ</em>-codifferential of <em>ω</em> is defined as <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ω</mi><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>δ</mi><mo>(</mo><mi>ρ</mi><mi>ω</mi><mo>)</mo></math></span> with a given positive function <span><math><mi>ρ</mi><mo>=</mo><mi>ρ</mi><mo>(</mo><mo>|</mo><mi>ω</mi><mo>|</mo><mo>)</mo></math></span>.</div><div>We evolve a given closed form <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> by the <em>nonlinear heat flow system</em> <span><math><mover><mrow><mi>ω</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mi>d</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ω</mi></math></span> for a time dependent exterior form <span><math><mi>ω</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> on <em>M</em>. Under an ellipticity condition on the function <em>ρ</em>, we show that the nonlinear heat flow system with initial condition <span><math><mi>ω</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> has a unique solution for all times, which converges to a <em>ρ</em>-harmonic form in the cohomology class of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. This yields a <em>nonlinear Hodge theorem</em> that every cohomology class of <em>M</em> has a unique <em>ρ</em>-harmonic representative.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bifurcation curves for the one-dimensional perturbed Gelfand problem with the Minkowski-curvature operator","authors":"Shao-Yuan Huang ,&nbsp;Shin-Hwa Wang","doi":"10.1016/j.jde.2024.10.002","DOIUrl":"10.1016/j.jde.2024.10.002","url":null,"abstract":"<div><div>In this paper, we study the shapes of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand problem with the Minkowski-curvature operator<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><msup><mrow><mo>(</mo><mfrac><mrow><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mi>λ</mi><mi>exp</mi><mo>⁡</mo><mrow><mo>(</mo><mfrac><mrow><mi>a</mi><mi>u</mi></mrow><mrow><mi>a</mi><mo>+</mo><mi>u</mi></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mrow><mtext></mtext><mspace></mspace></mrow><mo>−</mo><mi>L</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mi>L</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mo>−</mo><mi>L</mi><mo>)</mo><mo>=</mo><mi>u</mi><mo>(</mo><mi>L</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span>where <span><math><mi>λ</mi><mo>&gt;</mo><mn>0</mn></math></span> is a bifurcation parameter and <span><math><mi>a</mi><mo>,</mo><mi>L</mi><mo>&gt;</mo><mn>0</mn></math></span> are evolution parameters. We determine the shapes of the bifurcation curves for different positive values <em>a</em> and <em>L</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432938","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}

{"title":"On the sharp Hessian integrability conjecture in the plane","authors":"Thialita M. Nascimento,&nbsp;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>&lt;</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":null,"pages":null},"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}

{"title":"On the Cauchy problem for a combined mCH-Novikov integrable equation with linear dispersion","authors":"Zhenyu Wan ,&nbsp;Ying Wang ,&nbsp;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":null,"pages":null},"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}

{"title":"On some regularity properties of mixed local and nonlocal elliptic equations","authors":"Xifeng Su ,&nbsp;Enrico Valdinoci ,&nbsp;Yuanhong Wei ,&nbsp;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":null,"pages":null},"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}