Pub Date : 2023-11-06DOI: 10.1007/s00028-023-00924-9
Abderrahim Charkaoui, Nour Eddine Alaa
{"title":"An $$L^1$$-theory for a nonlinear temporal periodic problem involving p(x)-growth structure with a strong dependence on gradients","authors":"Abderrahim Charkaoui, Nour Eddine Alaa","doi":"10.1007/s00028-023-00924-9","DOIUrl":"https://doi.org/10.1007/s00028-023-00924-9","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135635034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-06DOI: 10.1007/s00028-023-00920-z
Giovanni Franzina, Bruno Volzone
Abstract Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.
摘要采用Brasco (Adv Math 394:108029, 2022)的方法，研究了光滑边界开有界集上有符号分数阶多孔介质方程的长时间行为。考虑齐次外部狄利克雷边界条件。我们证明了如果初始基准具有足够小的能量，那么一旦适当地重新标度，解收敛于次线性分数阶Lane-Emden方程的非平凡常符号解。在此基础上给出了一个非局部的充分能判据，这对于确定极限轮廓的正解或负解具有重要意义。
{"title":"Large time behavior of signed fractional porous media equations on bounded domains","authors":"Giovanni Franzina, Bruno Volzone","doi":"10.1007/s00028-023-00920-z","DOIUrl":"https://doi.org/10.1007/s00028-023-00920-z","url":null,"abstract":"Abstract Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135635714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-28DOI: 10.1007/s00028-023-00923-w
Ruben Jakob
{"title":"The Willmore flow of Hopf-tori in the 3-sphere","authors":"Ruben Jakob","doi":"10.1007/s00028-023-00923-w","DOIUrl":"https://doi.org/10.1007/s00028-023-00923-w","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"140 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-26DOI: 10.1007/s00028-023-00916-9
Stefano Pagliarani, Giacomo Lucertini, Andrea Pascucci
Abstract We consider a class of degenerate equations in non-divergence form satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin diffusions.
{"title":"Optimal regularity for degenerate Kolmogorov equations in non-divergence form with rough-in-time coefficients","authors":"Stefano Pagliarani, Giacomo Lucertini, Andrea Pascucci","doi":"10.1007/s00028-023-00916-9","DOIUrl":"https://doi.org/10.1007/s00028-023-00916-9","url":null,"abstract":"Abstract We consider a class of degenerate equations in non-divergence form satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin diffusions.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136381954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-26DOI: 10.1007/s00028-023-00922-x
Torebek, Berikbol T.
This paper studies the large-time behavior of solutions to the quasilinear inhomogeneous parabolic equation with combined nonlinearities. This equation is a natural extension of the heat equations with combined nonlinearities considered by Jleli-Samet-Souplet (Proc AMS, 2020). Firstly, we focus on an interesting phenomenon of discontinuity of the critical exponents. In particular, we will fill the gap in the results of Jleli-Samet-Souplet for the critical case. We are also interested in the influence of the forcing term on the critical behavior of the considered problem, so we will define another critical exponent depending on the forcing term.
{"title":"Critical exponents for the p-Laplace heat equations with combined nonlinearities","authors":"Torebek, Berikbol T.","doi":"10.1007/s00028-023-00922-x","DOIUrl":"https://doi.org/10.1007/s00028-023-00922-x","url":null,"abstract":"This paper studies the large-time behavior of solutions to the quasilinear inhomogeneous parabolic equation with combined nonlinearities. This equation is a natural extension of the heat equations with combined nonlinearities considered by Jleli-Samet-Souplet (Proc AMS, 2020). Firstly, we focus on an interesting phenomenon of discontinuity of the critical exponents. In particular, we will fill the gap in the results of Jleli-Samet-Souplet for the critical case. We are also interested in the influence of the forcing term on the critical behavior of the considered problem, so we will define another critical exponent depending on the forcing term.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136377036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-24DOI: 10.1007/s00028-023-00914-x
Ophélie Cuvillier, Francesco Fanelli, Elena Salguero
In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain $$mathbb {T}^d$$ , for space dimensions $$d=2,3$$ . We admit the average turbulent kinetic energy k to vanish in part of the domain, i.e. we consider the case $$k ge 0$$ ; in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces $$H^s$$ , for any $$s>1+d/2$$ . We expect this regularity to be optimal, due to the degeneracy of the system when $$k approx 0$$ . We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the nonlinear terms involved in the computations.
本文研究了周期域$$mathbb {T}^d$$中空间维为$$d=2,3$$的Kolmogorov湍流双方程模型的适定性。我们承认平均湍流动能k在部分区域内消失，即我们考虑$$k ge 0$$;在这种情况下，方程的抛物线结构变得简并。对于该系统，我们证明了Sobolev空间$$H^s$$中对于任意$$s>1+d/2$$的局部适定性结果。我们期望这个规则是最优的，因为当$$k approx 0$$。我们还证明了一个延拓准则，并给出了解的寿命的下界。结果的证明是基于Littlewood-Paley分析和环面上的准微分演算，以及计算中涉及的非线性项的精确换易子分解。
{"title":"Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaces","authors":"Ophélie Cuvillier, Francesco Fanelli, Elena Salguero","doi":"10.1007/s00028-023-00914-x","DOIUrl":"https://doi.org/10.1007/s00028-023-00914-x","url":null,"abstract":"In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain $$mathbb {T}^d$$ , for space dimensions $$d=2,3$$ . We admit the average turbulent kinetic energy k to vanish in part of the domain, i.e. we consider the case $$k ge 0$$ ; in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces $$H^s$$ , for any $$s>1+d/2$$ . We expect this regularity to be optimal, due to the degeneracy of the system when $$k approx 0$$ . We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the nonlinear terms involved in the computations.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1007/s00028-023-00917-8
J. M. Mazón, A. Molino, J. Toledo
Abstract This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian, $$begin{aligned} frac{partial v}{partial t} - Delta _1 u ni 0 hbox { in } (0, infty ) times Omega , quad vin gamma (u), end{aligned}$$ ∂v∂t-Δ1u∋0in(0,∞)×Ω,v∈γ(u), and initial data in $$L^1(Omega )$$ L1(Ω) , where $$Omega $$ Ω is a bounded smooth domain in $${mathbb {R}}^N$$ RN and $$gamma $$ γ is a maximal monotone graph in $${mathbb {R}}times {mathbb {R}}$$ R×R . We prove that, under certain assumptions on the graph $$gamma $$ γ , there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.
研究一类双非线性1-拉普拉斯方程的Neumann问题。 $$begin{aligned} frac{partial v}{partial t} - Delta _1 u ni 0 hbox { in } (0, infty ) times Omega , quad vin gamma (u), end{aligned}$$ ∂v∂t - Δ 1 u∈0 in(0，∞)× Ω， v∈γ (u)，初始数据in $$L^1(Omega )$$ l1 (Ω)，其中 $$Omega $$ 中的有界光滑域Ω $${mathbb {R}}^N$$ R N和 $$gamma $$ γ是中的极大单调图 $${mathbb {R}}times {mathbb {R}}$$ R × R。我们在图上的某些假设下证明了这一点 $$gamma $$ γ，解存在唯一性。此外，我们证明了这些解与总变分流的诺伊曼问题的解是一致的。我们证明这些假设是必要的。
{"title":"Doubly nonlinear equations for the 1-Laplacian","authors":"J. M. Mazón, A. Molino, J. Toledo","doi":"10.1007/s00028-023-00917-8","DOIUrl":"https://doi.org/10.1007/s00028-023-00917-8","url":null,"abstract":"Abstract This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian, $$begin{aligned} frac{partial v}{partial t} - Delta _1 u ni 0 hbox { in } (0, infty ) times Omega , quad vin gamma (u), end{aligned}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>∋</mml:mo> <mml:mn>0</mml:mn> <mml:mspace /> <mml:mtext>in</mml:mtext> <mml:mspace /> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>v</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>γ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> and initial data in $$L^1(Omega )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where $$Omega $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>Ω</mml:mi> </mml:math> is a bounded smooth domain in $${mathbb {R}}^N$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> and $$gamma $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>γ</mml:mi> </mml:math> is a maximal monotone graph in $${mathbb {R}}times {mathbb {R}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . We prove that, under certain assumptions on the graph $$gamma $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>γ</mml:mi> </mml:math> , there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"22 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135994235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-14DOI: 10.1007/s00028-023-00915-w
Mikhail Anikushin
{"title":"Frequency theorem and inertial manifolds for neutral delay equations","authors":"Mikhail Anikushin","doi":"10.1007/s00028-023-00915-w","DOIUrl":"https://doi.org/10.1007/s00028-023-00915-w","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135800300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-06DOI: 10.1007/s00028-023-00918-7
Ning-An Lai, Nico Michele Schiavone
{"title":"Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques","authors":"Ning-An Lai, Nico Michele Schiavone","doi":"10.1007/s00028-023-00918-7","DOIUrl":"https://doi.org/10.1007/s00028-023-00918-7","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135352549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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