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Weak and parabolic solutions of advection–diffusion equations with rough velocity field 具有粗糙速度场的平流扩散方程的弱解和抛物线解
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2023-12-16 DOI: 10.1007/s00028-023-00919-6
Paolo Bonicatto, Gennaro Ciampa, Gianluca Crippa
{"title":"Weak and parabolic solutions of advection–diffusion equations with rough velocity field","authors":"Paolo Bonicatto, Gennaro Ciampa, Gianluca Crippa","doi":"10.1007/s00028-023-00919-6","DOIUrl":"https://doi.org/10.1007/s00028-023-00919-6","url":null,"abstract":"<p>We study the Cauchy problem for the advection–diffusion equation <span>(partial _t u + {{,mathrm{textrm{div}},}}(uvarvec{b}) = Delta u)</span> associated with a merely integrable divergence-free vector field <span>(varvec{b})</span> defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"16 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138685646","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}
引用次数: 0
Asymptotic behavior of solutions for nonlinear parabolic problems with Marcinkiewicz data 具有Marcinkiewicz数据的非线性抛物型问题解的渐近性质
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2023-11-28 DOI: 10.1007/s00028-023-00929-4
Lucio Boccardo, Luigi Orsina, Maria Michaela Porzio
{"title":"Asymptotic behavior of solutions for nonlinear parabolic problems with Marcinkiewicz data","authors":"Lucio Boccardo, Luigi Orsina, Maria Michaela Porzio","doi":"10.1007/s00028-023-00929-4","DOIUrl":"https://doi.org/10.1007/s00028-023-00929-4","url":null,"abstract":"<p>In this paper we prove the asymptotic behavior, as <i>t</i> tends to zero, of solutions of nonlinear parabolic equations with initial data belonging to Marcinkiewicz spaces. Namely, that if the initial datum <span>(u_{0})</span> belongs to <span>(M^{m}(Omega ))</span>, then </p><span>$$begin{aligned} Vert u(t)Vert _{scriptstyle L^{r}(Omega )}^{*} le {mathcal {C}},frac{Vert u_{0}Vert _{scriptstyle L^{m}(Omega )}^{*}}{t^{frac{N}{2}left( frac{1}{m} - frac{1}{r}right) }}, qquad forall ,t &gt; 0, end{aligned}$$</span><p>thus extending to Marcinkiewicz spaces the results which hold for data in Lebesgue spaces.\u0000</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"33 2","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138518585","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}
引用次数: 0
Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel–Lizorkin space $$F^{d+1}_{1, infty }(mathbb {R}^d)$$ 端点临界triiebel - lizorkin空间中不可压缩欧拉方程解的时间正则性 $$F^{d+1}_{1, infty }(mathbb {R}^d)$$
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2023-11-28 DOI: 10.1007/s00028-023-00927-6
Hee Chul Pak
{"title":"Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel–Lizorkin space $$F^{d+1}_{1, infty }(mathbb {R}^d)$$","authors":"Hee Chul Pak","doi":"10.1007/s00028-023-00927-6","DOIUrl":"https://doi.org/10.1007/s00028-023-00927-6","url":null,"abstract":"<p>An evidence of temporal discontinuity of the solution in <span>(F^s_{1, infty }(mathbb {R}^d))</span> is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in related spaces are also discussed.\u0000</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"90 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138515707","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}
引用次数: 0
On a quasilinear fully parabolic predator–prey model with indirect pursuit-evasion interaction 一类具有追捕-逃避间接相互作用的拟线性全抛物型捕食者-猎物模型
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2023-11-28 DOI: 10.1007/s00028-023-00931-w
Chuanjia Wan, Pan Zheng, Wenhai Shan
{"title":"On a quasilinear fully parabolic predator–prey model with indirect pursuit-evasion interaction","authors":"Chuanjia Wan, Pan Zheng, Wenhai Shan","doi":"10.1007/s00028-023-00931-w","DOIUrl":"https://doi.org/10.1007/s00028-023-00931-w","url":null,"abstract":"<p>In this paper, we study the quasilinear fully parabolic predator–prey model with indirect pursuit-evasion interaction </p><span>$$begin{aligned} begin{aligned} left{ begin{aligned}&amp;u_t=nabla cdot left( D_{1}(u)nabla uright) -chi nabla cdot left( S_{1}(u)nabla zright) +uleft( alpha v-a_{1} -b_{1}uright) ,&amp;x in varOmega , t&gt;0, &amp;v_t=nabla cdot left( D_{2}(v)nabla vright) +xi nabla cdot left( S_{2}(v)nabla {w}right) +vleft( a_{2} -b_{2} v-uright) ,&amp;x in varOmega , t&gt;0, &amp;{w_t}=Delta w+beta {u}-gamma {w},&amp;x in varOmega , t&gt;0,&amp;{z_t}=Delta z+delta {v}-rho z,&amp;x in varOmega , t&gt;0, end{aligned} right. end{aligned} end{aligned}$$</span><p>under homogeneous Neumann boundary conditions in a smoothly bounded domain <span>(varOmega subset mathbb {R}^{n}(nge 1))</span>, where <span>( chi , xi , alpha , beta , gamma , delta , rho , a_{1},a_{2},)</span> <span>(b_{1},b_{2})</span> are positive parameters, the functions <span>(D_{i} in C^{2}([0,infty )))</span> and <span>(S_{i}in C^{2}([0,infty )))</span> with <span>(S_{i}(0)=0(i=1,2))</span>. Firstly, under certain suitable conditions, we prove that the system admits a unique globally bounded classical solution when <span>(nle 4)</span>. Moreover, we investigate the asymptotic stability and precise convergence rates of globally bounded solutions by constructing appropriate Lyapunov functionals. Finally, we present numerical simulations that not only support our theoretical results, but also involve new and interesting phenomena.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"77 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138542749","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}
引用次数: 0
Boundedness of the conformal hyperboloidal energy for a wave-Klein–Gordon model 波-克莱因-戈登模型共形双曲能量的有界性
3区 数学
Journal of Evolution Equations Pub Date : 2023-11-09 DOI: 10.1007/s00028-023-00925-8
Philippe G. LeFloch, Jesús Oliver, Yoshio Tsutsumi
{"title":"Boundedness of the conformal hyperboloidal energy for a wave-Klein–Gordon model","authors":"Philippe G. LeFloch, Jesús Oliver, Yoshio Tsutsumi","doi":"10.1007/s00028-023-00925-8","DOIUrl":"https://doi.org/10.1007/s00028-023-00925-8","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":" 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241265","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}
引用次数: 2
A pathwise regularization by noise phenomenon for the evolutionary p-Laplace equation 演化p-拉普拉斯方程的噪声现象路径正则化
3区 数学
Journal of Evolution Equations Pub Date : 2023-11-09 DOI: 10.1007/s00028-023-00926-7
Florian Bechtold, Jörn Wichmann
{"title":"A pathwise regularization by noise phenomenon for the evolutionary p-Laplace equation","authors":"Florian Bechtold, Jörn Wichmann","doi":"10.1007/s00028-023-00926-7","DOIUrl":"https://doi.org/10.1007/s00028-023-00926-7","url":null,"abstract":"Abstract We study an evolutionary p -Laplace problem whose potential is subject to a translation in time. Provided the trajectory along which the potential is translated admits a sufficiently regular local time, we establish existence of solutions to the problem for singular potentials for which a priori bounds in classical approaches break down, thereby establishing a pathwise regularization by noise phenomena for this nonlinear problem.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":" 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192783","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}
引用次数: 3
An $$L^1$$-theory for a nonlinear temporal periodic problem involving p(x)-growth structure with a strong dependence on gradients 涉及p(x)生长结构的非线性时间周期问题的$$L^1$$ -理论
3区 数学
Journal of Evolution Equations Pub Date : 2023-11-06 DOI: 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}
引用次数: 0
Large time behavior of signed fractional porous media equations on bounded domains 有界域上有符号分数多孔介质方程的大时间行为
3区 数学
Journal of Evolution Equations Pub Date : 2023-11-06 DOI: 10.1007/s00028-023-00920-z
Giovanni Franzina, Bruno Volzone
{"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}
引用次数: 0
The Willmore flow of Hopf-tori in the 3-sphere Hopf-tori在3球中的Willmore流
3区 数学
Journal of Evolution Equations Pub Date : 2023-10-28 DOI: 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}
引用次数: 7
Optimal regularity for degenerate Kolmogorov equations in non-divergence form with rough-in-time coefficients 具有粗糙时间系数的非散度形式退化Kolmogorov方程的最优正则性
3区 数学
Journal of Evolution Equations Pub Date : 2023-10-26 DOI: 10.1007/s00028-023-00916-9
Stefano Pagliarani, Giacomo Lucertini, Andrea Pascucci
{"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}
引用次数: 2
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