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Rate of convergence in the large diffusion limit for the heat equation with a dynamical boundary condition 具有动力边界条件的热方程大扩散极限下的收敛速度
Asymptot. Anal. Pub Date : 2018-06-16 DOI: 10.3233/ASY-181517
M. Fila, Kazuhiro Ishige, Tatsuki Kawakami, J. Lankeit
{"title":"Rate of convergence in the large diffusion limit for the heat equation with a dynamical boundary condition","authors":"M. Fila, Kazuhiro Ishige, Tatsuki Kawakami, J. Lankeit","doi":"10.3233/ASY-181517","DOIUrl":"https://doi.org/10.3233/ASY-181517","url":null,"abstract":"We study the heat equation on a half-space or on an exterior domain with a linear dynamical boundary condition. Our main aim is to establish the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"34 1","pages":"37-57"},"PeriodicalIF":0.0,"publicationDate":"2018-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73511977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Well-posedness, regularity and asymptotic analyses for a fractional phase field system 分数相场系统的适定性、正则性及渐近分析
Asymptot. Anal. Pub Date : 2018-06-12 DOI: 10.3233/ASY-191524
P. Colli, G. Gilardi
{"title":"Well-posedness, regularity and asymptotic analyses for a fractional phase field system","authors":"P. Colli, G. Gilardi","doi":"10.3233/ASY-191524","DOIUrl":"https://doi.org/10.3233/ASY-191524","url":null,"abstract":"This paper is concerned with a non-conserved phase field system of Caginalp type in which the main operators are fractional versions of two fixed linear operators $A$ and $B$. The operators $A$ and $B$ are supposed to be densely defined, unbounded, self-adjoint, monotone in the Hilbert space $L^2(Omega)$, for some bounded and smooth domain $Omega$, and have compact resolvents. Our definition of the fractional powers of operators uses the approach via spectral theory. A nonlinearity of double-well type occurs in the phase equation and either a regular or logarithmic potential, as well as a non-differentiable potential involving an indicator function, is admitted in our approach. We show general well-posedness and regularity results, extending the corresponding results that are known for the non-fractional elliptic operators with zero Neumann conditions or other boundary conditions like Dirichlet or Robin ones. Then, we investigate the longtime behavior of the system, by fully characterizing every element of the $omega$-limit as a stationary solution. In the final part of the paper we study the asymptotic behavior of the system as the parameter $sigma$ appearing in the operator $B^{2sigma}$ that plays in the phase equation decreasingly tends to zero. We can prove convergence to a phase relaxation problem at the limit, in which an additional term containing the projection of the phase variable on the kernel of $B$ appears.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"7 1","pages":"93-128"},"PeriodicalIF":0.0,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88752024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
On the asymptotic behaviour of the pure Neumann problem in cylinder-like domains and its applications 纯Neumann问题在类柱域上的渐近性质及其应用
Asymptot. Anal. Pub Date : 2018-06-07 DOI: 10.3233/ASY-181462
M. Chipot, S. Zube
{"title":"On the asymptotic behaviour of the pure Neumann problem in cylinder-like domains and its applications","authors":"M. Chipot, S. Zube","doi":"10.3233/ASY-181462","DOIUrl":"https://doi.org/10.3233/ASY-181462","url":null,"abstract":"We consider in this paper the pure Neumann problem in n-dimensional cylinder-like domains. We are interested in the asymptotic behaviour of the solution of this kind of problems when the domain becomes infinite in p-directions, 1 ≤ p < n. We show that this solution converges exponentially to the solution of a Neumann problem in the corresponding unbounded domain. We distinguish between the case p = 1 and 1 < p < n the latter requiring a more involved analysis. For p = 1 we consider also the special situation when the domain and the initial data are periodic.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"1 1","pages":"163-185"},"PeriodicalIF":0.0,"publicationDate":"2018-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83595065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Characterizations of anisotropic high order Sobolev spaces 各向异性高阶Sobolev空间的表征
Asymptot. Anal. Pub Date : 2018-05-23 DOI: 10.3233/ASY-181515
N. Lam, Ali Maalaoui, A. Pinamonti
{"title":"Characterizations of anisotropic high order Sobolev spaces","authors":"N. Lam, Ali Maalaoui, A. Pinamonti","doi":"10.3233/ASY-181515","DOIUrl":"https://doi.org/10.3233/ASY-181515","url":null,"abstract":"We establish two types of characterizations for high order anisotropic Sobolev spaces. In particular, we prove high order anisotropic versions of Bourgain-Brezis- Mironescu's formula and Nguyen's formula.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"35 1","pages":"239-260"},"PeriodicalIF":0.0,"publicationDate":"2018-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83513529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
On stabilization of solutions of higher order evolution inequalities 高阶演化不等式解的镇定性
Asymptot. Anal. Pub Date : 2018-03-18 DOI: 10.3233/asy-191522
A. Kon'kov, A. Shishkov
{"title":"On stabilization of solutions of higher order evolution inequalities","authors":"A. Kon'kov, A. Shishkov","doi":"10.3233/asy-191522","DOIUrl":"https://doi.org/10.3233/asy-191522","url":null,"abstract":"We obtain sharp conditions guaranteeing that every non-negative weak solution of the inequality $$ sum_{|alpha| = m} \u0000partial^alpha \u0000a_alpha (x, t, u) \u0000- \u0000u_t \u0000ge \u0000f (x, t) g (u) \u0000quad \u0000mbox{in} {mathbb R}_+^{n+1} = {mathbb R}^n times (0, infty), \u0000quad \u0000m,n ge 1, $$ stabilizes to zero as $t to infty$. These conditions generalize the well-known Keller-Osserman condition on the grows of the function $g$ at infinity.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"17 1","pages":"1-17"},"PeriodicalIF":0.0,"publicationDate":"2018-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84772821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem without monotonicity condition 无单调条件下具有连通障碍的偏微分方程系统和切换问题的黏度解
Asymptot. Anal. Pub Date : 2018-02-13 DOI: 10.3233/ASY-181508
S. Hamadène, M. Mnif, Sarra Neffati
{"title":"Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem without monotonicity condition","authors":"S. Hamadène, M. Mnif, Sarra Neffati","doi":"10.3233/ASY-181508","DOIUrl":"https://doi.org/10.3233/ASY-181508","url":null,"abstract":"We show the existence and uniqueness of a continuous viscosity solution of a system of partial differential equations (PDEs for short) without assuming the usual monotonicity conditions on the driver function as in Hamad`ene and Morlais's article cite{hamadene2013viscosity}. Our method strongly relies on the link between PDEs and reflected backward stochastic differential equations with interconnected obstacles for which we already know that the solution exists and is unique for general drivers.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"1 1","pages":"123-136"},"PeriodicalIF":0.0,"publicationDate":"2018-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83911887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Low-lying eigenvalues of semiclassical Schrödinger operator with degenerate wells 具有退化井的半经典Schrödinger算子的低洼特征值
Asymptot. Anal. Pub Date : 2018-02-08 DOI: 10.3233/ASY-181493
J. Bony, N. Popoff
{"title":"Low-lying eigenvalues of semiclassical Schrödinger operator with degenerate wells","authors":"J. Bony, N. Popoff","doi":"10.3233/ASY-181493","DOIUrl":"https://doi.org/10.3233/ASY-181493","url":null,"abstract":"In this article, we consider the semiclassical Schr\"odinger operator $P = - h^{2} Delta + V$ in $mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $lambda_{k} ( P )$ as $h to 0$. First, we give a necessary and sufficient criterion upon $V^{-1} ( 0 )$ for $lambda_{1} ( P ) h^{- 2}$ to be bounded. When $d = 1$ and $V^{-1} ( 0 ) = { 0 }$, we are able to control the eigenvalues $lambda_{k} ( P )$ for monotonous potentials by a quantity linked to an interval $I_{h}$, determined by an implicit relation involving $V$ and $h$. Next, we consider the case where $V$ has a flat minimum, in the sense that it vanishes to infinite order. We give the asymptotic of the eigenvalues: they behave as the eigenvalues of the Dirichlet Laplacian on $I_{h}$. Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"2 1","pages":"23-36"},"PeriodicalIF":0.0,"publicationDate":"2018-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81979795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Turbulence in active fluids caused by self-propulsion 由自我推进引起的主动流体中的乱流
Asymptot. Anal. Pub Date : 2018-02-05 DOI: 10.3233/ASY-181510
C. Bui, H. Löwen, J. Saal
{"title":"Turbulence in active fluids caused by self-propulsion","authors":"C. Bui, H. Löwen, J. Saal","doi":"10.3233/ASY-181510","DOIUrl":"https://doi.org/10.3233/ASY-181510","url":null,"abstract":"A rigoros analytical justification of turbulence observed in active fluids and caused by self-propulsion is presented. We prove existence of unstable wave modes for the generalized Stokes and Navier-Stokes systems by developing an approach in spaces of Fourier transformed Radon measures.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"53 1","pages":"195-209"},"PeriodicalIF":0.0,"publicationDate":"2018-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88154477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Pressure reconstruction for weak solutions of the two-phase incompressible Navier-Stokes equations with surface tension 带表面张力的两相不可压缩Navier-Stokes方程弱解的压力重建
Asymptot. Anal. Pub Date : 2018-01-15 DOI: 10.3233/ASY-181507
H. Abels, J. Daube, C. Kraus
{"title":"Pressure reconstruction for weak solutions of the two-phase incompressible Navier-Stokes equations with surface tension","authors":"H. Abels, J. Daube, C. Kraus","doi":"10.3233/ASY-181507","DOIUrl":"https://doi.org/10.3233/ASY-181507","url":null,"abstract":"For the two-phase incompressible Navier--Stokes equations with surface tension, we derive an appropriate weak formulation incorporating a variational formulation using divergence-free test functions. We prove a consistency result to justify our definition and, under reasonable regularity assumptions, we reconstruct the pressure function from the weak formulation.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"47 1","pages":"51-86"},"PeriodicalIF":0.0,"publicationDate":"2018-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91300571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains 无界域上非散度型椭圆方程的混合边值问题
Asymptot. Anal. Pub Date : 2018-01-02 DOI: 10.3233/ASY-181469
Dat Cao, Akif I. Ibraguimov, A. Nazarov
{"title":"Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains","authors":"Dat Cao, Akif I. Ibraguimov, A. Nazarov","doi":"10.3233/ASY-181469","DOIUrl":"https://doi.org/10.3233/ASY-181469","url":null,"abstract":"We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragm'en-Lindel\"of type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the \"thickness\" of its Dirichlet portion. The result is formulated in terms of so-called $s$-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain \"admissibility\" condition in the sequence of layers converging to infinity.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"2 1","pages":"75-90"},"PeriodicalIF":0.0,"publicationDate":"2018-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90329417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
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