M. Fila, Kazuhiro Ishige, Tatsuki Kawakami, J. Lankeit
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
We obtain sharp conditions guaranteeing that every non-negative weak solution of the inequality $$ sum_{|alpha| = m} partial^alpha a_alpha (x, t, u) - u_t ge f (x, t) g (u) quad mbox{in} {mathbb R}_+^{n+1} = {mathbb R}^n times (0, infty), quad m,n ge 1, $$ 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.
我们得到了保证不等式$$ sum_{|alpha| = m} partial^alpha a_alpha (x, t, u) - u_t ge f (x, t) g (u) quad mbox{in} {mathbb R}_+^{n+1} = {mathbb R}^n times (0, infty), quad m,n ge 1, $$的所有非负弱解稳定于零的尖锐条件$t to infty$。这些条件推广了著名的Keller-Osserman条件关于函数$g$在无穷远处的增长。
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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