We consider the Oberbeck--Boussinesq approximation driven by an inhomogeneous temperature distribution on the boundary of a bounded fluid domain. The relevant boundary conditions are perturbed by a non--local term arising in the incompressible limit of the Navier--Stokes--Fourier system. The long time behaviour of the resulting initial/boundary value problem is investigated.
{"title":"The Oberbeck–Boussinesq approximation and Rayleigh–Bénard convection revisited","authors":"E. Feireisl, Elisabetta Rocca, Giulio Schimperna","doi":"10.3934/dcds.2024032","DOIUrl":"https://doi.org/10.3934/dcds.2024032","url":null,"abstract":"We consider the Oberbeck--Boussinesq approximation driven by an inhomogeneous temperature distribution on the boundary of a bounded fluid domain. The relevant boundary conditions are perturbed by a non--local term arising in the incompressible limit of the Navier--Stokes--Fourier system. The long time behaviour of the resulting initial/boundary value problem is investigated.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140459512","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}
We study the solutions to a nonlocal 1-Laplacian equation given by $$ 2text{P.V.}int_{mathbb{R}^N}frac{u(x)-u(y)}{|u(x)-u(y)|} frac{dy}{|x-y|^{N+s}}=f(x) quad textmd{for } xin Omega, $$ with Dirichlet boundary condition $u(x)=0$ in $mathbb R^Nbackslash Omega$ and nonnegative $L^1$-data. By investigating the asymptotic behaviour of renormalized solutions $u_p$ to the nonlocal $p$-Laplacian equations as $p$ goes to $1^+$, we introduce a suitable definition of solutions and prove that the limit function $u$ of ${u_p}$ is a solution of the nonlocal $1$-Laplacian equation above.
{"title":"On the solutions of nonlocal 1-Laplacian equation with $ L^1 $-data","authors":"Dingding Li, Chao Zhang","doi":"10.3934/dcds.2023148","DOIUrl":"https://doi.org/10.3934/dcds.2023148","url":null,"abstract":"We study the solutions to a nonlocal 1-Laplacian equation given by $$ 2text{P.V.}int_{mathbb{R}^N}frac{u(x)-u(y)}{|u(x)-u(y)|} frac{dy}{|x-y|^{N+s}}=f(x) quad textmd{for } xin Omega, $$ with Dirichlet boundary condition $u(x)=0$ in $mathbb R^Nbackslash Omega$ and nonnegative $L^1$-data. By investigating the asymptotic behaviour of renormalized solutions $u_p$ to the nonlocal $p$-Laplacian equations as $p$ goes to $1^+$, we introduce a suitable definition of solutions and prove that the limit function $u$ of ${u_p}$ is a solution of the nonlocal $1$-Laplacian equation above.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139295676","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}
We consider the dynamics of a boosted soliton evolving under the cubic NLS with an external potential. We show that for sufficiently large velocities, the soliton is effectively transmitted through the potential. This result extends work of Holmer, Marzuola, and Zworski, who considered the case of a delta potential with no bound states, and work of Datchev and Holmer, who considered the case of a delta potential with a linear bound state.
{"title":"Transmission of fast solitons for the NLS with an external potential","authors":"Christopher C. Hogan, Jason Murphy","doi":"10.3934/dcds.2023142","DOIUrl":"https://doi.org/10.3934/dcds.2023142","url":null,"abstract":"We consider the dynamics of a boosted soliton evolving under the cubic NLS with an external potential. We show that for sufficiently large velocities, the soliton is effectively transmitted through the potential. This result extends work of Holmer, Marzuola, and Zworski, who considered the case of a delta potential with no bound states, and work of Datchev and Holmer, who considered the case of a delta potential with a linear bound state.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139321036","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}
We consider H"older continuous $GL(d,mathbb R)$-valued cocycles, and more generally linear cocycles, over an accessible volume-preserving center-bunched partially hyperbolic diffeomorphism. We study the regularity of a conjugacy between two cocycles. We establish continuity of a measurable conjugacy between {em any} constant $GL(d,mathbb R)$-valued cocycle and its perturbation. We deduce this from our main technical result on continuity of a measurable conjugacy between a fiber bunched linear cocycle and a cocycle with a certain block-triangular structure. The latter class covers constant cocycles with one Lyapunov exponent. We also establish a result of independent interest on continuity of measurable solutions for twisted vector-valued cohomological equations over partially hyperbolic systems. In addition, we give more general versions of earlier results on regularity of invariant subbudles, Riemannian metrics, and conformal structures.
{"title":"On regularity of conjugacy between linear cocycles over partially hyperbolic systems","authors":"B. Kalinin, V. Sadovskaya","doi":"10.3934/dcds.2023145","DOIUrl":"https://doi.org/10.3934/dcds.2023145","url":null,"abstract":"We consider H\"older continuous $GL(d,mathbb R)$-valued cocycles, and more generally linear cocycles, over an accessible volume-preserving center-bunched partially hyperbolic diffeomorphism. We study the regularity of a conjugacy between two cocycles. We establish continuity of a measurable conjugacy between {em any} constant $GL(d,mathbb R)$-valued cocycle and its perturbation. We deduce this from our main technical result on continuity of a measurable conjugacy between a fiber bunched linear cocycle and a cocycle with a certain block-triangular structure. The latter class covers constant cocycles with one Lyapunov exponent. We also establish a result of independent interest on continuity of measurable solutions for twisted vector-valued cohomological equations over partially hyperbolic systems. In addition, we give more general versions of earlier results on regularity of invariant subbudles, Riemannian metrics, and conformal structures.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139339396","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}
For an infinite discrete group $G$ acting on a compact metric space $X$, we introduce several weak versions of equicontinuity along subsets of $G$ and show that if a minimal system $(X,G)$ admits an invariant measure then $(X,G)$ is distal if and only if it is pairwise IP$^*$-equicontinuous; if the product system $(Xtimes X,G)$ of a minimal system $(X,G)$ has a dense set of minimal points, then $(X,G)$ is distal if and only if it is pairwise IP$^*$-equicontinuous if and only if it is pairwise central$^*$-equicontinuous; if $(X,G)$ is a minimal system with $G$ being abelian, then $(X,G)$ is a system of order $infty$ if and only if it is pairwise FIP$^*$-equicontinuous.
{"title":"Characterizations of distality via weak equicontinuity","authors":"Jian Li, Y. Yang","doi":"10.3934/dcds.2023096","DOIUrl":"https://doi.org/10.3934/dcds.2023096","url":null,"abstract":"For an infinite discrete group $G$ acting on a compact metric space $X$, we introduce several weak versions of equicontinuity along subsets of $G$ and show that if a minimal system $(X,G)$ admits an invariant measure then $(X,G)$ is distal if and only if it is pairwise IP$^*$-equicontinuous; if the product system $(Xtimes X,G)$ of a minimal system $(X,G)$ has a dense set of minimal points, then $(X,G)$ is distal if and only if it is pairwise IP$^*$-equicontinuous if and only if it is pairwise central$^*$-equicontinuous; if $(X,G)$ is a minimal system with $G$ being abelian, then $(X,G)$ is a system of order $infty$ if and only if it is pairwise FIP$^*$-equicontinuous.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81816682","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}
In"IP-sets and polynomial recurrence", Bergelson, Furstenberg, and McCutcheon established the following far reaching extension of Khintchine's recurrence theorem: For any invertible probability preserving system $(X,mathcal A,mu,T)$, any non-constant polynomial $pinmathbb Z[x]$ with $p(0)=0$, any $Ainmathcal A$, and any $epsilon>0$, the set $$R_epsilon^p(A)={ninmathbb N,|,mu(Acap T^{-p(n)}A)>mu^2(A)-epsilon}$$ is IP$^*$, meaning that for any increasing sequence $(n_k)_{kinmathbb N}$ in $mathbb N$, $$text{FS}((n_k)_{kinmathbb N})cap R_epsilon^p(A)neq emptyset,$$ where $$text{FS}((n_k)_{kinmathbb N})={sum_{jin F}n_j,|,Fsubseteq mathbb N,text{ is finite}text{ and }Fneqemptyset}={n_{k_1}+cdots+n_{k_t},|,k_11$ and $p(0)=0$ there is an invertible probability preserving system $(X,mathcal A,mu,T)$, a set $Ainmathcal A$, and an $epsilon>0$ for which the set $R_epsilon^p(A)$ is not IP$_0^*$.
{"title":"Failure of Khintchine-type results along the polynomial image of IP0 sets","authors":"Rigoberto Zelada","doi":"10.3934/dcds.2023152","DOIUrl":"https://doi.org/10.3934/dcds.2023152","url":null,"abstract":"In\"IP-sets and polynomial recurrence\", Bergelson, Furstenberg, and McCutcheon established the following far reaching extension of Khintchine's recurrence theorem: For any invertible probability preserving system $(X,mathcal A,mu,T)$, any non-constant polynomial $pinmathbb Z[x]$ with $p(0)=0$, any $Ainmathcal A$, and any $epsilon>0$, the set $$R_epsilon^p(A)={ninmathbb N,|,mu(Acap T^{-p(n)}A)>mu^2(A)-epsilon}$$ is IP$^*$, meaning that for any increasing sequence $(n_k)_{kinmathbb N}$ in $mathbb N$, $$text{FS}((n_k)_{kinmathbb N})cap R_epsilon^p(A)neq emptyset,$$ where $$text{FS}((n_k)_{kinmathbb N})={sum_{jin F}n_j,|,Fsubseteq mathbb N,text{ is finite}text{ and }Fneqemptyset}={n_{k_1}+cdots+n_{k_t},|,k_1<cdots<k_t,,tinmathbb N}.$$ In view of the potential new applications to combinatorics, this result has led to the question of whether a further strengthening of Khintchine's recurrence theorem holds, namely whether the set $R_epsilon^p(A)$ is IP$_0^*$ meaning that there exists a $tinmathbb N$ such that for any finite sequence $n_1<cdots<n_t$ in $mathbb N$, $${sum_{jin F}n_j,|,Fsubseteq {1,...,t}text{ and }Fneq emptyset}cap R_epsilon^p(A)neq emptyset.$$ In this paper we give a negative answer to this question by showing that for any given polynomial $pinmathbb Z[x]$ with deg$(p)>1$ and $p(0)=0$ there is an invertible probability preserving system $(X,mathcal A,mu,T)$, a set $Ainmathcal A$, and an $epsilon>0$ for which the set $R_epsilon^p(A)$ is not IP$_0^*$.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139358733","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}
We establish the uniqueness and the asymptotic stability of the invariant measure for the two dimensional Navier Stokes equations driven by a multiplicative noise which is either bounded or with a sublinear or a linear growth. We work on an effectively elliptic setting, that is we require that the range of the covariance operator contains the unstable directions. We exploit the generalized asymptotic coupling techniques of Glatt Holtz,Mattingly,Richards(2017) and Kulik,Scheutzow(2018), used by these authors for the stochastic Navier Stokes equations with additive noise. Here we show how these methods are flexible enough to deal with multiplicative noise as well. A crucial role in our argument is played by the Foias Prodi estimate in expected valued, which has a different form (exponential or polynomial decay) according to the growth condition of the multiplicative noise.
{"title":"Uniqueness of the invariant measure and asymptotic stability for the 2D Navier-Stokes equations with multiplicative noise","authors":"B. Ferrario, M. Zanella","doi":"10.3934/dcds.2023102","DOIUrl":"https://doi.org/10.3934/dcds.2023102","url":null,"abstract":"We establish the uniqueness and the asymptotic stability of the invariant measure for the two dimensional Navier Stokes equations driven by a multiplicative noise which is either bounded or with a sublinear or a linear growth. We work on an effectively elliptic setting, that is we require that the range of the covariance operator contains the unstable directions. We exploit the generalized asymptotic coupling techniques of Glatt Holtz,Mattingly,Richards(2017) and Kulik,Scheutzow(2018), used by these authors for the stochastic Navier Stokes equations with additive noise. Here we show how these methods are flexible enough to deal with multiplicative noise as well. A crucial role in our argument is played by the Foias Prodi estimate in expected valued, which has a different form (exponential or polynomial decay) according to the growth condition of the multiplicative noise.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139361803","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}
The stress concentration is a common phenomenon in the study of fluid-solid model. In this paper, we investigate the boundary gradient estimates and the second order derivatives estimates for the Stokes flow when the rigid particles approach the boundary of the matrix in dimension three. We classify the effect on the blow-up rates of the stress from the prescribed various boundary data: locally constant case and locally polynomial case. Our results hold for general convex inclusions, including two important cases in practice, spherical inclusions and ellipsoidal inclusions. The blow-up rates of the Cauchy stress in the narrow region are also obtained. We establish the corresponding estimates in higher dimensions greater than three.
{"title":"Stress blow-up analysis when suspending rigid particles approach boundary in 3D Stokes flow","authors":"Haigang Li, Longjuan Xu, Peihao Zhang","doi":"10.3934/dcds.2023141","DOIUrl":"https://doi.org/10.3934/dcds.2023141","url":null,"abstract":"The stress concentration is a common phenomenon in the study of fluid-solid model. In this paper, we investigate the boundary gradient estimates and the second order derivatives estimates for the Stokes flow when the rigid particles approach the boundary of the matrix in dimension three. We classify the effect on the blow-up rates of the stress from the prescribed various boundary data: locally constant case and locally polynomial case. Our results hold for general convex inclusions, including two important cases in practice, spherical inclusions and ellipsoidal inclusions. The blow-up rates of the Cauchy stress in the narrow region are also obtained. We establish the corresponding estimates in higher dimensions greater than three.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139370729","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}
This paper belongs to a series of papers devoted to the study of the structure of the non-wandering set of an A-diffeomorphism. We study such set $NW(f)$ for an $Omega$-stable diffeomorphism $f$, given on a closed connected 3-manifold $M^3$. Namely, we prove that if all basic sets in $NW(f)$ are trivial except attractors, then every non-trivial attractor is either one-dimensional non-orientable or two-dimensional expanding.
{"title":"On a structure of non-wandering set of an $ Omega $-stable 3-diffeomorphism possessing a hyperbolic attractor","authors":"M. Barinova, O. Pochinka, E. Yakovlev","doi":"10.3934/dcds.2023094","DOIUrl":"https://doi.org/10.3934/dcds.2023094","url":null,"abstract":"This paper belongs to a series of papers devoted to the study of the structure of the non-wandering set of an A-diffeomorphism. We study such set $NW(f)$ for an $Omega$-stable diffeomorphism $f$, given on a closed connected 3-manifold $M^3$. Namely, we prove that if all basic sets in $NW(f)$ are trivial except attractors, then every non-trivial attractor is either one-dimensional non-orientable or two-dimensional expanding.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86472323","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}
In this work, we study the local wellposedness of the solution to a nonlinear elliptic-dispersive coupled system which serves as a model for a Micro-Electro-Mechanical System (MEMS). A simple electrostatically actuated MEMS capacitor device consists of two parallel plates separated by a gas-filled thin gap. The nonlinear elliptic-dispersive coupled system modelling the device combines a linear elliptic equation for the gas pressure with a semilinear dispersive equation for the gap width. We show the local-in-time existence of strict solutions for the system, by combining elliptic regularity results for the elliptic equation, Lipschitz continuous dependence of its solution on that of the dispersive equation, and then local-in-time existence for a resulting abstract dispersive problem. Semigroup approaches are key to solve the abstract dispersive problem.
{"title":"Wellposedness of an elliptic-dispersive coupled system for MEMS","authors":"H. Gimperlein, Runan He, A. Lacey","doi":"10.3934/dcds.2023055","DOIUrl":"https://doi.org/10.3934/dcds.2023055","url":null,"abstract":"In this work, we study the local wellposedness of the solution to a nonlinear elliptic-dispersive coupled system which serves as a model for a Micro-Electro-Mechanical System (MEMS). A simple electrostatically actuated MEMS capacitor device consists of two parallel plates separated by a gas-filled thin gap. The nonlinear elliptic-dispersive coupled system modelling the device combines a linear elliptic equation for the gas pressure with a semilinear dispersive equation for the gap width. We show the local-in-time existence of strict solutions for the system, by combining elliptic regularity results for the elliptic equation, Lipschitz continuous dependence of its solution on that of the dispersive equation, and then local-in-time existence for a resulting abstract dispersive problem. Semigroup approaches are key to solve the abstract dispersive problem.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80819241","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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