Piermarco Cannarsa, Wei Cheng, Jiahui Hong, Kaizhi Wang
On a smooth closed manifold $M$, we introduce a novel theory of maximal slope curves for any pair $(phi,H)$ with $phi$ a semiconcave function and $H$ a Hamiltonian. By using the notion of maximal slope curve from gradient flow theory, the intrinsic singular characteristics constructed in [Cannarsa, P.; Cheng, W., textit{Generalized characteristics and Lax-Oleinik operators: global theory}. Calc. Var. Partial Differential Equations 56 (2017), no. 5, 56:12], the smooth approximation method developed in [Cannarsa, P.; Yu, Y. textit{Singular dynamics for semiconcave functions}. J. Eur. Math. Soc. 11 (2009), no. 5, 999--1024], and the broken characteristics studied in [Khanin, K.; Sobolevski, A., textit{On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations}. Arch. Ration. Mech. Anal. 219 (2016), no. 2, 861--885], we prove the existence and stability of such maximal slope curves and discuss certain new weak KAM features. We also prove that maximal slope curves for any pair $(phi,H)$ are exactly broken characteristics which have right derivatives everywhere. Applying this theory, we establish a global variational construction of strict singular characteristics and broken characteristics. Moreover, we prove a result on the global propagation of cut points along generalized characteristics, as well as a result on the propagation of singular points along strict singular characteristics, for weak KAM solutions. We also obtain the continuity equation along strict singular characteristics which clarifies the mass transport nature in the problem of propagation of singularities.
在光滑闭合流形 $M$ 上，我们为任意一对 $(phi,H)$（其中 $phi$ 为半凹函数，$H$ 为哈密尔顿函数）引入了一种新的最大斜曲线理论。通过使用梯度流理论中的最大斜率曲线概念，[Cannarsa, P.; Cheng, W., textit{Generalized characteristics and Lax-Oleinik operators: global theory}.Calc. Var.Var.Partial Differential Equations 56 (2017), no.5, 56:12], the smoothapproximation method developed in [Cannarsa, P.; Yu, Y. textit{Singulardynamics for semiconcave functions}.J. Eur.11 (2009), no.11 (2009), no.5,999--1024], and the broken characteristics studied in [Khanin, K.; Sobolevski,A., textit{On dynamics of Lagrangian trajectories for Hamilton-Jacobiequations}. Arch.Arch.Ration.Mech.Anal.219 (2016), no. 2, 861--885], we provethe existence and stability of such maximal slope curves and discuss certainnew weak KAM features.我们还证明了任何一对$(phi,H)$的最大斜率曲线都是精确破碎的特征，其右导数无处不在。应用这一理论，我们建立了严格奇异特征和破碎特征的全局变分构造。此外，我们还证明了弱 KAM 解的切点沿广义特征全局传播的结果，以及奇异点沿严格奇异特征传播的结果。我们还得到了沿严格奇异特征的连续性方程，从而澄清了奇异点传播问题中的质量传输性质。
{"title":"Variational construction of singular characteristics and propagation of singularities","authors":"Piermarco Cannarsa, Wei Cheng, Jiahui Hong, Kaizhi Wang","doi":"arxiv-2409.00961","DOIUrl":"https://doi.org/arxiv-2409.00961","url":null,"abstract":"On a smooth closed manifold $M$, we introduce a novel theory of maximal slope\u0000curves for any pair $(phi,H)$ with $phi$ a semiconcave function and $H$ a\u0000Hamiltonian. By using the notion of maximal slope curve from gradient flow theory, the\u0000intrinsic singular characteristics constructed in [Cannarsa, P.; Cheng, W.,\u0000textit{Generalized characteristics and Lax-Oleinik operators: global theory}.\u0000Calc. Var. Partial Differential Equations 56 (2017), no. 5, 56:12], the smooth\u0000approximation method developed in [Cannarsa, P.; Yu, Y. textit{Singular\u0000dynamics for semiconcave functions}. J. Eur. Math. Soc. 11 (2009), no. 5,\u0000999--1024], and the broken characteristics studied in [Khanin, K.; Sobolevski,\u0000A., textit{On dynamics of Lagrangian trajectories for Hamilton-Jacobi\u0000equations}. Arch. Ration. Mech. Anal. 219 (2016), no. 2, 861--885], we prove\u0000the existence and stability of such maximal slope curves and discuss certain\u0000new weak KAM features. We also prove that maximal slope curves for any pair\u0000$(phi,H)$ are exactly broken characteristics which have right derivatives\u0000everywhere. Applying this theory, we establish a global variational construction of\u0000strict singular characteristics and broken characteristics. Moreover, we prove\u0000a result on the global propagation of cut points along generalized\u0000characteristics, as well as a result on the propagation of singular points\u0000along strict singular characteristics, for weak KAM solutions. We also obtain\u0000the continuity equation along strict singular characteristics which clarifies\u0000the mass transport nature in the problem of propagation of singularities.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195945","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}
Many decision-making algorithms draw inspiration from the inner workings of individual biological systems. However, it remains unclear whether collective behavior among biological species can also lead to solutions for computational tasks. By studying the coexistence of species that interact through simple rules on a network, we demonstrate that the underlying dynamical system can recover near-optimal solutions to the maximum independent set problem -- a fundamental, computationally hard problem in graph theory. Furthermore, we observe that the optimality of these solutions is improved when the competitive pressure in the system is gradually increased. We explain this phenomenon by showing that the cascade of bifurcation points, which occurs with rising competitive pressure in our dynamical system, naturally gives rise to Katz centrality-based node removal in the network. By formalizing this connection, we propose a biologically inspired discrete algorithm for approximating the maximum independent set problem on a graph. Our results indicate that complex systems may collectively possess the capacity to perform non-trivial computations, with implications spanning biology, economics, and other fields.
{"title":"Finding Large Independent Sets in Networks Using Competitive Dynamics","authors":"Niek Mooij, Ivan Kryven","doi":"arxiv-2409.01336","DOIUrl":"https://doi.org/arxiv-2409.01336","url":null,"abstract":"Many decision-making algorithms draw inspiration from the inner workings of\u0000individual biological systems. However, it remains unclear whether collective\u0000behavior among biological species can also lead to solutions for computational\u0000tasks. By studying the coexistence of species that interact through simple\u0000rules on a network, we demonstrate that the underlying dynamical system can\u0000recover near-optimal solutions to the maximum independent set problem -- a\u0000fundamental, computationally hard problem in graph theory. Furthermore, we\u0000observe that the optimality of these solutions is improved when the competitive\u0000pressure in the system is gradually increased. We explain this phenomenon by\u0000showing that the cascade of bifurcation points, which occurs with rising\u0000competitive pressure in our dynamical system, naturally gives rise to Katz\u0000centrality-based node removal in the network. By formalizing this connection,\u0000we propose a biologically inspired discrete algorithm for approximating the\u0000maximum independent set problem on a graph. Our results indicate that complex\u0000systems may collectively possess the capacity to perform non-trivial\u0000computations, with implications spanning biology, economics, and other fields.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195961","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 paper, we proved that for every non-degenerate $C^3$ compact star-shaped hypersurface $Sigma$ in $mathbb{R}^{8}$ which carries no prime closed characteristic of Maslov-type index $-1$, there exist at least four prime closed characteristics on $Sigma$.
{"title":"Four closed characteristics on compact star-shaped hypersurfaces in $mathbb{R}^{8}$","authors":"Huagui Duan, Dong Xie","doi":"arxiv-2409.04460","DOIUrl":"https://doi.org/arxiv-2409.04460","url":null,"abstract":"In this paper, we proved that for every non-degenerate $C^3$ compact\u0000star-shaped hypersurface $Sigma$ in $mathbb{R}^{8}$ which carries no prime\u0000closed characteristic of Maslov-type index $-1$, there exist at least four\u0000prime closed characteristics on $Sigma$.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195949","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}
The heterochaos baker maps are piecewise affine maps on the square or the cube that are one of the simplest partially hyperbolic systems. The Dyck shift is a well-known example of a subshift that has two fully supported ergodic measures of maximal entropy (MMEs). We show that the two ergodic MMEs of the Dyck shift are represented as asymptotic distributions of sets of periodic points of different multipliers. We transfer this result to the heterochaos baker maps, and show that their two ergodic MMEs are represented as asymptotic distributions of sets of periodic points of different unstable dimensions.
{"title":"Distributions of periodic points for the Dyck shift and the heterochaos baker maps","authors":"Hiroki Takahasi","doi":"arxiv-2409.01261","DOIUrl":"https://doi.org/arxiv-2409.01261","url":null,"abstract":"The heterochaos baker maps are piecewise affine maps on the square or the\u0000cube that are one of the simplest partially hyperbolic systems. The Dyck shift\u0000is a well-known example of a subshift that has two fully supported ergodic\u0000measures of maximal entropy (MMEs). We show that the two ergodic MMEs of the\u0000Dyck shift are represented as asymptotic distributions of sets of periodic\u0000points of different multipliers. We transfer this result to the heterochaos\u0000baker maps, and show that their two ergodic MMEs are represented as asymptotic\u0000distributions of sets of periodic points of different unstable dimensions.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195917","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}
Let $(M,g)$ be a closed, connected and orientable Riemannian manifold with nonnegative Ricci curvature. Consider a Lagrangian $L(x,v):TMtoR$ defined by $L(x,v):=frac 12g_x(v,v)-omega(v)+c$, where $cinR$ and $omega$ is a closed 1-form. From the perspective of differential geometry, we estimate the Laplacian of the weak KAM solution $u$ to the associated Hamilton-Jacobi equation $H(x,du)=c[L]$ in the barrier sense. This analysis enables us to prove that each weak KAM solution $u$ is constant if and only if $omega$ is a harmonic 1-form. Furthermore, we explore several applications to the Mather quotient and Ma~n'e's Lagrangian.
{"title":"A geometric approach to Mather quotient problem","authors":"Wei Cheng, Wenxue Wei","doi":"arxiv-2409.00958","DOIUrl":"https://doi.org/arxiv-2409.00958","url":null,"abstract":"Let $(M,g)$ be a closed, connected and orientable Riemannian manifold with\u0000nonnegative Ricci curvature. Consider a Lagrangian $L(x,v):TMtoR$ defined by\u0000$L(x,v):=frac 12g_x(v,v)-omega(v)+c$, where $cinR$ and $omega$ is a closed\u00001-form. From the perspective of differential geometry, we estimate the\u0000Laplacian of the weak KAM solution $u$ to the associated Hamilton-Jacobi\u0000equation $H(x,du)=c[L]$ in the barrier sense. This analysis enables us to prove\u0000that each weak KAM solution $u$ is constant if and only if $omega$ is a\u0000harmonic 1-form. Furthermore, we explore several applications to the Mather\u0000quotient and Ma~n'e's Lagrangian.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195921","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 paper, we study the existence and properties of conformal measures on limit sets of (anti)holomorphic correspondences. We show that if the critical exponent satisfies $1leq delta_{operatorname{crit}}(x) <+infty,$ the correspondence $F$ is (relatively) hyperbolic on the limit set $Lambda_+(x)$, and $Lambda_+(x)$ is minimal, then $Lambda_+(x)$ admits a non-atomic conformal measure for $F$ and the Hausdorff dimension of $Lambda_+(x)$ is strictly less than 2. As a special case, this shows that for a parameter $a$ in the interior of a hyperbolic component of the modular Mandelbrot set, the limit set of the Bullett--Penrose correspondence $F_a$ has a non-atomic conformal measure and its Hausdorff dimension is strictly less than 2. The same results hold for the LLMM correspondences, under some extra assumptions on its defining function $f$.
{"title":"Conformal measures of (anti)holomorphic correspondences","authors":"Nils Hemmingsson, Xiaoran Li, Zhiqiang Li","doi":"arxiv-2409.01361","DOIUrl":"https://doi.org/arxiv-2409.01361","url":null,"abstract":"In this paper, we study the existence and properties of conformal measures on\u0000limit sets of (anti)holomorphic correspondences. We show that if the critical\u0000exponent satisfies $1leq delta_{operatorname{crit}}(x) <+infty,$ the\u0000correspondence $F$ is (relatively) hyperbolic on the limit set $Lambda_+(x)$,\u0000and $Lambda_+(x)$ is minimal, then $Lambda_+(x)$ admits a non-atomic\u0000conformal measure for $F$ and the Hausdorff dimension of $Lambda_+(x)$ is\u0000strictly less than 2. As a special case, this shows that for a parameter $a$ in\u0000the interior of a hyperbolic component of the modular Mandelbrot set, the limit\u0000set of the Bullett--Penrose correspondence $F_a$ has a non-atomic conformal\u0000measure and its Hausdorff dimension is strictly less than 2. The same results\u0000hold for the LLMM correspondences, under some extra assumptions on its defining\u0000function $f$.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195918","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 emergent dynamics of a rapidly yawing spheroidal swimmer interacting with a viscous shear flow. We show that the rapid yawing generates non-axisymmetric emergent effects, with the active swimmer behaving as an effective passive particle with two orthogonal planes of symmetry. The shape of the equivalent effective particle is different to the average shape of the active particle. Moreover, despite having two planes of symmetry, the equivalent passive particle is not an ellipsoid in general, except for specific scenarios in which the effective shape is a spheroid. We use a multiple scales analysis for systems to derive the emergent swimmer behaviour, which requires solving a nonautonomous nonlinear 3D dynamical system, and we validate our analysis via comparison to numerical simulations.
{"title":"Rapidly yawing spheroids in viscous shear flow: Emergent loss of symmetry","authors":"Mohit P. Dalwadi","doi":"arxiv-2409.01273","DOIUrl":"https://doi.org/arxiv-2409.01273","url":null,"abstract":"We investigate the emergent dynamics of a rapidly yawing spheroidal swimmer\u0000interacting with a viscous shear flow. We show that the rapid yawing generates\u0000non-axisymmetric emergent effects, with the active swimmer behaving as an\u0000effective passive particle with two orthogonal planes of symmetry. The shape of\u0000the equivalent effective particle is different to the average shape of the\u0000active particle. Moreover, despite having two planes of symmetry, the\u0000equivalent passive particle is not an ellipsoid in general, except for specific\u0000scenarios in which the effective shape is a spheroid. We use a multiple scales\u0000analysis for systems to derive the emergent swimmer behaviour, which requires\u0000solving a nonautonomous nonlinear 3D dynamical system, and we validate our\u0000analysis via comparison to numerical simulations.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195955","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}
Collisional-radiative (CR) models describe the atomic processes in a plasma by tracking the population density in the ground and excited states for each charge state of the atom or ion. These models predict important plasma properties such as charge state distributions and radiative emissivity and opacity. Accurate CR modeling is essential in radiative plasma modeling for magnetic fusion, especially when significant amount of impurities are introduced into the plasmas. In radiative plasma simulations, a CR model, which is a set of high-dimensional stiff ordinary differential equations (ODE), needs to be solved on each grid point in the configuration space, which can overwhelm the plasma simulation cost. In this work, we propose a deep learning method that discovers the latent space and learns its corresponding latent dynamics, which can capture the essential physics to make accurate predictions at much lower online computational cost. To facilitate coupling of the latent space CR dynamics with the plasma simulation model in physical variables, our latent space in the autoencoder must be a grey box, consisting of a physical latent space and a data-driven or blackbox latent space. It has been demonstrated that the proposed architecture can accurately predict both the full-order CR dynamics and the critical physical quantity of interest, the so-called radiative power loss rate.
{"title":"Latent Space Dynamics Learning for Stiff Collisional-radiative Models","authors":"Xuping Xie, Qi Tang, Xianzhu Tang","doi":"arxiv-2409.05893","DOIUrl":"https://doi.org/arxiv-2409.05893","url":null,"abstract":"Collisional-radiative (CR) models describe the atomic processes in a plasma\u0000by tracking the population density in the ground and excited states for each\u0000charge state of the atom or ion. These models predict important plasma\u0000properties such as charge state distributions and radiative emissivity and\u0000opacity. Accurate CR modeling is essential in radiative plasma modeling for\u0000magnetic fusion, especially when significant amount of impurities are\u0000introduced into the plasmas. In radiative plasma simulations, a CR model, which\u0000is a set of high-dimensional stiff ordinary differential equations (ODE), needs\u0000to be solved on each grid point in the configuration space, which can overwhelm\u0000the plasma simulation cost. In this work, we propose a deep learning method\u0000that discovers the latent space and learns its corresponding latent dynamics,\u0000which can capture the essential physics to make accurate predictions at much\u0000lower online computational cost. To facilitate coupling of the latent space CR\u0000dynamics with the plasma simulation model in physical variables, our latent\u0000space in the autoencoder must be a grey box, consisting of a physical latent\u0000space and a data-driven or blackbox latent space. It has been demonstrated that\u0000the proposed architecture can accurately predict both the full-order CR\u0000dynamics and the critical physical quantity of interest, the so-called\u0000radiative power loss rate.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195950","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}
To study any dynamical system it is useful to find a partition that allows essentially faithful encoding (injective, up to a small exceptional set) into a subshift. Most topological and measure-theoretic systems can be represented by Bratteli-Vershik (or adic, or BV) systems. So it is natural to ask when can a BV system be encoded essentially faithfully. We show here that for BV diagrams defined by homogeneous positive integer multivariable polynomials, and a wide family of their generalizations, which we call polynomial shape diagrams, for every choice of the edge ordering the coding according to initial path segments of a fixed finite length is injective off of a negligible exceptional set.
{"title":"Polynomial shape adic systems are inherently expansive","authors":"Sarah Frick, Karl Petersen, Sandi Shields","doi":"arxiv-2409.00762","DOIUrl":"https://doi.org/arxiv-2409.00762","url":null,"abstract":"To study any dynamical system it is useful to find a partition that allows\u0000essentially faithful encoding (injective, up to a small exceptional set) into a\u0000subshift. Most topological and measure-theoretic systems can be represented by\u0000Bratteli-Vershik (or adic, or BV) systems. So it is natural to ask when can a\u0000BV system be encoded essentially faithfully. We show here that for BV diagrams\u0000defined by homogeneous positive integer multivariable polynomials, and a wide\u0000family of their generalizations, which we call polynomial shape diagrams, for\u0000every choice of the edge ordering the coding according to initial path segments\u0000of a fixed finite length is injective off of a negligible exceptional set.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195953","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 paper we prove that the solution to the primitive equations is predicted by the corresponding data assimilation(DA) equations in $H^2$. Although, the DA equation does not include the direct information about the base solution and its initial conditions, the solution to the DA equation exponentially convergence to the base(original) solution when the external forces are known even before they are observed. Additionally, when the external force is not completely known but its spatially dense observations are available, then the DA is stable, $i.e.$ the DA solution lies in a sufficiently small neighborhood of the base solution.
{"title":"Data Assimilation to the Primitive Equations in $H^2$","authors":"Ken Furukawa","doi":"arxiv-2409.00579","DOIUrl":"https://doi.org/arxiv-2409.00579","url":null,"abstract":"In this paper we prove that the solution to the primitive equations is\u0000predicted by the corresponding data assimilation(DA) equations in $H^2$.\u0000Although, the DA equation does not include the direct information about the\u0000base solution and its initial conditions, the solution to the DA equation\u0000exponentially convergence to the base(original) solution when the external\u0000forces are known even before they are observed. Additionally, when the external\u0000force is not completely known but its spatially dense observations are\u0000available, then the DA is stable, $i.e.$ the DA solution lies in a sufficiently\u0000small neighborhood of the base solution.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"116 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195947","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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