The Craik-Leibovich equation (CL) serves as the theoretical model for Langmuir circulation. We show that the CL equation can be reduced to the dual space of a certain Lie algebra central extension. On this space, the CL equation can be rewritten as a Hamiltonian equation corresponding to the kinetic energy. Additionally, we provide an explanation of the appearance of this central extension structure through an averaging theory for Langmuir circulation. Lastly, we prove a stability theorem for two-dimensional steady flows of the CL equation. The paper also contains two examples of stable steady CL flows.
{"title":"On the Hamiltonian and geometric structure of Langmuir circulation","authors":"Cheng Yang","doi":"10.3934/cam.2023004","DOIUrl":"https://doi.org/10.3934/cam.2023004","url":null,"abstract":"The Craik-Leibovich equation (CL) serves as the theoretical model for Langmuir circulation. We show that the CL equation can be reduced to the dual space of a certain Lie algebra central extension. On this space, the CL equation can be rewritten as a Hamiltonian equation corresponding to the kinetic energy. Additionally, we provide an explanation of the appearance of this central extension structure through an averaging theory for Langmuir circulation. Lastly, we prove a stability theorem for two-dimensional steady flows of the CL equation. The paper also contains two examples of stable steady CL flows.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117164597","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 quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.
{"title":"Quantization of Hamiltonian and non-Hamiltonian systems","authors":"S. Rashkovskiy","doi":"10.3934/cam.2023014","DOIUrl":"https://doi.org/10.3934/cam.2023014","url":null,"abstract":"\u0000The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.\u0000","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124196814","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, the 1-D compressible non-isentropic Euler equations with the source term $ betarho|u|^ alpha u $ in a bounded domain are considered. First, we study the existence of steady flows which can keep the upstream supersonic or subsonic state. Then, by wave decomposition and uniform prior estimations, we prove the global existence and stability of smooth solutions under small perturbations around the steady supersonic flow. Moreover, we get that the smooth supersonic solution is a temporal periodic solution with the same period as the boundary, after a certain start-up time, once the boundary conditions are temporal periodic.
研究了源项为$ betarho|u|^ alpha u $的有界域上的一维可压缩非等熵欧拉方程。首先，研究了能保持上游超声速或亚音速状态的定常流的存在性。然后，通过波分解和均匀先验估计，证明了稳定超声速流动周围小扰动下光滑解的全局存在性和稳定性。在一定的启动时间后，一旦边界条件为时间周期，则光滑超声速解是与边界周期相同的时间周期解。
{"title":"Global existence and stability of temporal periodic solution to non-isentropic compressible Euler equations with a source term","authors":"Shuyue Ma, Jiawei Sun, Huimin Yu","doi":"10.3934/cam.2023013","DOIUrl":"https://doi.org/10.3934/cam.2023013","url":null,"abstract":"In this paper, the 1-D compressible non-isentropic Euler equations with the source term $ betarho|u|^ alpha u $ in a bounded domain are considered. First, we study the existence of steady flows which can keep the upstream supersonic or subsonic state. Then, by wave decomposition and uniform prior estimations, we prove the global existence and stability of smooth solutions under small perturbations around the steady supersonic flow. Moreover, we get that the smooth supersonic solution is a temporal periodic solution with the same period as the boundary, after a certain start-up time, once the boundary conditions are temporal periodic.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125899337","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 a partially synchronizable system for a coupled system of wave equations with different wave speeds in the framework of classical solutions in one dimensional. A partially synchronizable system is defined as a system with at least one partial synchronized solutions. In fact, we cannot consider partial synchronization as the case that the system has the same wave speeds, because the influence of different wave speeds cause only some of the function in a given space being in a partially synchronized state, rather than all functions. Therefore, we can only consider under what conditions the coupled system can have partially synchronized solutions. We will consider it in two ways. On the one hand, under the necessary conditions, we obtain an unclosed characteristic equation associated with the partially synchronizable state. We add conditions to the wave speed matrix and coupling matrix to make the equation closed. From this, the characteristic function can be obtained, and all partially synchronized solutions are obtained; then we obtain the conditions under which the initial value should be satisfied. On the other hand, we consider a system of three variables first, where there are only two synchronized variables. By subtracting them to obtain a new variable, the problem can be transformed into the problem wherein the system that satisfies the new variable should have only zero solutions. Then solving this problem can lead to obtaining the conditions required for a partially synchronized solution. After extending it to the case of $ N $ variables, similar conclusions can be obtained.
本文在一维经典解的框架下，研究了不同波速波动方程耦合系统的部分可同步系统。部分可同步系统被定义为具有至少一个部分同步解决方案的系统。事实上，我们不能把部分同步看作系统具有相同波速的情况，因为不同波速的影响只会使给定空间中的部分函数处于部分同步状态，而不是所有函数都处于部分同步状态。因此，我们只能考虑在什么条件下耦合系统可以有部分同步解。我们将从两个方面来考虑。一方面，在必要条件下，我们得到了与部分可同步状态相关的不闭合特征方程。在波速矩阵和耦合矩阵中加入条件，使方程闭合。由此得到特征函数，并得到所有部分同步解;然后得到满足初值的条件。另一方面，我们首先考虑一个有三个变量的系统，其中只有两个同步变量。将它们相减得到一个新变量，将问题转化为满足新变量的方程组只有零解的问题。然后，解决这个问题可以导致获得部分同步解决方案所需的条件。将其推广到$ N $变量的情况下，可以得到类似的结论。
{"title":"On a partially synchronizable system for a coupled system of wave equations in one dimension","authors":"Yachun Li, Chen-main Wang","doi":"10.3934/cam.2023023","DOIUrl":"https://doi.org/10.3934/cam.2023023","url":null,"abstract":"In this paper, we study a partially synchronizable system for a coupled system of wave equations with different wave speeds in the framework of classical solutions in one dimensional. A partially synchronizable system is defined as a system with at least one partial synchronized solutions. In fact, we cannot consider partial synchronization as the case that the system has the same wave speeds, because the influence of different wave speeds cause only some of the function in a given space being in a partially synchronized state, rather than all functions. Therefore, we can only consider under what conditions the coupled system can have partially synchronized solutions. We will consider it in two ways. On the one hand, under the necessary conditions, we obtain an unclosed characteristic equation associated with the partially synchronizable state. We add conditions to the wave speed matrix and coupling matrix to make the equation closed. From this, the characteristic function can be obtained, and all partially synchronized solutions are obtained; then we obtain the conditions under which the initial value should be satisfied. On the other hand, we consider a system of three variables first, where there are only two synchronized variables. By subtracting them to obtain a new variable, the problem can be transformed into the problem wherein the system that satisfies the new variable should have only zero solutions. Then solving this problem can lead to obtaining the conditions required for a partially synchronized solution. After extending it to the case of $ N $ variables, similar conclusions can be obtained.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122197573","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}
Previously, boundary control problems for parabolic type equations were considered. A portion of the thin rod boundary has a temperature-controlled heater. Its mode of operation should be found so that the average temperature in some region reaches a certain value. In this article, we consider the boundary control problem for the pseudo-parabolic equation. The value of the solution with the control parameter is given in the boundary of the interval. Control constraints are given such that the average value of the solution in considered domain takes a given value. The auxiliary problem is solved by the method of separation of variables, and the problem under consideration is reduced to the Volterra integral equation. The existence theorem of admissible control is proved by the Laplace transform method.
{"title":"On a boundary control problem for a pseudo-parabolic equation","authors":"F. Dekhkonov","doi":"10.3934/cam.2023015","DOIUrl":"https://doi.org/10.3934/cam.2023015","url":null,"abstract":"Previously, boundary control problems for parabolic type equations were considered. A portion of the thin rod boundary has a temperature-controlled heater. Its mode of operation should be found so that the average temperature in some region reaches a certain value. In this article, we consider the boundary control problem for the pseudo-parabolic equation. The value of the solution with the control parameter is given in the boundary of the interval. Control constraints are given such that the average value of the solution in considered domain takes a given value. The auxiliary problem is solved by the method of separation of variables, and the problem under consideration is reduced to the Volterra integral equation. The existence theorem of admissible control is proved by the Laplace transform method.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"535 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127640400","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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