{"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}
{"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}
{"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}
{"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}
{"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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