Pub Date : 2024-07-15DOI: 10.4310/cms.2024.v22.n5.a9
Yan Gui, Cheng Wang, Jingrun Chen
Magnetization dynamics in ferromagnetic materials is modeled by the Landau-Lifshitz (LL) equation, a nonlinear system of partial differential equations. Among the numerical approaches, semi-implicit schemes are widely used in the micromagnetics simulation, due to a nice compromise between accuracy and efficiency. At each time step, only a linear system needs to be solved and a projection is then applied to preserve the length of magnetization. However, this linear system contains variable coefficients and a non-symmetric structure, and thus an efficient linear solver is highly desired. If the damping parameter becomes large, it has been realized that efficient solvers are only available to a linear system with constant, symmetric, and positive definite (SPD) structure. In this work, based on the implicit-explicit Runge-Kutta (IMEX-RK) time discretization, we introduce an artificial damping term, which is treated implicitly. The remaining terms are treated explicitly. This strategy leads to a semi-implicit scheme with the following properties: (1) only a few linear systems with constant and SPD structure needs to be solved at each time step; (2) it works for the LL equation with arbitrary damping parameter; (3) high-order accuracy can be obtained with high-order IMEX-RK time discretization. Numerically, second-order and third-order IMEX-RK methods are designed in both the 1-D and 3-D domains. A comparison with the backward differentiation formula scheme is undertaken, in terms of accuracy and efficiency. The robustness of both numerical methods is tested on the first benchmark problem from National Institute of Standards and Technology. The linearized stability estimate and optimal rate convergence analysis are provided for an alternate IMEX-RK2 numerical scheme as well.
{"title":"Implicit-explicit Runge-Kutta methods for Landau-Lifshitz equation with arbitrary damping","authors":"Yan Gui, Cheng Wang, Jingrun Chen","doi":"10.4310/cms.2024.v22.n5.a9","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a9","url":null,"abstract":"Magnetization dynamics in ferromagnetic materials is modeled by the Landau-Lifshitz (LL) equation, a nonlinear system of partial differential equations. Among the numerical approaches, semi-implicit schemes are widely used in the micromagnetics simulation, due to a nice compromise between accuracy and efficiency. At each time step, only a linear system needs to be solved and a projection is then applied to preserve the length of magnetization. However, this linear system contains variable coefficients and a non-symmetric structure, and thus an efficient linear solver is highly desired. If the damping parameter becomes large, it has been realized that efficient solvers are only available to a linear system with constant, symmetric, and positive definite (SPD) structure. In this work, based on the implicit-explicit Runge-Kutta (IMEX-RK) time discretization, we introduce an artificial damping term, which is treated implicitly. The remaining terms are treated explicitly. This strategy leads to a semi-implicit scheme with the following properties: (1) only a few linear systems with constant and SPD structure needs to be solved at each time step; (2) it works for the LL equation with arbitrary damping parameter; (3) high-order accuracy can be obtained with high-order IMEX-RK time discretization. Numerically, second-order and third-order IMEX-RK methods are designed in both the 1-D and 3-D domains. A comparison with the backward differentiation formula scheme is undertaken, in terms of accuracy and efficiency. The robustness of both numerical methods is tested on the first benchmark problem from National Institute of Standards and Technology. The linearized stability estimate and optimal rate convergence analysis are provided for an alternate IMEX-RK2 numerical scheme as well.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"70 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.4310/cms.2024.v22.n5.a5
Yishu Gong, Alexander Kiselev
Chemotaxis, which involves the directed movement of cells in response to a chemical gradient, plays a crucial role in a broad variety of biological processes. Examples include bacterial motion, the development of single-cell or multicellular organisms, and immune responses. Chemotaxis directs bacteria’s movement to find food (e.g., glucose) by swimming toward the highest concentration of food molecules. In multicellular organisms, chemotaxis is critical to early development (e.g., movement of sperm towards the egg during fertilization). Chemotaxis also helps mobilize phagocytic and immune cells at sites of infection, tissue injury, and thus facilitates immune reactions. In this paper, we study a PDE system that describes chemotactic processes in one dimension, which may correspond to a thin channel, the setting relevant in many applications: for example, spermatozoa progression to the ovum inside a Fallopian tube or immune response in a blood vessel. Our objective is to obtain qualitatively precise estimates on how chemotaxis improves reaction efficiency, when compared to purely diffusive situation. The techniques we use to achieve this goal include a variety of comparison principles and analysis of mass transport for a class of Fokker–Planck operators.
{"title":"Chemotactic reaction enhancement in one dimension","authors":"Yishu Gong, Alexander Kiselev","doi":"10.4310/cms.2024.v22.n5.a5","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a5","url":null,"abstract":"Chemotaxis, which involves the directed movement of cells in response to a chemical gradient, plays a crucial role in a broad variety of biological processes. Examples include bacterial motion, the development of single-cell or multicellular organisms, and immune responses. Chemotaxis directs bacteria’s movement to find food (e.g., glucose) by swimming toward the highest concentration of food molecules. In multicellular organisms, chemotaxis is critical to early development (e.g., movement of sperm towards the egg during fertilization). Chemotaxis also helps mobilize phagocytic and immune cells at sites of infection, tissue injury, and thus facilitates immune reactions. In this paper, we study a PDE system that describes chemotactic processes in one dimension, which may correspond to a thin channel, the setting relevant in many applications: for example, spermatozoa progression to the ovum inside a Fallopian tube or immune response in a blood vessel. Our objective is to obtain qualitatively precise estimates on how chemotaxis improves reaction efficiency, when compared to purely diffusive situation. The techniques we use to achieve this goal include a variety of comparison principles and analysis of mass transport for a class of Fokker–Planck operators.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"2 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.4310/cms.2024.v22.n5.a8
Zhicheng Tong, Yong Li
As is well known the classical Hartman–Grobman theorem states that a $C^1$ mapping can be $C^0$ linearized near its hyperbolic fixed point in $mathbb{R}^n$. However, it is quite nontrivial to guarantee the local homeomorphism to be differentiable. Recently, the regularity assumption on derivative of the mapping has been weakened to Hölder’s type, significantly improving the work of $C^infty$, but still unknown for only differentiable case. We will try to touch this question in this paper. Without Hölder’s type, we first consider the existence and regularity of weak-stable manifolds for homeomorphisms with contraction in a Banach space, and further study linearization of mappings near hyperbolic fixed points. More precisely, we propose an Integrability Condition for regularity on linearization which is proved to be sharp, and establish a differentiable Hartman–Grobman theorem via modulus of continuity in a general Banach space. Thus we provide an almost complete answer to the question mentioned above.
{"title":"Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space","authors":"Zhicheng Tong, Yong Li","doi":"10.4310/cms.2024.v22.n5.a8","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a8","url":null,"abstract":"As is well known the classical Hartman–Grobman theorem states that a $C^1$ mapping can be $C^0$ linearized near its hyperbolic fixed point in $mathbb{R}^n$. However, it is quite nontrivial to guarantee the local homeomorphism to be differentiable. Recently, the regularity assumption on derivative of the mapping has been weakened to Hölder’s type, significantly improving the work of $C^infty$, but still unknown for only differentiable case. We will try to touch this question in this paper. Without Hölder’s type, we first consider the existence and regularity of weak-stable manifolds for homeomorphisms with contraction in a Banach space, and further study linearization of mappings near hyperbolic fixed points. More precisely, we propose an Integrability Condition for regularity on linearization which is proved to be sharp, and establish a differentiable Hartman–Grobman theorem via modulus of continuity in a general Banach space. Thus we provide an almost complete answer to the question mentioned above.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"28 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.4310/cms.2024.v22.n5.a2
Chongyao Chen, Ziang Chen, Jianfeng Lu
In this work, we establish a representation theorem for multivariable totally symmetric functions: a multisymmetric continuous function must be the composition of a continuous function and a set of generators of the multisymmetric polynomials. We then study the singularity and geometry of the generators, and show that the regularity may become worse after applying the decomposition.
{"title":"Representation theorem for multivariable totally symmetric functions","authors":"Chongyao Chen, Ziang Chen, Jianfeng Lu","doi":"10.4310/cms.2024.v22.n5.a2","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a2","url":null,"abstract":"In this work, we establish a representation theorem for multivariable totally symmetric functions: a multisymmetric continuous function must be the composition of a continuous function and a set of generators of the multisymmetric polynomials. We then study the singularity and geometry of the generators, and show that the regularity may become worse after applying the decomposition.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"63 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a7
Ben Duan, Haoran Zheng
The Euler–Poisson equations with varying background charges in finitely long flat nozzles are investigated, for which two and only two transonic shock solutions are constructed. In href{https://dx.doi.org/10.4310/CMS.2012.v10.n2.a1}{[textrm{T. Luo and Z.P. Xin, Commun. Math. Sci., 10:419–462, 2012}]}, Luo and Xin established the wellposedness of steady Euler–Poisson equations for the constant background charge. Motivated by their pioneering work and combined with the special physical character of semiconductor devices, we propose the transonic shock problem in which the density of the background charge is a piecewise constant function and its discontinuity is determined only by shock fronts. The existence and non-uniqueness of transonic shock solutions are obtained via the method of shock matching.
研究了有限长平喷嘴中背景电荷变化的欧拉-泊松方程，构建了两个且仅有两个跨音速冲击解。在 href{https://dx.doi.org/10.4310/CMS.2012.v10.n2.a1}{[textrm{T. Luo and Z.P. Xin, Commun. Math. Sci.在他们开创性工作的激励下，结合半导体器件的特殊物理特性，我们提出了跨音速冲击问题，其中背景电荷密度是一个片断恒定函数，其不连续性仅由冲击前沿决定。通过冲击匹配方法，我们得到了跨音速冲击解的存在性和非唯一性。
{"title":"Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges","authors":"Ben Duan, Haoran Zheng","doi":"10.4310/cms.2024.v22.n3.a7","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a7","url":null,"abstract":"The Euler–Poisson equations with varying background charges in finitely long flat nozzles are investigated, for which two and only two transonic shock solutions are constructed. In href{https://dx.doi.org/10.4310/CMS.2012.v10.n2.a1}{[textrm{T. Luo and Z.P. Xin, Commun. Math. Sci., 10:419–462, 2012}]}, Luo and Xin established the wellposedness of steady Euler–Poisson equations for the constant background charge. Motivated by their pioneering work and combined with the special physical character of semiconductor devices, we propose the transonic shock problem in which the density of the background charge is a piecewise constant function and its discontinuity is determined only by shock fronts. The existence and non-uniqueness of transonic shock solutions are obtained via the method of shock matching.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"152 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a10
Felisia Angela Chiarello, Simone Göttlich, Thomas Schillinger, Andrea Tosin
We present a formal kinetic derivation of a second order macroscopic traffic model from a stochastic particle model. The macroscopic model is given by a system of hyperbolic partial differential equations (PDEs) with a discontinuous flux function, in which the traffic density and the headway are the averaged quantities. A numerical study illustrates the performance of the second order model compared to the particle approach. We also analyse numerically uncertain traffic accidents by considering statistical measures of the solution to the PDEs.
{"title":"Hydrodynamic traffic flow models including random accidents: A kinetic derivation","authors":"Felisia Angela Chiarello, Simone Göttlich, Thomas Schillinger, Andrea Tosin","doi":"10.4310/cms.2024.v22.n3.a10","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a10","url":null,"abstract":"We present a formal kinetic derivation of a second order macroscopic traffic model from a stochastic particle model. The macroscopic model is given by a system of hyperbolic partial differential equations (PDEs) with a discontinuous flux function, in which the traffic density and the headway are the averaged quantities. A numerical study illustrates the performance of the second order model compared to the particle approach. We also analyse numerically uncertain traffic accidents by considering statistical measures of the solution to the PDEs.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"134 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a9
Mauro Garavello, Francesca Marcellini
This paper deals with the Cauchy problem for a PDE‑ODE model, where a system of two conservation laws, namely the Two-Phase macroscopic model proposed in [Rinaldo M. Colombo, Francesca Marcellini, and Michel Rascle, $href{https://doi.org/10.1137/090752468}{textrm{SIAM J. Appl. Math., 70(7):2652–2666, 2010}}$], is coupled with an ordinary differential equation describing the trajectory of an autonomous vehicle (AV), which aims to control the traffic flow. Under suitable assumptions, we prove a global-in-time existence result.
本文讨论了一个 PDE-ODE 模型的考奇问题，在该模型中，两个守恒定律系统，即[Rinaldo M. Colombo, Francesca Marcellini, and Michel Rascle, $href{https://doi.org/10.1137/090752468}{textrm{SIAM J. Appl.在合适的假设条件下，我们证明了全局时间内的存在性结果。
{"title":"Autonomous vehicles driving traffic: The Cauchy problem","authors":"Mauro Garavello, Francesca Marcellini","doi":"10.4310/cms.2024.v22.n3.a9","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a9","url":null,"abstract":"This paper deals with the Cauchy problem for a PDE‑ODE model, where a system of two conservation laws, namely the Two-Phase macroscopic model proposed in [Rinaldo M. Colombo, Francesca Marcellini, and Michel Rascle, $href{https://doi.org/10.1137/090752468}{textrm{SIAM J. Appl. Math., 70(7):2652–2666, 2010}}$], is coupled with an ordinary differential equation describing the trajectory of an autonomous vehicle (AV), which aims to control the traffic flow. Under suitable assumptions, we prove a global-in-time existence result.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"42 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a5
Gang Bao, Lei Zhang
This paper considers an obstacle scattering problem in a chiral medium under circularly polarized oblique plane wave incidence, which can be represented as a combination of a left-circularly polarized plane wave and a right-circularly polarized one. We apply a reduced model problem with coupled oblique derivative boundary conditions, describing the cross-coupling effect of electric and magnetic fields. A novel boundary integral equation is constructed by introducing single-layer potential operators and the corresponding normal and tangential derivative operators. The corresponding properties are obtained by splitting techniques to overcome the singularity of integral operators. A numerical method for solving the boundary integral equation is developed, whose convergence is proved. Numerical results are presented to show the performance of the proposed method.
{"title":"Analysis and computation for the scattering problem of electromagnetic waves in chiral media","authors":"Gang Bao, Lei Zhang","doi":"10.4310/cms.2024.v22.n3.a5","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a5","url":null,"abstract":"This paper considers an obstacle scattering problem in a chiral medium under circularly polarized oblique plane wave incidence, which can be represented as a combination of a left-circularly polarized plane wave and a right-circularly polarized one. We apply a reduced model problem with coupled oblique derivative boundary conditions, describing the cross-coupling effect of electric and magnetic fields. A novel boundary integral equation is constructed by introducing single-layer potential operators and the corresponding normal and tangential derivative operators. The corresponding properties are obtained by splitting techniques to overcome the singularity of integral operators. A numerical method for solving the boundary integral equation is developed, whose convergence is proved. Numerical results are presented to show the performance of the proposed method.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"6 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a1
Jiyong Li
Recently, the long-time numerical simulation and error analysis of PDEs with weak nonlinearity (or small potentials) become an interesting topic. However, the existing results of long-time error analysis mostly focus on the single equations. In this paper, for the Klein–Gordon–Schrödinger equation (KGSE) with a small coupling constant $varepsilon in (0,1]$, we propose an exponential wave integrator Fourier pseudo-spectral (EWIFP) method by reformulating the KGSE into a coupled nonlinear Schrödinger system (CNLSS). Through careful and rigorous analysis, we establish improved error bounds for the numerical solution at $O(h^m + varepsilon tau^2)$ in the long-time domain up to $O(1/varepsilon)$ where $m$ is determined by the regularity conditions, h is the mesh size and τ is the time step, respectively. Compared with the existing results, our analysis shows the long-time errors of numerical solution for the KGSE. In error analysis, in addition to the classical tools such as energy method and cut-off technique, we also adopt the regularity compensation oscillation (RCO) technique which has been developed recently to analyze the accumulation of errors carefully. The numerical experiments support our error estimates and demonstrate the long-term stability of discrete mass and energy. To the best of our knowledge, there has not been any relevant long-time error analysis for the KGSE and any improved uniform error bounds for an exponential wave integrator. Our work is novel and provides a reference for analyzing the improved error bounds of the numerical methods for other coupled equations.
{"title":"Improved uniform error bounds of an exponential wave integrator method for the Klein–Gordon–Schrödinger equation with the small coupling constant","authors":"Jiyong Li","doi":"10.4310/cms.2024.v22.n3.a1","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a1","url":null,"abstract":"Recently, the long-time numerical simulation and error analysis of PDEs with weak nonlinearity (or small potentials) become an interesting topic. However, the existing results of long-time error analysis mostly focus on the single equations. In this paper, for the Klein–Gordon–Schrödinger equation (KGSE) with a small coupling constant $varepsilon in (0,1]$, we propose an exponential wave integrator Fourier pseudo-spectral (EWIFP) method by reformulating the KGSE into a coupled nonlinear Schrödinger system (CNLSS). Through careful and rigorous analysis, we establish improved error bounds for the numerical solution at $O(h^m + varepsilon tau^2)$ in the long-time domain up to $O(1/varepsilon)$ where $m$ is determined by the regularity conditions, h is the mesh size and τ is the time step, respectively. Compared with the existing results, our analysis shows the long-time errors of numerical solution for the KGSE. In error analysis, in addition to the classical tools such as energy method and cut-off technique, we also adopt the regularity compensation oscillation (RCO) technique which has been developed recently to analyze the accumulation of errors carefully. The numerical experiments support our error estimates and demonstrate the long-term stability of discrete mass and energy. To the best of our knowledge, there has not been any relevant long-time error analysis for the KGSE and any improved uniform error bounds for an exponential wave integrator. Our work is novel and provides a reference for analyzing the improved error bounds of the numerical methods for other coupled equations.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"114 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140035267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a8
Shijin Deng, Wenjun Wang, Feng Xie, Xiongfeng Yang
In this paper, we study the global existence and the large-time behavior of solutions to the Cauchy problem of the full Euler system with damping and radiative effects around some constant equilibrium states. It is well-known that the solutions may blow up in finite time without the additional damping and radiative effects, and the global existence of the solutions obtained in this paper shows that these two effects together prevent the formation of the singularity when the initial perturbation is small. Combining the Green’s function method and energy estimates, we consider the pointwise structures of the solutions to obtain a precise description of the system. The construction of the Green’s function includes three steps: singularity removal, long wave-short wave decomposition and weighted energy estimate. Finally, we achieve the pointwise estimates of the solutions in the small perturbation framework by Duhamel’s principle, the pointwise structure of the Green’s function established for the linearized equations and bounded estimates for higher order derivatives of the solutions together.
{"title":"Global existence and asymptotic behavior of the full Euler system with damping and radiative effects in $mathbb{R}^3$","authors":"Shijin Deng, Wenjun Wang, Feng Xie, Xiongfeng Yang","doi":"10.4310/cms.2024.v22.n3.a8","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a8","url":null,"abstract":"In this paper, we study the global existence and the large-time behavior of solutions to the Cauchy problem of the full Euler system with damping and radiative effects around some constant equilibrium states. It is well-known that the solutions may blow up in finite time without the additional damping and radiative effects, and the global existence of the solutions obtained in this paper shows that these two effects together prevent the formation of the singularity when the initial perturbation is small. Combining the Green’s function method and energy estimates, we consider the pointwise structures of the solutions to obtain a precise description of the system. The construction of the Green’s function includes three steps: singularity removal, long wave-short wave decomposition and weighted energy estimate. Finally, we achieve the pointwise estimates of the solutions in the small perturbation framework by Duhamel’s principle, the pointwise structure of the Green’s function established for the linearized equations and bounded estimates for higher order derivatives of the solutions together.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"9 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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