Pub Date : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a1
Maxime Bouchereau, Philippe Chartier, Mohammed Lemou, Florian Méhats
Ordinary Differential Equations are generally too complex to be solved analytically. Approximations thereof can be obtained by general purpose numerical methods. However, even though accurate schemes have been developed, they remain computationally expensive: In this paper, we resort to the theory of modified equations in order to obtain “on the fly” cheap numerical approximations. The recipe consists in approximating, prior to that, the modified field associated to the modified equation by neural networks. Elementary convergence results are then established and the efficiency of the technique is demonstrated on experiments.
{"title":"Machine learning methods for autonomous ordinary differential equations","authors":"Maxime Bouchereau, Philippe Chartier, Mohammed Lemou, Florian Méhats","doi":"10.4310/cms.2024.v22.n6.a1","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a1","url":null,"abstract":"Ordinary Differential Equations are generally too complex to be solved analytically. Approximations thereof can be obtained by general purpose numerical methods. However, even though accurate schemes have been developed, they remain computationally expensive: In this paper, we resort to the theory of modified equations in order to obtain “on the fly” cheap numerical approximations. The recipe consists in approximating, prior to that, the modified field associated to the modified equation by <i>neural networks</i>. Elementary convergence results are then established and the efficiency of the technique is demonstrated on experiments.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"7 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744411","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 : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a5
Honghua Chen, Geng Lai, Wancheng Sheng
This paper studies the two-dimensional (2D) Zeldovich-von Neumann-Döring (ZND) combustion equations with initial data, which are a combination of an axisymmetric flow in a ring and vacuum in the remaining domain. Existence of a global-in-time smooth solution to the initial value problem is obtained by the method of characteristic decomposition, provided that the initial data satisfy some sufficient conditions. The large-time behavior of the solution is also studied. As a result, at any time, the ring continues to expand until the gas burns out in infinite time for the system. The solution describes a phenomenon of the expansion of 2D reacting flows with swirl in vacuum or a phenomenon of “fire whirl”.
{"title":"Global smooth solutions to the two-dimensional axisymmetric Zeldovich-von Neumann-Döring combustion equations with swirl","authors":"Honghua Chen, Geng Lai, Wancheng Sheng","doi":"10.4310/cms.2024.v22.n6.a5","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a5","url":null,"abstract":"This paper studies the two-dimensional (2D) Zeldovich-von Neumann-Döring (ZND) combustion equations with initial data, which are a combination of an axisymmetric flow in a ring and vacuum in the remaining domain. Existence of a global-in-time smooth solution to the initial value problem is obtained by the method of characteristic decomposition, provided that the initial data satisfy some sufficient conditions. The large-time behavior of the solution is also studied. As a result, at any time, the ring continues to expand until the gas burns out in infinite time for the system. The solution describes a phenomenon of the expansion of 2D reacting flows with swirl in vacuum or a phenomenon of “fire whirl”.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744414","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 : 2024-07-18DOI: 10.4310/cms.2024.v22.n6.a12
Mirza Karamehmedović, Faouzi Triki
We consider the localization in the eigenfunctions of regular Sturm-Liouville operators. After deriving non-asymptotic and asymptotic lower and upper bounds on the localization coefficient of the eigenfunctions, we characterize the landscape function in terms of the first eigenfunction. Several numerical experiments are provided to illustrate the obtained theoretical results.
{"title":"Localization and the landscape function for regular Sturm-Liouville operators","authors":"Mirza Karamehmedović, Faouzi Triki","doi":"10.4310/cms.2024.v22.n6.a12","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n6.a12","url":null,"abstract":"We consider the localization in the eigenfunctions of regular Sturm-Liouville operators. After deriving non-asymptotic and asymptotic lower and upper bounds on the localization coefficient of the eigenfunctions, we characterize the landscape function in terms of the first eigenfunction. Several numerical experiments are provided to illustrate the obtained theoretical results.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"64 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744422","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.a11
Guido Cavallaro, Carlo Marchioro
We study existence and uniqueness of the solution to the gravitational Vlasov–Poisson system evolving in $mathbb{R}^3$. It is assumed that initially the particles are distributed according to a spatial density with a power-law decay in space, allowing for unbounded mass, and an exponential decay in velocities given by a Maxwell–Boltzmann law. We extend a classical result which holds for systems with finite total mass.
{"title":"The gravitational Vlasov-Poisson system with infinite mass and velocities in $mathbb{R}^3$","authors":"Guido Cavallaro, Carlo Marchioro","doi":"10.4310/cms.2024.v22.n5.a11","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a11","url":null,"abstract":"We study existence and uniqueness of the solution to the gravitational Vlasov–Poisson system evolving in $mathbb{R}^3$. It is assumed that initially the particles are distributed according to a spatial density with a power-law decay in space, allowing for unbounded mass, and an exponential decay in velocities given by a Maxwell–Boltzmann law. We extend a classical result which holds for systems with finite total mass.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"46 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719564","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.a6
Ruisheng Qi, Meng Cai, Xiaojie Wang
The first aim of this paper is to examine existence, uniqueness and regularity for the stochastic Cahn–Hilliard equation with additive noise in space dimension $dleq 3$. By applying a spectral Galerkin method to the infinite dimensional equation, we elaborate the well-posedness and regularity of the finite dimensional approximate problem. The key idea lies in transforming the stochastic problem with additive noise into an equivalent random equation. The regularity of the solution to the equivalent random equation is obtained, in one dimension, with the aid of the Gagliardo–Nirenberg inequality and is done in two and three dimensions, by the energy argument. Further, the approximate solution is shown to be strongly convergent to the unique mild solution of the original stochastic equation, whose spatio-temporal regularity can be attained by similar arguments. In addition, a fully discrete approximation of such problem is investigated, performed by the spectral Galerkin method in space and the backward Euler method in time. The previously obtained regularity results help us to identify strong convergence rates of the fully discrete scheme. Numerical examples are finally included to confirm the theoretical findings.
{"title":"Strong convergence rates of a fully discrete scheme for the stochastic Cahn-Hilliard equation with additive noise","authors":"Ruisheng Qi, Meng Cai, Xiaojie Wang","doi":"10.4310/cms.2024.v22.n5.a6","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a6","url":null,"abstract":"The first aim of this paper is to examine existence, uniqueness and regularity for the stochastic Cahn–Hilliard equation with additive noise in space dimension $dleq 3$. By applying a spectral Galerkin method to the infinite dimensional equation, we elaborate the well-posedness and regularity of the finite dimensional approximate problem. The key idea lies in transforming the stochastic problem with additive noise into an equivalent random equation. The regularity of the solution to the equivalent random equation is obtained, in one dimension, with the aid of the Gagliardo–Nirenberg inequality and is done in two and three dimensions, by the energy argument. Further, the approximate solution is shown to be strongly convergent to the unique mild solution of the original stochastic equation, whose spatio-temporal regularity can be attained by similar arguments. In addition, a fully discrete approximation of such problem is investigated, performed by the spectral Galerkin method in space and the backward Euler method in time. The previously obtained regularity results help us to identify strong convergence rates of the fully discrete scheme. Numerical examples are finally included to confirm the theoretical findings.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"2012 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719561","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.a4
Ying Sun, Jianwen Zhang
This paper is concerned with an initial and boundary value problem for planar compressible magnetohydrodynamics with temperature-dependent transport coefficients. In the case when the viscosity $mu (theta) = lambda (theta) = theta^alpha$, the magnetic diffusivity $nu (theta ) = theta^alpha$ and the heat-conductivity $kappa (theta ) = theta^beta$ with $alpha ,beta in[0, infty)$, we prove the global existence of strong solution under some restrictions on the growth exponent $alpha$ and the initial norms. As a byproduct, the exponential stability of the solution is obtained. It is worth pointing out that the initial data could be large if $alpha geq 0$ is small, and the growth exponent of heat-conductivity $beta geq 0$ can be arbitrarily large.
{"title":"Global existence and exponential stability of planar magnetohydrodynamics with temperature-dependent transport coefficients","authors":"Ying Sun, Jianwen Zhang","doi":"10.4310/cms.2024.v22.n5.a4","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a4","url":null,"abstract":"This paper is concerned with an initial and boundary value problem for planar compressible magnetohydrodynamics with temperature-dependent transport coefficients. In the case when the viscosity $mu (theta) = lambda (theta) = theta^alpha$, the magnetic diffusivity $nu (theta ) = theta^alpha$ and the heat-conductivity $kappa (theta ) = theta^beta$ with $alpha ,beta in[0, infty)$, we prove the global existence of strong solution under some restrictions on the growth exponent $alpha$ and the initial norms. As a byproduct, the exponential stability of the solution is obtained. It is worth pointing out that the initial data could be large if $alpha geq 0$ is small, and the growth exponent of heat-conductivity $beta geq 0$ can be arbitrarily large.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"12 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719558","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.a7
Zhengguang Guo, Zunzun Zhang, Caidi Zhao
We study the global regularity for the three dimensional incompressible magnetohydrodynamic-Boussinesq (MHD-Boussinesq) equations. By establishing some sufficient regularity conditions in terms of partial components of velocity and magnetic fields, we prove that solutions to the MHD-Boussinesq equations without thermal diffusivity will not blow-up in any finite time.
{"title":"Global regularity criteria of the 3D MHD-Boussinesq equations without thermal diffusion","authors":"Zhengguang Guo, Zunzun Zhang, Caidi Zhao","doi":"10.4310/cms.2024.v22.n5.a7","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a7","url":null,"abstract":"We study the global regularity for the three dimensional incompressible magnetohydrodynamic-Boussinesq (MHD-Boussinesq) equations. By establishing some sufficient regularity conditions in terms of partial components of velocity and magnetic fields, we prove that solutions to the MHD-Boussinesq equations without thermal diffusivity will not blow-up in any finite time.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719565","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.a3
Xinghong Pan, Chao-Jiang Xu
For the three dimensional Prandtl boundary layer equations, we will show that for arbitrary $M$ and sufficiently small $epsilon$, the lifespan of the Gevrey-2 solution is at least of size $epsilon^{-M}$ if the initial data lies in suitable Gevrey-2 spaces with size of $epsilon$.
{"title":"Long-time existence of Gevrey-2 solutions to the 3D Prandtl boundary layer equations","authors":"Xinghong Pan, Chao-Jiang Xu","doi":"10.4310/cms.2024.v22.n5.a3","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a3","url":null,"abstract":"For the three dimensional Prandtl boundary layer equations, we will show that for arbitrary $M$ and sufficiently small $epsilon$, the lifespan of the Gevrey-2 solution is at least of size $epsilon^{-M}$ if the initial data lies in suitable Gevrey-2 spaces with size of $epsilon$.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719557","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.a1
Chunhua Jin
The chemotaxis-fluid model was proposed by Goldstein et al. in 2005 to characterize the bacterial swimming phenomenon in incompressible fluid. For the three-dimensional case, the global existence of bounded solutions to chemotaxis(–Navier)–Stokes model has always been an open problem. Therefore, researchers have been led to seek alternative avenues by turning their attention to the model with slow diffusion ($Delta n^m$ with $m gt 1$). Even with slow diffusion, the problem is not easy to solve. In particular, the closer $m$ is to $1$, the more difficult the study becomes. In this paper, we put forward a new method to prove the global existence and boundedness of weak solutions for any $m gt 1$. The new method allows us to obtain higher regularity when $m$ is close to 1. Subsequently, we also prove that the weak solution converges to the constant equilibrium point $(overline n_0, 0, 0)$ in the sense of $L^infty$-norm for $1lt m leq frac{5}{3}$. Based on this, we prove that the weak solution becomes smooth after a certain time and eventually becomes a classical solution.
{"title":"Global boundedness and eventual regularity of chemotaxis-fluid model driven by porous medium diffusion","authors":"Chunhua Jin","doi":"10.4310/cms.2024.v22.n5.a1","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a1","url":null,"abstract":"The chemotaxis-fluid model was proposed by Goldstein et al. in 2005 to characterize the bacterial swimming phenomenon in incompressible fluid. For the three-dimensional case, the global existence of bounded solutions to chemotaxis(–Navier)–Stokes model has always been an open problem. Therefore, researchers have been led to seek alternative avenues by turning their attention to the model with slow diffusion ($Delta n^m$ with $m gt 1$). Even with slow diffusion, the problem is not easy to solve. In particular, the closer $m$ is to $1$, the more difficult the study becomes. In this paper, we put forward a new method to prove the global existence and boundedness of weak solutions for any $m gt 1$. The new method allows us to obtain higher regularity when $m$ is close to 1. Subsequently, we also prove that the weak solution converges to the constant equilibrium point $(overline n_0, 0, 0)$ in the sense of $L^infty$-norm for $1lt m leq frac{5}{3}$. Based on this, we prove that the weak solution becomes smooth after a certain time and eventually becomes a classical solution.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"6 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719560","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.a10
Tiejun Li, Tiannan Xiao, Guoguo Yang
We revisited the central limit theorem (CLT) for stochastic gradient descent (SGD) type methods, including the vanilla SGD, momentum SGD and Nesterov accelerated SGD methods with constant or vanishing damping parameters. By taking advantage of Lyapunov function technique and $L^p$ bound estimates, we established the CLT under more general conditions on learning rates for broader classes of SGD methods as compared to previous results. The CLT for the time average was also investigated, and we found that it held in the linear case, while it was not generally true in nonlinear situation. Numerical tests were also carried out to verify our theoretical analysis.
{"title":"Revisiting the central limit theorems for the SGD-type methods","authors":"Tiejun Li, Tiannan Xiao, Guoguo Yang","doi":"10.4310/cms.2024.v22.n5.a10","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a10","url":null,"abstract":"We revisited the central limit theorem (CLT) for stochastic gradient descent (SGD) type methods, including the vanilla SGD, momentum SGD and Nesterov accelerated SGD methods with constant or vanishing damping parameters. By taking advantage of Lyapunov function technique and $L^p$ bound estimates, we established the CLT under more general conditions on learning rates for broader classes of SGD methods as compared to previous results. The CLT for the time average was also investigated, and we found that it held in the linear case, while it was not generally true in nonlinear situation. Numerical tests were also carried out to verify our theoretical analysis.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"322 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719563","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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