Pub Date : 2024-07-04DOI: 10.1007/s00021-024-00879-y
Piotr Kacprzyk, Wojciech M. Zaja̧czkowski
We consider the motion of an incompressible magnetohydrodynamics with resistivity in a domain bounded by a free surface which is coupled through the free surface with an electromagnetic field generated by a magnetic field prescribed on an exterior fixed boundary. On the free surface, transmission conditions for the electromagnetic field are imposed. As transmission condition we assume jumps of tangent components of magnetic and electric fields on the free surface. We prove local existence of solutions such that velocity and magnetic fields belong to (H^{2+alpha ,1+alpha /2}), (alpha >5/8).
{"title":"Existence of Local Solutions to a Free Boundary Problem for Incompressible Viscous Magnetohydrodynamics","authors":"Piotr Kacprzyk, Wojciech M. Zaja̧czkowski","doi":"10.1007/s00021-024-00879-y","DOIUrl":"https://doi.org/10.1007/s00021-024-00879-y","url":null,"abstract":"<p>We consider the motion of an incompressible magnetohydrodynamics with resistivity in a domain bounded by a free surface which is coupled through the free surface with an electromagnetic field generated by a magnetic field prescribed on an exterior fixed boundary. On the free surface, transmission conditions for the electromagnetic field are imposed. As transmission condition we assume jumps of tangent components of magnetic and electric fields on the free surface. We prove local existence of solutions such that velocity and magnetic fields belong to <span>(H^{2+alpha ,1+alpha /2})</span>, <span>(alpha >5/8)</span>.</p>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-10DOI: 10.1007/s00021-024-00882-3
Dania Ati, Rahma Agroum, Jonas Koko
{"title":"A Priori Error Analysis and Finite Element Approximations for a Coupled Model Under Nonlinear Slip Boundary Conditions","authors":"Dania Ati, Rahma Agroum, Jonas Koko","doi":"10.1007/s00021-024-00882-3","DOIUrl":"https://doi.org/10.1007/s00021-024-00882-3","url":null,"abstract":"","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141361955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-07DOI: 10.1007/s00021-024-00883-2
Vesa Julin, Domenico Angelo La Manna
{"title":"A Priori Estimates for the Motion of Charged Liquid Drop: A Dynamic Approach via Free Boundary Euler Equations","authors":"Vesa Julin, Domenico Angelo La Manna","doi":"10.1007/s00021-024-00883-2","DOIUrl":"https://doi.org/10.1007/s00021-024-00883-2","url":null,"abstract":"","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141375105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-05DOI: 10.1007/s00021-024-00881-4
R. Benabidallah, F. Ebobisse
{"title":"On the Steady Flows of Viscous Compressible Magnetohydrodynamic Equations in an Infinite Horizontal Layer","authors":"R. Benabidallah, F. Ebobisse","doi":"10.1007/s00021-024-00881-4","DOIUrl":"https://doi.org/10.1007/s00021-024-00881-4","url":null,"abstract":"","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141381761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-23DOI: 10.1007/s00021-024-00880-5
Zhaonan Luo, Wei Luo, Zhaoyang Yin
{"title":"The Optimal $${{varvec{L}}^2}$$ Decay Rate of the Velocity for the General FENE Dumbbell Model","authors":"Zhaonan Luo, Wei Luo, Zhaoyang Yin","doi":"10.1007/s00021-024-00880-5","DOIUrl":"https://doi.org/10.1007/s00021-024-00880-5","url":null,"abstract":"","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141105303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-19DOI: 10.1007/s00021-024-00871-6
Qin Duan, Xiangdi Huang
In this paper, we consider the 3-D compressible isentropic Navier–Stokes equations with constant shear viscosity (mu ) and the bulk one (lambda =brho ^beta ), here b is a positive constant, (beta ge 0). This model was first introduced and well studied by Vaigant and Kazhikhov (Sib Math J 36(6):1283–1316, 1995) in 2D domain. In this paper, under the assumption that (gamma >1), the local existence of weak solutions with higher regularity for the 3D periodic domain is established in the presence of vacuum without any smallness on the initial data. This generalize the previous paper (Desjardins in Commun Partial Differ Equ 22(5):977–1008, 1997; Huang and Yan in J Math Phys 62(11):111504, 2021) to variable viscosity coefficients. Also this is the first result concerning the local weak solution with high regularity for the Kazhikhov model in 3D case.
在本文中,我们考虑了具有恒定剪切粘度的三维可压缩等熵纳维-斯托克斯方程(3-D compressible isentropic Navier-Stokes equations with constant shear viscosity (mu ) and the bulk one (lambda =brho ^beta ),这里b是一个正常数,(beta ge 0).该模型由 Vaigant 和 Kazhikhov(Sib Math J 36(6):1283-1316, 1995)在二维域中首次提出并进行了深入研究。在本文中,在 (gamma >1)的假设下,建立了三维周期域在真空存在下具有更高正则性的弱解的局部存在性,而对初始数据没有任何小的影响。这将之前的论文(Desjardins 在 Commun Partial Differ Equ 22(5):977-1008, 1997; Huang and Yan 在 J Math Phys 62(11):111504, 2021)推广到了可变粘性系数。这也是第一个关于卡齐霍夫模型在三维情况下具有高正则性的局部弱解的结果。
{"title":"Local Weak Solution of the Isentropic Compressible Navier–Stokes Equations with Variable Viscosity","authors":"Qin Duan, Xiangdi Huang","doi":"10.1007/s00021-024-00871-6","DOIUrl":"https://doi.org/10.1007/s00021-024-00871-6","url":null,"abstract":"<p>In this paper, we consider the 3-D compressible isentropic Navier–Stokes equations with constant shear viscosity <span>(mu )</span> and the bulk one <span>(lambda =brho ^beta )</span>, here <i>b</i> is a positive constant, <span>(beta ge 0)</span>. This model was first introduced and well studied by Vaigant and Kazhikhov (Sib Math J 36(6):1283–1316, 1995) in 2D domain. In this paper, under the assumption that <span>(gamma >1)</span>, the local existence of weak solutions with higher regularity for the 3D periodic domain is established in the presence of vacuum without any smallness on the initial data. This generalize the previous paper (Desjardins in Commun Partial Differ Equ 22(5):977–1008, 1997; Huang and Yan in J Math Phys 62(11):111504, 2021) to variable viscosity coefficients. Also this is the first result concerning the local weak solution with high regularity for the Kazhikhov model in 3D case.\u0000</p>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141064015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-17DOI: 10.1007/s00021-024-00878-z
Ankit Kumar, Kush Kinra, Manil T. Mohan
In this article, we establish the Wong–Zakai approximation result for a class of stochastic partial differential equations (SPDEs) with fully local monotone coefficients perturbed by a multiplicative Wiener noise. This class of SPDEs encompasses various fluid dynamic models and also includes quasi-linear SPDEs, the convection–diffusion equation, the Cahn–Hilliard equation, and the two-dimensional liquid crystal model. It has been established that the class of SPDEs in question is well-posed, however, the existence of a unique solution to the associated approximating system cannot be inferred from the solvability of the original system. We employ a Faedo–Galerkin approximation method, compactness arguments, and Prokhorov’s and Skorokhod’s representation theorems to ensure the existence of a probabilistically weak solution for the approximating system. Furthermore, we also demonstrate that the solution is pathwise unique. Moreover, the classical Yamada–Watanabe theorem allows us to conclude the existence of a probabilistically strong solution (analytically weak solution) for the approximating system. Subsequently, we establish the Wong–Zakai approximation result for a class of SPDEs with fully local monotone coefficients. We utilize the Wong–Zakai approximation to establish the topological support of the distribution of solutions to the SPDEs with fully local monotone coefficients. Finally, we explore the physically relevant stochastic fluid dynamics models that are covered by this work’s functional framework.
{"title":"Wong–Zakai Approximation for a Class of SPDEs with Fully Local Monotone Coefficients and Its Application","authors":"Ankit Kumar, Kush Kinra, Manil T. Mohan","doi":"10.1007/s00021-024-00878-z","DOIUrl":"https://doi.org/10.1007/s00021-024-00878-z","url":null,"abstract":"<p>In this article, we establish the <i>Wong–Zakai approximation</i> result for a class of stochastic partial differential equations (SPDEs) with fully local monotone coefficients perturbed by a multiplicative Wiener noise. This class of SPDEs encompasses various fluid dynamic models and also includes quasi-linear SPDEs, the convection–diffusion equation, the Cahn–Hilliard equation, and the two-dimensional liquid crystal model. It has been established that the class of SPDEs in question is well-posed, however, the existence of a unique solution to the associated approximating system cannot be inferred from the solvability of the original system. We employ a Faedo–Galerkin approximation method, compactness arguments, and Prokhorov’s and Skorokhod’s representation theorems to ensure the existence of a <i>probabilistically weak solution</i> for the approximating system. Furthermore, we also demonstrate that the solution is pathwise unique. Moreover, the classical Yamada–Watanabe theorem allows us to conclude the existence of a <i>probabilistically strong solution</i> (analytically weak solution) for the approximating system. Subsequently, we establish the Wong–Zakai approximation result for a class of SPDEs with fully local monotone coefficients. We utilize the Wong–Zakai approximation to establish the topological support of the distribution of solutions to the SPDEs with fully local monotone coefficients. Finally, we explore the physically relevant stochastic fluid dynamics models that are covered by this work’s functional framework.</p>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141063272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-14DOI: 10.1007/s00021-024-00877-0
Juhi Jang, Pranava Chaitanya Jayanti, Igor Kukavica
We examine a micro-scale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282-287, 1959) which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in ({mathbb {T}}^3) for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.
{"title":"On the Mass Transfer in the 3D Pitaevskii Model","authors":"Juhi Jang, Pranava Chaitanya Jayanti, Igor Kukavica","doi":"10.1007/s00021-024-00877-0","DOIUrl":"https://doi.org/10.1007/s00021-024-00877-0","url":null,"abstract":"<p>We examine a micro-scale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282-287, 1959) which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in <span>({mathbb {T}}^3)</span> for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.</p>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141063271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-04DOI: 10.1007/s00021-024-00868-1
Yueqiang Shang, Jiali Zhu, Bo Zheng
We present and study a parallel grad-div stabilized finite element discretization algorithm based on entire-overlapping domain decomposition for the numerical simulation of Navier–Stokes equations. The algorithm is easy to implement on top of existing sequential software, in which each subproblem used to calculate a local solution in its designated subregion is actually a global problem with vast of degrees of freedom coming from its own subregion, and hence, can be solved independently with other subproblems. We derive error bounds of the approximate solution by employing the technical tool of local a priori estimate, and investigate the effect of grad-div stabilization term on the approximation solutions. Numerical comparisons, with both inf-sup stable and unstable mixed finite elements pairs for the velocity and pressure, show that our present algorithm has an amazing superiority to its counterpart without stabilization in the sense that accuracy of the approximate velocities could be improved by two orders of magnitude when the viscosity (nu ) is small. While compared with the usual standard serial grad-div stabilized finite element method, our algorithm saves lots of CPU time in computing a solution with comparable accuracy.
我们提出并研究了一种基于全重叠域分解的并行梯度-离散稳定有限元离散化算法,用于纳维-斯托克斯方程的数值模拟。该算法易于在现有顺序软件基础上实现,其中用于计算指定子区域局部解的每个子问题实际上都是一个全局问题,其自由度绝大部分来自自己的子区域,因此可以与其他子问题一起独立求解。我们利用局部先验估计的技术手段推导出近似解的误差边界,并研究了梯度稳定项对近似解的影响。通过对速度和压力的 inf-sup 稳定和不稳定混合有限元对进行数值比较,我们发现本算法比没有稳定化的算法具有惊人的优越性,即当粘度(nu )较小时,近似速度的精度可以提高两个数量级。与通常的标准串行梯度二维稳定有限元法相比,我们的算法在计算精度相当的解时节省了大量的 CPU 时间。
{"title":"A Parallel Finite Element Discretization Algorithm Based on Grad-Div Stabilization for the Navier–Stokes Equations","authors":"Yueqiang Shang, Jiali Zhu, Bo Zheng","doi":"10.1007/s00021-024-00868-1","DOIUrl":"https://doi.org/10.1007/s00021-024-00868-1","url":null,"abstract":"<p>We present and study a parallel grad-div stabilized finite element discretization algorithm based on entire-overlapping domain decomposition for the numerical simulation of Navier–Stokes equations. The algorithm is easy to implement on top of existing sequential software, in which each subproblem used to calculate a local solution in its designated subregion is actually a global problem with vast of degrees of freedom coming from its own subregion, and hence, can be solved independently with other subproblems. We derive error bounds of the approximate solution by employing the technical tool of local a priori estimate, and investigate the effect of grad-div stabilization term on the approximation solutions. Numerical comparisons, with both inf-sup stable and unstable mixed finite elements pairs for the velocity and pressure, show that our present algorithm has an amazing superiority to its counterpart without stabilization in the sense that accuracy of the approximate velocities could be improved by two orders of magnitude when the viscosity <span>(nu )</span> is small. While compared with the usual standard serial grad-div stabilized finite element method, our algorithm saves lots of CPU time in computing a solution with comparable accuracy.</p>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140882088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-02DOI: 10.1007/s00021-024-00875-2
Yazhou Chen, Bin Huang, Xiaoding Shi
We consider the initial-boundary-value problem of the isentropic compressible Navier–Stokes–Poisson equations subject to large and non-flat doping profile in 3D bounded domain with slip boundary condition and vacuum. The global well-posedness of classical solution is established with small initial energy but possibly large oscillations and vacuum. The steady state (except velocity) and the doping profile are allowed to be of large variation.
{"title":"Global Well-Posedness of Classical Solutions to the Compressible Navier–Stokes–Poisson Equations with Slip Boundary Conditions in 3D Bounded Domains","authors":"Yazhou Chen, Bin Huang, Xiaoding Shi","doi":"10.1007/s00021-024-00875-2","DOIUrl":"https://doi.org/10.1007/s00021-024-00875-2","url":null,"abstract":"<p>We consider the initial-boundary-value problem of the isentropic compressible Navier–Stokes–Poisson equations subject to large and non-flat doping profile in 3D bounded domain with slip boundary condition and vacuum. The global well-posedness of classical solution is established with small initial energy but possibly large oscillations and vacuum. The steady state (except velocity) and the doping profile are allowed to be of large variation.</p>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140836157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}