Journal of Mathematical Fluid Mechanics最新文献

{"title":"Energy Conservation for the Generalized Surface Quasi-geostrophic Equation","authors":"Yanqing Wang,&nbsp;Yulin Ye,&nbsp;Huan Yu","doi":"10.1007/s00021-023-00815-6","DOIUrl":"10.1007/s00021-023-00815-6","url":null,"abstract":"<div><p>In this paper, we consider the generalized surface quasi-geostrophic equation with the velocity <i>v</i> determined by <span>(v=mathcal {R}^{perp }Lambda ^{gamma -1}theta ,)</span> <span>(0&lt;gamma &lt; 2)</span>. It is shown that the <span>(L^p)</span>-norm of weak solutions is conserved provided <span>(theta in L^{p+1}left( 0,T; {B}^{frac{gamma }{3}}_{p+1, c(mathbb {N})}right) )</span> for <span>(0&lt;gamma &lt;frac{3}{2})</span> or <span>(theta in L^{p+1}left( 0,T; {{B}}^{alpha }_{p+1,infty }right) ~text {for any}~gamma -1&lt;alpha&lt;1 text { with} ~frac{3}{2}le gamma &lt;2)</span>. Therefore, the accurate relationships between the critical regularity for the energy conservation of the weak solutions and the regularity of velocity for the generalized surface quasi-geostrophic equation are presented.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42233467","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}

{"title":"Improved Well-Posedness for the Triple-Deck and Related Models via Concavity","authors":"David Gerard-Varet,&nbsp;Sameer Iyer,&nbsp;Yasunori Maekawa","doi":"10.1007/s00021-023-00809-4","DOIUrl":"10.1007/s00021-023-00809-4","url":null,"abstract":"<div><p>We establish linearized well-posedness of the Triple-Deck system in Gevrey-<span>(frac{3}{2})</span> regularity in the tangential variable, under concavity assumptions on the background flow. Due to the recent result (Dietert and Gerard-Varet in SIAM J Math Anal, 2021), one cannot expect a generic improvement of the result of Iyer and Vicol (Commun Pure Appl Math 74(8):1641–1684, 2021) to a weaker regularity class than real analyticity. Our approach exploits two ingredients, through an analysis of space-time modes on the Fourier–Laplace side: (i) stability estimates at the vorticity level, that involve the concavity assumption and a subtle iterative scheme adapted from Gerard-Varet et al. (Optimal Prandtl expansion around concave boundary layer, 2020. arXiv:2005.05022) (ii) smoothing properties of the Benjamin–Ono like equation satisfied by the Triple-Deck flow at infinity. Interestingly, our treatment of the vorticity equation also adapts to the so-called hydrostatic Navier–Stokes equations: we show for this system a similar Gevrey-<span>(frac{3}{2})</span> linear well-posedness result for concave data, improving at the linear level the recent work (Gérard-Varet et al. in Anal PDE 13(5):1417–1455, 2020).\u0000</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46511615","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}

{"title":"Localized Blow-Up Criterion for ( C^{ 1, alpha } ) Solutions to the 3D Incompressible Euler Equations","authors":"Dongho Chae,&nbsp;Jörg Wolf","doi":"10.1007/s00021-023-00813-8","DOIUrl":"10.1007/s00021-023-00813-8","url":null,"abstract":"<div><p>We prove a localized Beale–Kato–Majda type blow-up criterion for the 3D incompressible Euler equations in the Hölder space setting. More specifically, let <span>(vin C([0, T); C^{ 1, alpha } (Omega ))cap L^infty (0, T; L^2(Omega )))</span> be a solution to the Euler equations in a domain <span>(Omega subset {mathbb {R}}^3)</span>. If there exists a ball <span>(Bsubset Omega )</span> such that <span>( int limits nolimits _{0}^T Vert omega (s)Vert _{ BMO(B )} ds &lt; +infty , )</span> where <span>( omega = nabla times v)</span> stands for the vorticity, then <span>( vin C([0, T]; C^{ 1, alpha } (K)) )</span> for every compact subset <span>( K subset B )</span>. In the proof of this result, in order to handle the time evolution of the local Hölder norm of the vorticity we use the well-known Campanato space representation for the the Hölder space, and our argument relies on the Campanato space estimates for the solution to the corresponding transport equation.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4752455","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}

{"title":"Global Boundedness to a 3D Chemotaxis–Stokes System with Porous Medium Cell Diffusion and General Sensitivity Under Dirichlet Signal Boundary Condition","authors":"Yu Tian,&nbsp;Zhaoyin Xiang","doi":"10.1007/s00021-023-00812-9","DOIUrl":"10.1007/s00021-023-00812-9","url":null,"abstract":"<div><p>In this paper, we construct a globally bounded weak solution for the initial-boundary value problem of a three-dimensional chemotaxis–Stokes system with porous medium cell diffusion <span>(Delta n^m)</span> and inhomogeneous Dirichlet signal boundary for each <span>(m&gt;frac{13}{12})</span>. Compared with the quite well-developed solvability for the no-flux signal boundary value with <span>(m&gt;frac{7}{6})</span> (Winkler in Calc Var 54:3789–3828, 2015), to our best knowledge, this seems to be the first result on chemotaxis–fluid system with general matrix-valued sensitivity for such a Dirichlet signal boundary condition, under which even for the scalar sensitivity, we also extend the recent range <span>(m&gt;frac{7}{6})</span> (Wu and Xiang in J Differ Equ 315:122–158, 2022). Our proof will be based on a new observation on the boundary estimate and on a three-step induction argument. The same technique can be applied to the two-dimensional setting to confirm a similar conclusion for any <span>(m&gt;1)</span>.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-023-00812-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4391322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spatially Quasi-Periodic Solutions of the Euler Equation","authors":"Xu Sun,&nbsp;Peter Topalov","doi":"10.1007/s00021-023-00804-9","DOIUrl":"10.1007/s00021-023-00804-9","url":null,"abstract":"<div><p>We develop a framework for studying quasi-periodic maps and diffeomorphisms on <span>({mathbb {R}}^n)</span>. As an application, we prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on <span>({mathbb {R}}^n)</span>. In particular, the equation preserves the spatial quasi-periodicity of the initial data. Several results on the analytic dependence of solutions on the time and the initial data are proved.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-023-00804-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5151221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two-Phase Flows with Bulk–Surface Interaction: Thermodynamically Consistent Navier–Stokes–Cahn–Hilliard Models with Dynamic Boundary Conditions","authors":"Andrea Giorgini,&nbsp;Patrik Knopf","doi":"10.1007/s00021-023-00811-w","DOIUrl":"10.1007/s00021-023-00811-w","url":null,"abstract":"<div><p>We derive a novel thermodynamically consistent Navier–Stokes–Cahn–Hilliard system with dynamic boundary conditions. This model describes the motion of viscous incompressible binary fluids with different densities. In contrast to previous models in the literature, our new model allows for surface diffusion, a variable contact angle between the diffuse interface and the boundary, and mass transfer between bulk and surface. In particular, this transfer of material is subject to a mass conservation law including both a bulk and a surface contribution. The derivation is carried out by means of local energy dissipation laws and the Lagrange multiplier approach. Next, in the case of fluids with matched densities, we show the existence of global weak solutions in two and three dimensions as well as the uniqueness of weak solutions in two dimensions.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-023-00811-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5482625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Design of Global-in-Time Newton-Multigrid-Pressure Schur Complement Solvers for Incompressible Flow Problems","authors":"Christoph Lohmann,&nbsp;Stefan Turek","doi":"10.1007/s00021-023-00807-6","DOIUrl":"10.1007/s00021-023-00807-6","url":null,"abstract":"<div><p>In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton’s method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-023-00807-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5010235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convergence of the Fully Discrete Incremental Projection Scheme for Incompressible Flows","authors":"T. Gallouët,&nbsp;R. Herbin,&nbsp;J. C. Latché,&nbsp;D. Maltese","doi":"10.1007/s00021-023-00810-x","DOIUrl":"10.1007/s00021-023-00810-x","url":null,"abstract":"<div><p>The present paper addresses the convergence of a first-order in time incremental projection scheme for the time-dependent incompressible Navier–Stokes equations to a weak solution. We prove the convergence of the approximate solutions obtained by a semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non uniform rectangular meshes. Some first a priori estimates on the approximate solutions yield their existence. Compactness arguments, relying on these estimates, together with some estimates on the translates of the discrete time derivatives, are then developed to obtain convergence (up to the extraction of a subsequence), when the time step tends to zero in the semi-discrete scheme and when the space and time steps tend to zero in the fully discrete scheme; the approximate solutions are thus shown to converge to a limit function which is then shown to be a weak solution to the continuous problem by passing to the limit in these schemes.\u0000</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5010246","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}

{"title":"Dyadic Models for Fluid Equations: A Survey","authors":"Alexey Cheskidov,&nbsp;Mimi Dai,&nbsp;Susan Friedlander","doi":"10.1007/s00021-023-00799-3","DOIUrl":"10.1007/s00021-023-00799-3","url":null,"abstract":"<div><p>Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years, including the groundbreaking work of Ladyzhenskaya on the Navier–Stokes equations. However crucial questions such as the existence, uniqueness and regularity of the three dimensional Navier–Stokes equations remain open. Partly because of this mathematical challenge and partly motivated by the phenomena of turbulence, insights into the full PDEs have been sought via the study of simpler approximating systems that retain some of the original nonlinear features. One such simpler system is an infinite dimensional coupled set of nonlinear ordinary differential equations referred to a dyadic model. In this survey we provide a brief overview of dyadic models and describe recent results. In particular, we discuss results for certain dyadic models in the context of existence, uniqueness and regularity of solutions.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4759902","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}

{"title":"Euler–Lagrangian Approach to Stochastic Euler Equations in Sobolev Spaces","authors":"Christian Olivera,&nbsp;Juan D. Londoño","doi":"10.1007/s00021-023-00808-5","DOIUrl":"10.1007/s00021-023-00808-5","url":null,"abstract":"<div><p>The purpose of this paper is to establish the equivalence between Lagrangian and classical formulations for the stochastic incompressible Euler equations, the proof is based on Ito–Wentzell–Kunita formula and stochastic analysis techniques. Moreover, we prove a local existence result for the Lagrangian formulation in suitable Sobolev Spaces.\u0000</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4579510","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}