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1.
We consider the incompressible Navier–Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time Hölder continuous solutions. Our proof uses a stochastic representation formula to obtain a decay estimate for heat flows in Hölder spaces, and a stochastic Lagrangian formulation of the Navier–Stokes equations. 相似文献
2.
We construct solutions for 2- and 3-D stochastic nonhomogeneous incompressible Navier-Stokes equations with general multiplicative noise. These equations model the velocity of a mixture of incompressible fluids of varying density, influenced by random external forces that involve feedback; that is, multiplicative noise. Weak solutions for the corresponding deterministic equations were first found by Kazhikhov [A.V. Kazhikhov, Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid, Soviet Phys. Dokl. 19 (6) (1974) 331-332; English translation of the paper in: Dokl. Akad. Nauk SSSR 216 (6) (1974) 1240-1243]. A stochastic version with additive noise was solved by Yashima [H.F. Yashima, Equations de Navier-Stokes stochastiques non homogènes et applications, Thesis, Scuola Normale Superiore, Pisa, 1992].The methods here extend the Loeb space techniques used to obtain the first general solutions of the stochastic Navier-Stokes equations with multiplicative noise in the homogeneous case [M. Capiński, N.J. Cutland, Stochastic Navier-Stokes equations, Applicandae Math. 25 (1991) 59-85]. The solutions display more regularity in the 2D case. The methods also give a simpler proof of the basic existence result of Kazhikhov. 相似文献
3.
Global Strong Solution to the 3D Incompressible Navier-Stokes Equations with General Initial Data
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Tingting Zheng & Peixin Zhang 《数学研究》2015,48(3):250-255
We study the existence of global strong solution to an initial-boundary value
(or initial value) problem for the 3D nonhomogeneous incompressible Navier-Stokes
equations. In this study, the initial density is suitably small (or the viscosity coefficient
suitably large) and the initial vacuum is allowed. Results show that the unique solution
of the Navier-Stokes equations can be found. 相似文献
4.
Thomas Slawig 《Journal of Differential Equations》2005,219(1):116-143
We consider control problems with a general cost functional where the state equations are the stationary, incompressible Navier-Stokes equations with shear-dependent viscosity. The equations are quasi-linear. The control function is given as the inhomogeneity of the momentum equation. In this paper, we study a general class of viscosity functions which correspond to shear-thinning or shear-thickening behavior. The basic results concerning existence, uniqueness, boundedness, and regularity of the solutions of the state equations are reviewed. The main topic of the paper is the proof of Gâteaux differentiability, which extends known results. It is shown that the derivative is the unique solution to a linearized equation. Moreover, necessary first-order optimality conditions are stated, and the existence of a solution of a class of control problems is shown. 相似文献
5.
Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero. 相似文献
6.
GLOBAL SOLUTIONS FOR SOME OLDROYD MODELS OF NON-NEWTONIAN FLOWS 总被引:9,自引:0,他引:9
P. L. LIONS 《数学年刊B辑(英文版)》2000,21(2):131-146
The authors consider here some Oldroyd models of non-Newtonian flows consisting of a strong coupling between incompressible
Navier-Stokes equations and some transport equations. It is proved that there exist global weak solutiosn for general initial
conditions. The existence proof relies upon showing the propagation in time of the compactness of solutions. 相似文献
7.
Maria Carmela Lombardo 《Rendiconti del Circolo Matematico di Palermo》2001,50(2):299-311
We consider the time dependent incompressible Navier-Stokes equations on an half plane. For analytic initial data, existence
and uniqueness of the solution are proved using the Abstract Cauchy-Kovalevskaya Theorem in Banach spaces. The time interval
of existence is proved to be independent of the viscosity. 相似文献
8.
Martina Hofmanová 《随机分析与应用》2013,31(1):100-121
A new proof of existence of weak solutions to stochastic differential equations with continuous coefficients based on ideas from infinite-dimensional stochastic analysis is presented. The proof is fairly elementary, in particular, neither theorems on representation of martingales by stochastic integrals nor results on almost sure representation for tight sequences of random variables are needed. 相似文献
9.
In this paper we study a widely used zero equation model of turbulence. The governing equations are derived by applying to the incompressible Navier-Stokes equations the Reynolds time averaging procedure. We achieve closure by employing the eddy viscosity concept. Using the Implicit Function Theorem we obtain an existence and uniquencess result. We also discuss the existence of nonsingular solutions. Finally, we present an algorithm for solving the modeled equations. 相似文献
10.
We consider three-dimensional incompressible Navier-Stokes equations(NS) with different viscous coefficients in the vertical and horizontal variables. In particular, when one of these viscous coefficients is large enough compared with the initial data, we prove the global well-posedness of this system. In fact, we obtain the existence of a global strong solution to(NS) when the initial data verifies an anisotropic smallness condition which takes into account the different roles of the horizontal and vertical viscosity. 相似文献
11.
Rabi N. Bhattacharya Larry Chen Scott Dobson Ronald B. Guenther Chris Orum Mina Ossiander Enrique Thomann Edward C. Waymire 《Transactions of the American Mathematical Society》2003,355(12):5003-5040
A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.
12.
Lorenzo Toniazzi 《Journal of Mathematical Analysis and Applications》2019,469(2):594-622
Space–time fractional evolution equations are a powerful tool to model diffusion displaying space–time heterogeneity. We prove existence, uniqueness and stochastic representation of classical solutions for an extension of Caputo evolution equations featuring time-nonlocal initial conditions. We discuss the interpretation of the new stochastic representation. As part of the proof a new result about inhomogeneous Caputo evolution equations is proven. 相似文献
13.
A new seven-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained.And its stationary solutions,the existence of attractor and the global stability of the equations are firmly proved.At the same time,several issues such as some basic dynamical behaviors and routs to chaos are shown numerically by changing Reynolds number.The system exhibits a stochastic behavior approached through an involved sequence of bifurcations. 相似文献
14.
In this paper, the authors consider the zero-viscosity limit of the three dimensional incompressible steady Navier-Stokes equations in a half space R+×R2. The result shows that the solution of three dimensional incompressible steady Navier-Stokes equations converges to the solution of three dimensional incompressible steady Euler equations in Sobolev space as the viscosity coefficient going to zero. The method is based on a new weighted energy estimates and Nash-Moser itera... 相似文献
15.
B. Schmalfuβ 《随机分析与应用》2013,31(6):1075-1101
In the theory of stochastic differential equations we can distinguish between two kinds of attractors. The first one is the attractor (measure attractor) with respect to the Markov semigroup generated by a stochastic differential equation. The second meaning of attractors (random attractors) is to be understood with respect to each trajectory of the random equation. The aim of this paper is to bring together the two meanings of attractors. In particular, we show the existence of measure attractors if random attractors exist. We can also show the uniqueness of the stationary distributions of the stochastic Navier-Stokes equation if the viscosity is large 相似文献
16.
Xicheng Zhang 《Journal of Functional Analysis》2010,258(4):1361-1425
In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier-Stokes equations are also investigated. 相似文献
17.
Mina Ossiander 《Probability Theory and Related Fields》2005,133(2):267-298
A new probabilistic representation is presented for solutions of the incompressible Navier-Stokes equations in R3 with given forcing and initial velocity. This representation expresses solutions as scaled conditional expectations of functionals of a Markov process indexed by the nodes of a binary tree. It gives existence and uniqueness of weak solutions for all time under relatively simple conditions on the forcing and initial data. These conditions involve comparison of the forcing and initial data with majorizing kernels. 相似文献
18.
The Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions is considered.The existence of the global attractor is proved and the long time behavior of the trajectories,namely,the convergence to steady states,is studied. 相似文献
19.
Fanghua Lin 《纯数学与应用数学通讯》1998,51(3):241-257
Here we give a self-contained new proof of the partial regularity theorems for solutions of incompressible Navier-Stokes equations in three spatial dimensions. These results were originally due to Scheffer and Caffarelli, Kohn, and Nirenberg. Our proof is much more direct and simpler. © 1998 John Wiley & Sons, Inc. 相似文献
20.
It is showed that, as the Mach number goes to zero, the weak solution of the compressible Navier-Stokes equations in the whole space with general initial data converges to the strong solution of the incompressible Navier-Stokes equations as long as the later exists. The proof of the result relies on the new modulated energy functional and the Strichartz's estimate of linear wave equation. 相似文献