共查询到20条相似文献,搜索用时 31 毫秒
1.
We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations
with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in
terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these
criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations.
Received: 11 April 2000; in final firm: 26 November 2000 / Published online: 28 February 2002 相似文献
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J. Fontbona 《Probability Theory and Related Fields》2006,136(1):102-156
We develop a probabilistic interpretation of local mild solutions of the three dimensional Navier-Stokes equation in the Lp spaces, when the initial vorticity field is integrable. This is done by associating a generalized nonlinear diffusion of
the McKean-Vlasov type with the solution of the corresponding vortex equation. We then construct trajectorial (chaotic) stochastic
particle approximations of this nonlinear process. These results provide the first complete proof of convergence of a stochastic
vortex method for the Navier-Stokes equation in three dimensions, and rectify the algorithm conjectured by Esposito and Pulvirenti
in 1989. Our techniques rely on a fine regularity study of the vortex equation in the supercritical Lp spaces, and on an extension of the classic McKean-Vlasov model, which incorporates the derivative of the stochastic flow
of the nonlinear process to explain the vortex stretching phenomenon proper to dimension three.
Supported by Fondecyt Project 1040689 and Nucleus Millennium Information and Randomness ICM P01-005. 相似文献
4.
Dongho Chae 《Journal of Differential Equations》2006,227(2):640-651
We obtain new continuation principle of the local classical solutions of the 3D Euler equations, where the regularity condition of the direction field of the vorticiy and the integrability condition of the magnitude of the vorticity are incorporated simultaneously. The regularity of the vorticity direction field is most appropriately measured by the Triebel-Lizorkin type of norm. Similar result is also obtained for the inviscid 2D quasi-geostrophic equation. 相似文献
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The finite difference approximation of a nonstationary pseudo-advected
vorticity equation is proved to yield generalized solutions to the
two-dimensional stationary Euler equations with nonvanishing
vorticity. This result is obtained by the simultaneous limiting of
lattice scale and time.Received: 15 May 2002 相似文献
6.
The MAC discretization scheme for the incompressible Navier-Stokes equations is interpreted as a covolume approximation to the equations. Using some results from earlier papers dealing with covolume error estimates for div-curl equation systems, and under certain conditions on the data and the solutions of the Navier-Stokes equations, we obtain first-order error estimates for both the vorticity and the pressure.
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Jingyu Li 《Journal of Mathematical Analysis and Applications》2008,348(1):244-254
We study the regularity of the solutions for the wave equations with potentials that are time-dependent and singular. The size of the potentials is exactly a function of the spatial dimension rather than being small enough in the known results. Based on a weighted L2 estimate for the solutions, we prove the local regularity and the Strichartz estimates. The solvability of the equation is also studied. 相似文献
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林清泉 《中国科学A辑(英文版)》2002,45(12):1518-1522
The notion of weak solution for stochastic differential equation with terminal conditions is introduced. By Girsanov transformation,
the equivalence of existence of weak solutions for two-type equations is established. Several sufficient conditions for the
existence of the weak solutions for stochastic differential equation with terminal conditions are obtained, and the solution
existence condition for this type of equations is relaxed. Finally, an example is given to show that the result is an essential
extension of the one under Lipschitz condition ong with respect to (Y,Z). 相似文献
10.
This paper provides a new approach to study the solutions of a class of generalized Jacobi equations associated with the linearization of certain singular flows on Riemannian manifolds with dimension n + 1.A new class of generalized differential operators is defined.We investigate the kernel of the corresponding maximal operators by applying operator theory.It is shown that all nontrivial solutions to the generalized Jacobi equation are hyperbolic,in which there are n dimension solutions with exponential... 相似文献
11.
We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is C2,α on the complement of a closed set of Hausdorff dimension at most ? less than the dimension. The equation is assumed to be C1, and the constant ? > 0 depends only on the dimension and the ellipticity constants. The argument combines the W2,? estimates of Lin with a result of Savin on the C2,α regularity of viscosity solutions that are close to quadratic polynomials. © 2012 Wiley Periodicals, Inc. 相似文献
12.
Atanas Stefanov 《Transactions of the American Mathematical Society》2007,359(8):3589-3607
We consider the Schrödinger equation with derivative perturbation terms in one space dimension. For the linear equation, we show that the standard Strichartz estimates hold under specific smallness requirements on the potential. As an application, we establish existence of local solutions for quadratic derivative Schrödinger equations in one space dimension with small and rough Cauchy data.
13.
We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled -norm of the velocity with 1?3/p+2/q?2, 1?q?∞ is sufficiently small near z and if the scaled -norm of the magnetic field with 1?3/l+2/m?2, 1?m?∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations. 相似文献
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Steven Schochet 《纯数学与应用数学通讯》1996,49(9):911-965
Almost-sure convergence of a subsequence of the vorticity to a weak solution is proven for the point-vortex method for 2-D, inviscid, incompressible fluid flow. Here “almost-sure” is with respect to sequences of random components included in the initial position and strength of each vortex. The initial vorticity is assumed to be periodic and, depending on the initialization scheme, to lie in L log L or Lp with p > 2. The randomization of the initial data is not needed when the initial vorticity is nonnegative; such initial data also need not be periodic, and is only required to be a bounded measure lying in H−1. All these results are also valid for the “vortex-blob” method with the smoothing parameter vanishing at an arbitrary rate. The sense in which solutions of point-vortex dynamics are weak solutions of the Euler equations is also discussed. 相似文献
15.
Volker Elling 《Bulletin of the Brazilian Mathematical Society》2016,47(1):323-334
We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The regions of positive and negative vorticity form algebraic spirals. 相似文献
16.
We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L1(R2) for positive times is entirely determined by the trace of the vorticity at t=0, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa & Osada, and by Kato, this uniqueness property implies that the Cauchy problem for the vorticity equation in R2 is globally well-posed in the space of finite measures. In particular, this provides an example of a situation where the Navier-Stokes equation is well-posed for arbitrary data in a function space that is large enough to contain the initial data of some self-similar solutions. 相似文献
17.
Nils Ackermann 《Journal of Differential Equations》2009,246(4):1470-1499
The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to explicitly verify the condition, and we carry out the calculation in dimension one for a specific class of potentials. 相似文献
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Fattehallah Ghadi Vitoriano Ruas Mohamed Wakrim 《Journal of Computational and Applied Mathematics》2008
In this paper we present a new method to solve the 2D generalized Stokes problem in terms of the stream function and the vorticity. Such problem results, for instance, from the discretization of the evolutionary Stokes system. The difficulty arising from the lack of the boundary conditions for the vorticity is overcome by means of a suitable technique for uncoupling both variables. In order to apply the above technique to the Navier–Stokes equations we linearize the advective term in the vorticity transport equation as described in the development of the paper. We illustrate the good performance of our approach by means of numerical results, obtained for benchmark driven cavity problem solved with classical piecewise linear finite element. 相似文献
19.
When the right-hand side of an ordinary differential equation (ODE in short) is not Lipschitz, neither existence nor uniqueness of solutions remain valid. Nevertheless, adding to the differential equation a noise with nondegenerate intensity, we obtain a stochastic differential equation which has pathwise existence and uniqueness property. The goal of this short paper is to compare the limit of solutions to stochastic differential equation obtained by adding a noise of intensity ε to the generalized Filippov notion of solutions to the ODE. It is worth pointing out that our result does not depend on the dimension of the space while several related works in the literature are concerned with the one dimensional case. 相似文献
20.
The main purpose of this paper is to prove the well-posedness of the two-dimensional Boussinesq equations when the initial vorticity ω 0 ∈L1 (R 2 ) (or the finite Radon measure space). Using the stream function form of the equations and the Schauder fixed-point theorem to get the new proof of these results, we get that when the initial vorticity is smooth, there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations. 相似文献