共查询到20条相似文献,搜索用时 31 毫秒
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We prove the inviscid limit of the incompressible Navier–Stokes equations in the same topology of Besov spaces as the initial data. The proof is based on proving the continuous dependence of the Navier–Stokes equations uniformly with respect to the viscosity. To show the latter, we rely on some Bona–Smith type argument in the setting. Our obtained result implies a new result that the Cauchy problem of the Euler equations is locally well-posed in the borderline Besov space , , in the sense of Hadmard, which is an open problem left in recent works by Bourgain and Li in [3], [4] and by Misio?ek and Yoneda in [12], [13], [14]. 相似文献
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Let be a Noetherian local ring and M a finitely generated R-module. The invariants and of M were introduced in [3] and [17] in order to measure the non-Cohen–Macaulayness and the non-sequential-Cohen–Macaulayness of M, respectively. Let be the filtration of M such that is the largest submodule of M of dimension less than for all and . In this paper we prove that if , then there exists a constant c such that for all good parameter ideals of M with respect to this filtration. Here is the reducibility index of on M. This is an extension of the main results of [19], [20], [24]. 相似文献
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The Adimurthi–Druet [1] inequality is an improvement of the standard Moser–Trudinger inequality by adding a -type perturbation, quantified by , where is the first Dirichlet eigenvalue of Δ on a smooth bounded domain. It is known [3], [10], [14], [19] that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter α approaches . Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler–Lagrange equation, which take into account the fact that the problem becomes singular as . 相似文献
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Guoli Zhou 《Journal of Differential Equations》2019,266(11):7569-7637
We develop a new and general method to prove the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under non-periodic boundary conditions and driven by infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the time-uniform a priori estimates in some Sobolev space whose regularity is higher than the solution space. But this method can not be applied to the 3D stochastic PEs with the non-periodic boundary conditions. Therefore, the existence of universal attractor (weak attractor) was established in previous works (see [15], [16]). The main idea of our method is that we first derive that -almost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., using our method we only need to obtain time-uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system, while the common method need to establish time-uniform a priori estimates in a more regular functional space than the solution space. Take the stochastic PEs for example, as the unique strong solution to the stochastic PEs belongs to , in view of our method, we only need to obtain the time-uniform a priori estimates in the solution space to prove the existence of random attractor for this stochastic system, while the common method need to establish time-uniform a priori estimates for the solution in the functional space . However, time-uniform a priori estimates in for the solution to stochastic PEs are too difficult to be established. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by Wiener noises, fractional noises and even jump noises. In a forth coming paper, using this new method we [46] prove the existence of random attractor for the stochastic nematic liquid crystals equations. This is the first result about the long-time behavior of stochastic nematic liquid crystals equations. 相似文献
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Zihua Guo Xingxing Liu Luc Molinet Zhaoyang Yin 《Journal of Differential Equations》2019,266(2-3):1698-1707
We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space and even in the Besov space for . Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5], [14], [16]). 相似文献
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The free boundary problem for the three dimensional incompressible elastodynamics system is studied under the Rayleigh–Taylor sign condition. Both the columns of the elastic stress and the transpose of the deformation gradient are tangential to the boundary which moves with the velocity, and the pressure vanishes outside the flow domain. The linearized equation takes the form of wave equation in terms of the flow map in the Lagrangian coordinate, and the local-in-time existence of a unique smooth solution is proved using a geometric argument in the spirit of [19]. 相似文献
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We verify the critical case of Strauss' conjecture [30] concerning the blow-up of solutions to semilinear wave equations with variable coefficients in , where . The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when . The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou [43] and Zhou & Han [45]: exponential “eigenfunctions” of the Laplacian [37] are used to construct the test function for linear wave equation with variable coefficients and John's method of iterations [13] is augmented with the “slicing method” of Agemi, Kurokawa and Takamura [1] for lower bounds in the critical case. 相似文献
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We show uniqueness for overdetermined elliptic problems defined on topological disks Ω with boundary, i.e., positive solutions u to in so that and along ?Ω, the unit outward normal along ?Ω under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez–Mira generalized Hopf-type Theorem [19] to the realm of overdetermined elliptic problem.When is the standard sphere and f is a function so that and for any , we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki–Caffarelli–Nirenberg conjecture in for this choice of f. More precisely, this shows that if u is a positive solution to on a topological disk with boundary so that and along ?Ω, then Ω must be a geodesic disk and u is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D (cf. [33], [35]) for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains (cf. [28], also called Serrin Problem) in . 相似文献
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Natalí Ailín Cantizano Analía Silva 《Journal of Mathematical Analysis and Applications》2019,469(2):841-851
The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation in a bounded domain with Dirichlet condition, where is the well known p-fractional Laplacian and is the critical Sobolev exponent for the non local case. The proof follows the ideas of [28] and is based in the extension of the Concentration Compactness Principle for the p-fractional Laplacian [20] and Ekeland's variational Principle [7]. 相似文献
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In this paper we study the following type of the Schrödinger–Poisson–Slater equation with critical growth where and . For the case of . We develop a novel perturbation approach, together with the well-known Mountion–Pass theorem, to prove the existence of positive ground states. For the case of , we obtain the nonexistence of nontrivial solutions by restricting the range of μ and also study the existence of positive solutions by the constrained minimization method. For the case of , we use a truncation technique developed by Brezis and Oswald [9] together with a measure representation concentration-compactness principle due to Lions [27] to prove the existence of radial symmetrical positive solutions for with some . The above results nontrivially extend some theorems on the subcritical case obtained by Ianni and Ruiz [18] to the critical case. 相似文献
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We derive the sharp convergence rate in in periodic homogenization of second order parabolic systems with bounded measurable coefficients in Lipschitz cylinders. This extends the corresponding result for elliptic systems established in [20] to parabolic systems and improves the corresponding result in settings derived in [7], [28] for second order parabolic systems with time-dependent coefficients. 相似文献
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Kuijie Li Baoxiang Wang 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2019,36(6):1679-1707
This paper is concerned with the blowup criterion for mild solution to the incompressible Navier–Stokes equation in higher spatial dimensions . By establishing an ? regularity criterion in the spirit of [11], we show that if the mild solution u with initial data in , becomes singular at a finite time , then The corresponding result in 3D case has been obtained in [24]. As a by-product, we also prove a regularity criterion for the Leray–Hopf solution in the critical Besov space, which generalizes the results in [17], where blowup criterion in critical Lebesgue space is addressed. 相似文献