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We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to where denotes the standard weak Lebesgue space. 相似文献
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This paper deals with the following nonlinear elliptic equation where , is a bounded non-negative function in . By combining a finite reduction argument and local Pohozaev type of identities, we prove that if and has a stable critical point with and , then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions. 相似文献
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Benjamin C. Pooley 《Journal of Differential Equations》2018,264(8):5180-5196
It is known that in a classical setting, the Navier–Stokes equations can be reformulated in terms of so-called magnetization variables w that satisfy
(1)
and relate to the velocity u via a Leray projection . We will prove the equivalence of these formulations in the setting of weak solutions that are also in on the 3-dimensional torus.Our main focus is the proof of global well-posedness in for a new variant of (1), where is replaced by w in the second nonlinear term:(2)
This is based on a maximum principle, analogous to a similar property of the Burgers equations. 相似文献
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Zuoshunhua Shi 《Journal of Differential Equations》2018,264(3):1550-1580
In this paper, we mainly study the existence of self-similar solutions of stationary Navier–Stokes equations for dimension . For , if the external force is axisymmetric, scaling invariant, continuous away from the origin and small enough on the sphere , we shall prove that there exists a family of axisymmetric self-similar solutions which can be arbitrarily large in the class . Moreover, for axisymmetric external forces without swirl, corresponding to this family, the momentum flux of the flow along the symmetry axis can take any real number. However, there are no regular () axisymmetric self-similar solutions provided that the external force is a large multiple of some scaling invariant axisymmetric F which cannot be driven by a potential. In the case of dimension 4, there always exists at least one self-similar solution to the stationary Navier–Stokes equations with any scaling invariant external force in . 相似文献
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Douglas P. Hardin Michael C. Northington Alexander M. Powell 《Applied and Computational Harmonic Analysis》2018,44(2):294-311
A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators must satisfy , namely, . Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space ; our results provide an absolutely sharp improvement with . Our results are sharp in the sense that cannot be replaced by for any . 相似文献
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We compare the isoperimetric profiles of and of with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of and . Explicitly we show that and . We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions. 相似文献
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In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space . For this, we study first the solutions of forward–backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space . This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon. 相似文献
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We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in with . We prove the existence and uniqueness of the tempered random attractor that is compact in and attracts all tempered random subsets of with respect to the norm of . The main difficulty is to show the pullback asymptotic compactness of solutions in due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains. 相似文献
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Masoud Hassani 《Comptes Rendus Mathematique》2017,355(11):1133-1137
In this paper, we study the irreducible representation of in . This action preserves a quadratic form with signature . Thus, it acts conformally on the 3-dimensional Einstein universe . We describe the orbits induced in and its complement in . This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on [5]. 相似文献
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Let be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1) vanishes if and . In particular, square-integrable solutions ζ of (1) with vanish. As a consequence, we prove that: is a norm if with , for some with as well as . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let , , satisfy for some with . Then there exists a constant translation vector and a constant skew-symmetric matrix , such that . 相似文献
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We study the dynamics of infinitely many Cucker–Smale (C–S) flocking particles under the interplay of random communication and incompressible fluids. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the incompressible Navier–Stokes (N–S) equations. These two subsystems are coupled via the drag force. For this coupled model, we present the global existence of weak and strong solutions in . Under the extra regularity assumptions of the initial data, the unique solvability of strong solutions is also established in . In a large coupling regime and periodic spatial domain , we show that the velocities of C–S particles and fluids are asymptotically aligned to two constant velocities which may be different. 相似文献
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H. Inci 《Journal of Differential Equations》2018,264(4):2660-2683
In this paper we consider the inviscid SQG equation on the Sobolev spaces , . Using a geometric approach we show that for any the corresponding solution map, , is nowhere locally uniformly continuous. 相似文献
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In a previous work, it was shown how the linearized strain tensor field can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain , instead of the displacement vector field in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition on a portion of the boundary of Ω can be recast, again as boundary conditions on , but this time expressed only in terms of the new unknown . 相似文献