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JournalISSN: 0022-0396

Journal of Differential Equations 

About: Journal of Differential Equations is an academic journal. The journal publishes majorly in the area(s): Boundary value problem & Nonlinear system. It has an ISSN identifier of 0022-0396. Over the lifetime, 10932 publication(s) have been published receiving 301482 citation(s).

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Journal ArticleDOI
Abstract: On considere des solutions u∈H 1,p (Ω)∧L ∞ (Ω) (1

1,347 citations

Journal ArticleDOI
Abstract: In this paper, equations governing the time dependent flow of granular material under gravity are derived and analyzed. Formally these equations bear a strong resemblance to the Navier-Stokes equations for the flow of an incompressible, viscous fluid. However, the main result of this paper is that, depending on geometric and material parameters, the equations governing granular flow may lead to a violent instability analogous to that for u, = u XI up ;

934 citations

Journal ArticleDOI
Abstract: We construct a unique local regular solution in Lq(0, T; Lp) for a class of semilinear parabolic equations which includes the semilinear heat equation u t − Δu = ¦u¦ α u (α > 0) and the Navier-Stokes system. Here p and q are so chosen that the norm of Lq(0, T; Lp) is dimensionless or scaling invariant. The main relation between p and q for the semilinear heat equation is 1 q =( 1 p − 1 q ) n 2 ) , provided that initial data are in Lr with r = nα/2 > 1, where n is the space dimension. Applying our regular solutions to the Navier-Stokes system, we show that the k/2-dimensional Hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in L q (0, T; L p ), where k = 2 − q + nq p , p ⩾ n, 1 ⩽ q α . We show, moreover, that a turbulent solution is regular if it is in C((0, T); Ln).

749 citations

Journal ArticleDOI
Abstract: Discussion of the controllability of nonlinear systems described by the equation dx/dt - F(x,u). Concepts formulated by Chow (1939) and Lobry (1970) are applied to establish criteria for F and its derivatives to obtain qualitative information on sets which can be obtained from x which denotes a variable of state in an arbitrary, real, analytical manifold. It is shown that controllability implies strong accessibility for a large class of manifolds including Euclidean spaces.-

697 citations

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