Showing papers in "Nonlinear Analysis-theory Methods & Applications in 2001"
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TL;DR: In this article, the KS-equation was used as a model for cellular instabilities in a variety of situations, such as thin liquid lms on inclined planes, dendritic fronts in dilute binary alloys, and Alfven drift waves in plasmas.
Abstract: where we refer to ? 0 as the “anti-di ussion” parameter. Note that a more general form ut + 1uxxxx + 2uxx + 3uux = 0 can always be reduced to (1.1) by appropriate rescaling of t; x and u. Eq. (1.1) was derived independently by Kuramoto et al. [20–22] as a model for phase turbulence in reaction–di usion systems and by Sivashinsky [33] as a model for plane ame propagation, describing the combined in uence of di usion and thermal conduction of the gas on stability of a plane ame front. So far, it has been well understood that the KS-equation can also serve as a mathematical model for cellular instabilities in a variety of situations: the ow of thin liquid lms on inclined planes [30] (in the limit of large surface tension), dendritic fronts in dilute binary alloys [31], and Alfven drift waves in plasmas [23] (as a nonlinear saturation mechanism of the dissipative trapped ion modes).
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TL;DR: Meir-Keeler contractive maps are defined as maps that satisfy d(Tx, Ty) < φ(d(x, y)) for some L-function φ.
Abstract: Let (X, d) be a metric space. A map T : X → X is called Meir-Keeler contractive if ∀ > 0 ∃δ > 0 such that ≤ d(x, y) < + δ ⇒ d(Tx, Ty) < We introduce ”L-functions” and characterize Meir-Keeler contractive maps as maps that satisfy d(Tx, Ty) < φ(d(x, y)) for some L-function φ. This characterization makes it easy to compare such maps with those satisfying the Boyd-Wong’s condition.
TL;DR: In this paper, it was shown that many important results of classical Perron-Frobenius Theory can be extended from linear selfmappings of the standard cone in finite dimensional real space to concave self-appings of this cone, albeit the spectrum of these concave mappings is more intricate than that for linear mappings.
Abstract: Many important results of classical Perron–Frobenius Theory can be extended from linear selfmappings of the standard cone in finite dimensional real space to concave selfmappings of this cone. This is in particular true for minima of linear mappings, albeit the spectrum of these special concave mappings is more intricate than that for linear mappings. As classical Perron–Frobenius Theory has numerous applications there are many new applications for its concave extension. 1991 Mathematical Subject Classification: Primary 39A11, 15A48; secondary 47H07, 47H12.
TL;DR: In this paper, the authors established a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in R. They proved that if the pressure p associated to a weak solution u belongs to L 2 2 2−r ( (0, T ) ;. M2, dr ( R d )d), (0.2) where. M 2, dr is the critical Morrey-Campanato space for 0 < r < 1, then the weak solution is actually regular.
Abstract: In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in R. It is known that if a Leray weak solution u belongs to L 2 1−r ( (0, T ) ;L d r ) for some 0 ≤ r ≤ 1, (0.1) then u is regular. It is proved that if the pressure p associated to a Leray weak solution u belongs to L 2 2−r ( (0, T ) ; . M2, dr ( R d )d) , (0.2) where . M2, dr ( R d ) is the critical Morrey-Campanato space (a definition is given in the text) for 0 < r < 1, then the weak solution is actually regular. Since this space . M2, dr is wider than L d r and . Xr, the above regularity criterion (0.2) is an improvement of Zhou’s result.
TL;DR: In this article, asymptotic behaviors in time of solutions to the initial boundary value problems in the half space for a one-dimensional isentropic model system of compressible viscous gas were investigated.
Abstract: We consider asymptotic behaviors in time of solutions to the initial boundary value problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. In particular, we focus our attention on inflow(or outflow) problems where the velocity on the boundary is given as a constant inward (or outward) flow, and try to classify all asymptotic behaviors of the solutions. It turns out that depending on the data both on the boundary and at far field (especially depending on whether the state is subsonic, transonic, or supersonic), the asymptotic state variously consists of rarefaction waves, viscous shock waves, and also stationary boundary layer. Moreover, we give a survey of our recent results on some particular cases which justify our classification.
TL;DR: In this article, the structure of recursion operators of evolution equations has been discussed, and it is shown that the nonlocal part of a Nijenhuis operator contains the candidates of roots and coroots.
Abstract: In this paper we discuss the structure of recursion operators. We show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries. This enables us to compute recursion operators more systematically. Under certain conditions (which hold for all examples known to us) Nijenhuis operators are well defined, i.e., they give rise to hierarchies of infinitely many commuting symmetries of the operator. Moreover, the nonlocal part of a Nijenhuis operator contains the candidates of roots and coroots.