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Convergence towards self-similar asymptotic behavior for the dissipative quasi-geostrophic equations

José A. Carrillo, Lucas C. F. Ferreira (2006)

Banach Center Publications

This work proves the convergence in L¹(ℝ²) towards an Oseen vortex-like solution to the dissipative quasi-geostrophic equations for several sets of initial data with suitable decay at infinity. The relative entropy method applies in a direct way for solving this question in the case of signed initial data and the difficulty lies in showing the existence of unique global solutions for the class of initial data for which all properties needed in the entropy approach are met. However, the estimates...

Convexity estimates for flows by powers of the mean curvature

Felix Schulze (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the evolution of a closed, convex hypersurface in ℝ n + 1 in direction of its normal vector, where the speed equals a power k ≥ 1 of the mean curvature. We show that if initially the ratio of the biggest and smallest principal curvatures at every point is close enough to 1 , depending only on k and n , then this is maintained under the flow. As a consequence we obtain that, when rescaling appropriately as the flow contracts to a point, the evolving surfaces converge to the unit sphere.

Correctors and field fluctuations for the pϵ(x)-laplacian with rough exponents : The sublinear growth case

Silvia Jimenez (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

A corrector theory for the strong approximation of gradient fields inside periodic composites made from two materials with different power law behavior is provided. Each material component has a distinctly different exponent appearing in the constitutive law relating gradient to flux. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplification of applied...

Decay and asymptotic behavior of solutions of the Keller-Segel system of degenerate and nondegenerate type

Takayoshi Ogawa (2006)

Banach Center Publications

We classify the global behavior of weak solutions of the Keller-Segel system of degenerate and nondegenerate type. For the stronger degeneracy, the weak solution exists globally in time and has a uniform time decay under some extra conditions. If the degeneracy is weaker, the solution exhibits a finite time blow up if the data is nonnegative. The situation is very similar to the semilinear case. Some additional discussion is also presented.

Decay estimates of solutions of a nonlinearly damped semilinear wave equation

Aissa Guesmia, Salim A. Messaoudi (2005)

Annales Polonici Mathematici

We consider an initial boundary value problem for the equation u t t - Δ u - ∇ ϕ · ∇ u + f ( u ) + g ( u t ) = 0 . We first prove local and global existence results under suitable conditions on f and g. Then we show that weak solutions decay either algebraically or exponentially depending on the rate of growth of g. This result improves and includes earlier decay results established by the authors.

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