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It is established convergence to a particular equilibrium for weak solutions of abstract linear
equations of the second order in time associated with monotone operators with nontrivial kernel. Concerning nonlinear
hyperbolic equations with monotone and conservative potentials, it is proved a general asymptotic convergence result
in terms of weak and strong topologies of appropriate Hilbert spaces. It is also considered the stabilization of a
particular equilibrium via the introduction of an asymptotically...
We prove the convergence at a large scale of a non-local first order equation to an
anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.
We study convergence of solutions to stationary states in an astrophysical model of evolution of clouds of self-gravitating particles.
We study the behaviour of the solutions of the Cauchy problem
and prove that if initial data decay fast enough at infinity then the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. Certain generalizations are also mentioned.
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...
We study the evolution of a closed, convex hypersurface in in direction of its normal vector, where the speed equals a power 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 , depending only on and , 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.
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...
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