Breakdown of smoothness for the Muskat problem

In this paper we show that there exist analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down i.e. no longer belongs to $C^4$.Comment: 93 pages, 10 figures (6 added

Splash singularities for the one-phase Muskat problem in stable regimes

This paper shows finite time singularity formation for the Muskat problem in a stable regime. The framework we exhibit is with a dry region, where the density and the viscosity are set equal to 0 (the gradient of the pressure is equal to (0, 0)) in the complement of the fluid domain. The singularity is a splash-type: a smooth fluid boundary collapses due to two different particles evolve to collide at a single point. This is the first example of a splash singularity for a parabolic problem.Ministerio de Ciencia e InnovaciónEuropean Research CouncilNational Science FoundationOffice of Naval Researc

Contribución para la caracterización bioclimática del Norte de Portugal. La transición florística atlántico-mediterránea.

Contribución para la caracterización bioclimática del norte de Portugal. La transición florística atlántico-mediterránea. Este trabajo se incluye en la serie de trabajos ya iniciados por Crespí et al. (2001), con el objetivo de contribuir al conocimiento bioclimático del Norte de Portugal, a través de la información florística presente. En este sentido, y para proceder a una posible cartografia de las áreas biogeográficas naturales desde una perspectiva termopluviométrica, son ahora introducidos 39 nuevos táxones para la relación ya existente. Los táxones analizados en esta aportación son: Anthemis triumfetti f. flosculosa, Arnica montana subsp. atlantica, Aster aragonensis, Carduus bourgeanus, Carduus carpetanus, Carduus platypus subsp. platypus, Carduus platypus var. granatensis, Carduus pycnocephalus, Centaurea cyanus, Centaurea geresensi, Centaurea herminii subsp. herminii, Centaurea langeana, Centaurea rivularis, Centaurea triumfetti subsp. lingulata, Leucanthemopsis pallida subsp.flaveola, Leucanthemopsis pallida subsp. pulverulenta, Leuzea rhaponticoides, Phalacrocarpum hoffmannseggii, Santolina rosmarinifolia, Santolina semidentata, Arbutus unedo, Cistus ladanifer, Cistus laurifolius, Cistus populifolius, Cistus psilosepalus, Cistus salvifolius, Erica arborea, Erica australis, Erica ciliaris, Erica cinerea, Erica lusitanica, Erica scoparia, Erica tetralix, Erica umbellata, Halimium lasianthum subsp alyssoides, Halimium lasianthum subsp lasianthum, Halimium ocymoides, Halimium umbellatum subsp umbellatum, Halimium umbellatum subsp. viscosum. De modo a poder alcanzar los objetivos propuestos, se procede a la aplicación de una rutina programática multivariada, basada en un sistema de información geográfico (SIG), a partir del cual cada uno de los táxones estudiados es georeferenciado sobre cartografía ambiental (confeccionada y publicada por el Instituto para a Conservação da Natureza, del Ministerio del Ambiente portugués), comparándola después con las caracterizaciones fitoclimáticas de Franco (1994) y de Costa et al. (1998). Los resultados obtenidos confirman no sólo un acusado efecto gradual de transición entre el área de influencia bioclimática atlántica y la mediterránea, como además la existencia de una tendencia de comportamiento fitoclimático intermedio, no considerada hasta el momento en las caracterizaciones bioclimáticas ya publicadas.Approach on the climatic characterization for Northern of Portugal. The atlanticmediterranean floristic transition. Thirty-nine specific and infraspecific taxa with different occurrences in the continental portuguese area are referred in the present work. In the light of these behaviours, the climatic and geomorphological variability of the North of Portugal has been important environmental factors to explain the floristic diversity in this part of the country. In this case, the taxa analysed are: Anthemis triumfetti f. flosculosa, Arnica montana subsp. atlantica, Aster aragonensis, Carduus bourgeanus, Carduus carpetanus, Carduus platypus subsp. platypus, Carduus platypus var.granatensis, Carduus pycnocephalus, Centaurea cyanus, Centaurea geresensi, Centaurea herminii subsp. herminii, Centaurea langeana, Centaurea rivularis, Centaurea triumfetti subsp. lingulata, Leucanthemopsis pallida subsp. flaveola, Leucanthemopsis pallida subsp. pulverulenta, Leuzea rhaponticoides, Phalacrocarpum hoffmannseggii, Santolina rosmarinifolia, Santolina semidentata, Arbutus unedo, Cistus ladanifer, Cistus laurifolius, Cistus populifolius, Cistus psilosepalus, Cistus salvifolius, Erica arborea, Erica australis, Erica ciliaris, Erica cinerea, Erica lusitanica, Erica scoparia, Erica tetralix, Erica umbellata, Halimium lasianthum subsp alyssoides, Halimium lasianthum subsp lasianthum, Halimium ocymoides, Halimium umbellatum subsp umbellatum, Halimium umbellatum subsp. viscosum. Because of the appropriate management of the floristic richness, a multivariate statistic routine is applied on a Geographical Informatic System (GIS) the chorological, geomorphological and climatic characterization for each one are described in this paper. The geological substrate and the altitudinal classes are focused for the geomorphological approach, as well as two biogeographic classifications are used to correlate the occurrence of the taxa and their bioclimatic preferences. In the light of this analytical scheme, every taxa is geo-referenced upon an environmental cartography (elaborated by the Instituto para a Conservação da Natureza), compared with the Franco´s (1994) and Costa´s et al. (1998) phytoclimatic characterizations. The results obtained are pointing out the floristic progressive differentiation between the atlantic bioclimatic influence and the mediterranean one, as well as the existence of an intermediate bioclimatic tendency which had never been considered so far

Incompressible flow in porous media with fractional diffusion

In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with $\alpha\in (1,2]$, we obtain the existence of the global attractor for the solutions in the space $H^s$ for any $s > (N/2)+1-\alpha$

Finite time singularities for the free boundary incompressible Euler equations

In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity.Ministerio de Ciencia e InnovaciónEuropean Research CouncilNational Science Foundatio

Structural stability for the splash singularities of the water waves problem

In this paper we show a structural stability result for water waves. The main motivation for this result is that we would like to exhibit a water wave whose interface starts as a graph and ends in a splash. Numerical simulations lead to an approximate solution with the desired behaviour. The stability result will conclude that near the approximate solution to water waves there is an exact solution.Ministerio de Ciencia e InnovaciónInstituto de Ciencias Matemáticas Severo OchoaEuropean Research CouncilNational Science Foundatio

Splash singularity for water waves

We exhibit smooth initial data for the two-dimensional (2D) waterwave equation for which we prove that smoothness of the interface breaks down in finite time. Moreover, we show a stability result together with numerical evidence that there exist solutions of the 2D water-wave equation that start from a graph, turn over, and collapse in a splash singularity (self-intersecting curve in one point) in finite time.Ministerio de Ciencia e InnovaciónEuropean Research CouncilNational Science Foundatio

Finite time singularities for water waves with surface tension

Here we consider the 2D free boundary incompressible Euler equation with surface tension. We prove that the surface tension does not prevent a finite time splash or splat singularity, i.e. that the curve touches itself either in a point or along an arc. To do so, the main ingredients of the proof are a transformation to desingularize the curve and a priori energy estimates.Ministerio de Ciencia e InnovaciónEuropean Research CouncilNational Science Foundatio

Turning waves and breakdown for incompressible flows

We consider the evolution of an interface generated between two immiscible, incompressible, and irrotational fluids. Specifically we study the Muskat and water wave problems. We show that starting with a family of initial data given by (α, f0(α)), the interface reaches a regime in finite time in which is no longer a graph. Therefore there exists a time ∗t where the solution of the free boundary problem parameterized as s (α, f(α, t)) blows up:: k∂αfkL∞(t∗) = ∞. In particular, for the Muskat problem, this result allows us to reach an unstable regime, for which the Rayleigh-Taylor condition changes sign and the solution breaks down.Ministerio de Ciencia e InnovaciónEuropean Research CouncilNational Science FoundationOffice of Naval Researc