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Francisco Gancedo

Publications and source records attributed to Francisco Gancedo.

At least 37 records · Page 2Linked to original sources
arXiv2021

Global existence in the Lipschitz class for the N-Peskin problem

In this paper we study a toy model of the Peskin problem that captures the motion of the full Peskin problem in the normal direction and discards the tangential elastic stretching contributions. This model takes the form of a fully nonlinear scalar contour equation. The Peskin problem is a fluid-structure interaction problem that describes the motion of an elastic rod immersed in an incompressible Stokes fluid. We prove global in time existence of solution for initial data in the critical Lipschitz space. Using a new decomposition together with cancellation properties, pointwise methods allow us to obtain the desired estimates in the Lipschitz class. Moreover, we perform energy estimates in order to obtain that the solution lies in the space $L^2 \left( [0,T];H^{3/2} \right) $ to satisfy the contour equation pointwise.

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arXiv2019

Surface tension stabilization of the Rayleigh-Taylor instability for a fluid layer in a porous medium

This paper studies the dynamics of an incompressible fluid driven by gravity and capillarity forces in a porous medium. The main interest is the stabilization of the fluid in Rayleigh-Taylor unstable situations where the fluid lays on top of a dry region. An important feature considered here is that the layer of fluid is under an impervious wall. This physical situation have been widely study by mean of thin film approximations in the case of small characteristic high of the fluid considering its strong interaction with the fixed boundary. Here, instead of considering any simplification leading to asymptotic models, we deal with the complete free boundary problem. We prove that, if the fluid interface is smaller than an explicit constant, the solution is global in time and it becomes instantly analytic. In particular, the fluid does not form drops in finite time. Our results are stated in terms of Wiener spaces for the interface together with some non-standard Wiener-Sobolev anisotropic spaces required to describe the regularity of the fluid pressure and velocity. These Wiener-Sobolev spaces are of independent interest as they can be useful in other problems. Finally, let us remark that our techniques do not rely on the irrotational character of the fluid in the bulk and they can be applied to other free boundary problems.

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arXiv2019

Regularity results for viscous 3D Boussinesq temperature fronts

This paper is about the dynamics of non-diffusive temperature fronts evolving by the incompressible viscous Boussinesq system in $\mathbb{R}^3$. We provide local in time existence results for initial data of arbitrary size. Furthermore, we show global in time propagation of regularity for small initial data in critical spaces. The developed techniques allow to consider general fronts where the temperature is piecewise Hölder (not necessarily constant), which preserve their structure together with the regularity of the evolving interface.

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arXiv2019

On the local existence and blow-up for generalized SQG patches

We study patch solutions of a family of transport equations given by a parameter $α$, $0< α<2$, with the cases $α=0$ and $α=1$ corresponding to the Euler and the surface quasi-geostrophic equations respectively. In this paper, using several new cancellations, we provide the following new results. First, we prove local well-posedness for $H^{2}$ patches in the half-space setting for $0<α< 1/3$, allowing self-intersection with the fixed boundary. Furthermore, we are able to extend the range of $α$ for which finite time singularities have been shown in \cite{KYZ} and \cite{KRYZ}. Second, we establish that patches remain regular for $0<α<2$ as long as the arc-chord condition and the regularity of order $C^{1+δ}$ for $δ>α/2$ are time integrable. This finite-time singularity criterion holds for lower regularity than the regularity shown in numerical simulations in \cite{CFMR} and \cite{ScottDritschel} for surface quasi-geostrophic patches, where the curvature of the contour blows up numerically. This is the first proof of a finite-time singularity criterion lower than or equal to the regularity in the numerics. Finally, we also improve results in \cite{G} and in \cite{CCCGW}, giving local-wellposedness for patches in $H^{2}$ for $0<α< 1$ and in $H^3$ for $1<α<2$.

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arXiv2017

On The Muskat Problem With Viscosity Jump: Global In Time Results

The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contour equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in $L^\infty$. We prove global in time existence results for medium size initial stable data in critical spaces. We also enhance previous methods by showing smoothing (instant analyticity), improving the medium size constant in 3D, together with sharp decay rates of analytic norms. The found technique is twofold, giving ill-posedness in unstable situations for very low regular solutions.

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arXiv2017

Uniqueness for SQG patch solutions

This paper is about the evolution of a temperature front governed by the Surface quasi-geostrophic equation. The existence part of that program within the scale of Sobolev spaces was obtained by one of the authors [10]. Here we revisit that proof introducing some new tools and points of view which allow us to conclude the also needed uniqueness result.

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arXiv2017

Global regularity of 2D density patches for inhomogeneous Navier-Stokes

This paper is about Lions' open problem on density patches \cite{LIONS}: whether inhomogeneous incompressible Navier-Stokes equations preserve the initial regularity of the free boundary given by density patches. Using classical Sobolev spaces for the velocity, we first establish the propagation of $C^{1+γ}$ regularity with $0<γ<1$ in the case of positive density. Furthermore, we go beyond to show the persistence of a geometrical quantity such as the curvature. In addition, we obtain a proof for $C^{2+γ}$ regularity.

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arXiv2016

Global regularity for 2D Boussinesq temperature patches with no diffusion

This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space $\mathbb{R}^2$. We prove that for initial patches with $W^{2,\infty}$ boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound $\nabla^2u$, even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the $C^{1+γ}$ regularity result in [23], $0<γ<1$, using the scale of Sobolev spaces for the velocity. Furthermore, taking advantage of the new cancellations, we go beyond to show the persistence of regularity for $C^{2+γ}$ patches.

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arXiv2015

Global regularity for 2D Muskat equations with finite slope

We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class $W^{2,p}$, $1<p\leq \infty$. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.

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arXiv2015

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 found 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.

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arXiv2014

On the Muskat problem: global in time results in 2D and 3D

This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the estimates from our paper \cite{PDPB} to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature.

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arXiv2014

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.

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arXiv2013

Absence of splash singularities for SQG sharp fronts and the Muskat problem

In this paper for either the sharp front Surface Quasi-Geostrophic equation or the Muskat problem we rule out the "splash singularity" blow-up scenario; in other words we prove that the contours evolving from either of these systems can not intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem \cite{ADCPJ}. Our result confirms the numerical simulations in \cite{CFMR} where it is shown that the curvature blows up due to the contours collapsing at a point. Here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of the work \cite{DP3} in which squirt singularities are ruled out; here a positive volume of fluid between the contours can not be ejected in finite time.

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arXiv2012

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.

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arXiv2012

On a singular incompressible porous media equation

In this paper we study a singularly modified version of the incompressible porous media equation. We investigate the implications for the local well-posedness of the equations by modifying, with a fractional derivative, the constitutive relation between the scalar density and the convecting divergence free velocity vector. Our analysis is motivated by recent work \cite{CCCGW} where it is shown that for the surface quasi-geostrophic equation such a singular modification of the constitutive law for the velocity, quite surprisingly still yields a locally well-posed problem. In contrast, for the singular active scalar equation discussed in this paper, local well-posedness does not hold for smooth solutions, but it does hold for certain weak solutions.

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arXiv2012

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 $C^4$.

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