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Rafael Granero-Belinchón

Publications and source records attributed to Rafael Granero-Belinchón.

At least 37 records · Page 2Linked to original sources
arXiv2024

On the global well-posedness of interface dynamics for gravity Stokes flow

In this paper we establish the global-in-time well-posedness for an arbitrary $C^{1+γ}$, $0<γ<1$, initial internal wave for the free boundary gravity Stokes system in two dimensions. This classical well-posedness result is complemented with a weak solvability result in the case of $C^γ$ or Lipschitz interfaces. Furthermore, we also propose and study several one-dimensional models that capture different features of the full internal wave problem for the gravity Stokes system and show that all of them present finite time singularities. This fact evidences the fine structure of the non-linearity in the full system that allows for the free boundary problem to be globally well-posed while simplifications of it blow-up in finite time.

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arXiv2024

Well-posedness and decay for a nonlinear propagation wave model in atmospheric flows

In this note, we provide two results concerning the global well-posedness and decay of solutions to an asymptotic model describing the nonlinear wave propagation in the troposphere, namely, the morning glory phenomenon. The proof of the first result combines a pointwise estimate together with some interpolation inequalities to close the energy estimates in Sobolev spaces. The second proof relies on suitable Wiener-like functional spaces.

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arXiv2023

Global existence for certain fourth order evolution equations

In this paper we establish three global in time results for two fourth order nonlinear parabolic equations. The first of such equations involves the Hessian and appears in epitaxial growth. For such equation we give conditions ensuring the global existence of solution. For certain regime of the parameters, our size condition involves the norm in a critical space with respect to the scaling of the equation and improves previous existing results in the literature for this equation. The second of the equations under study is a thin film equation with a porous medium nonlinearity. For this equation we establish conditions leading to the global existence of solution.

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arXiv2023

Well-posedness and singularity formation for the Kolmogorov two-equation model of turbulence in 1-D

We study the Kolomogorov two-equation model of turbulence in one space dimension. Two are the main results of the paper. First of all, we establish a local well-posedness theory in Sobolev spaces even in the case of vanishing mean turbulent kinetic energy. Then, we show that, in general, those solutions must blow up in finite time. To the best of our knowledge, these results are the first establishing the well-posedness of the system for vanishing initial data and the occurence of finite time singularities for the model under study.

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arXiv2023

Long time interface dynamics for gravity Stokes flow

We study the dynamics of the interface given by two incompressible viscous fluids in the Stokes regime filling a 2D horizontally periodic strip. The fluids are subject to the gravity force and the density difference induces the dynamics of the interface. We derive the contour dynamics formulation for this problem through a $x_1$-periodic version of the Stokeslet. Using this new system, we show local-in-time well-posedness when the initial interface is described by a curve with no self-intersections and $C^{1+γ}$ Hölder regularity, $0<γ<1$. This well-posedness result holds regardless of the Rayleigh-Taylor stability of the physical system. In addition, global-in-time existence and decay to the flat stationary state is proved in the Rayleigh-Taylor stable regime for small initial data. Finally, in the Rayleigh-Taylor unstable regime, we construct a wide family of smooth solutions with exponential in time growth for an arbitrary large interval of existence. Remarkably, the initial data leading to this exponential growth possibly lack any symmetry.

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arXiv2022

Well-posedness theory for non-homogeneous incompressible fluids with odd viscosity

Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed ``odd viscosity'', becomes non-dissipative. At the mathematical level, this fact translates into a loss of derivatives at the level of \textsl{a priori} estimates: while the odd viscosity term depends on derivatives of the velocity field, no parabolic smoothing effect can be expected. In the present paper, we establish a well-posedness theory in Sobolev spaces for a system of incompressible non-homogeneous fluids with odd viscosity. The crucial point of the analysis is the introduction of a set of \emph{good unknowns}, which allow for the emerging of a hidden hyperbolic structure underlying the system of equations. It is exactly this hyperbolic structure which makes it possible to circumvent the derivative loss and propagate high enough Sobolev norms of the solution. The well-posedness result is local in time; two continuation criteria are also established.

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arXiv2022

Finite time blow-up for some parabolic systems arising in turbulence theory

We study a class of non-linear parabolic systems relevant in turbulence theory. Those systems can be viewed as simplified versions of the Prandtl one-equation and Kolmogorov two-equation models of turbulence. We restrict our attention to the case of one space dimension. We consider initial data for which the diffusion coefficients may vanish. We prove that, under this condition, those systems are locally well-posed in the class of Sobolev spaces of high enough regularity, but also that there exist smooth initial data for which the corresponding solutions blow up in finite time. We are able to put in evidence two different types of blow-up mechanism. In addition, the results are extended to the case of transport-diffusion systems, namely to the case when convection is taken into account.

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arXiv2022

A unified approach towards the impossibility of finite time vanishing depth for incompressible free boundary flows

In this paper we study the motion of an internal water wave and an internal wave in a porous medium. For these problems we establish that, if the free boundary and, in the case of the Euler equations, also the tangential velocity at the interface are sufficiently smooth, the depth cannot vanish in finite time. This results holds regardless of gravity and surface tension effects or, if applicable, the stratification in multiphase flows.

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arXiv2022

A nonlocal model describing tumor angiogenesis

In this paper we study the onset of angiogenesis and derive a new model to describe it. This new model takes the form of a nonlocal Burgers equation with both diffusive and dispersive terms. For a particular value of the parameters, the equation reduces to $$ \partial_t p-\frac{1}{2}(-Δ)^{(α-1)/2}H \partial_t p=-\frac{1}{2}(-Δ)^{α/2} p+ p\partial_x p-\partial_x p, $$ where $H$ denotes the Hilber transform. In addition to the derivation of the new model, we also prove a number of well-posedness results. Finally, some preliminary numerics are shown. These numerics suggest that the dynamics of the equation is rich enough to have solutions that blow up in finite time.

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arXiv2021

Interfaces in incompressible flows

The motion of both internal and surface waves in incompressible fluids under capillary and gravity forces is a major research topic. In particular, we review the derivation of some new models describing the dynamics of gravity-capillary nonlinear waves in incompressible flows. These models take the form of both bidirectional and unidirectional nonlinear and nonlocal wave equations. More precisely, with the goal of telling a more complete story, in this paper we present the results in the works \cite{GS,aurther2019rigorous,granero2019models,granero2019well,granero2020well,granero2021motion, granero2021global} together with some new results regarding the well-posedness of the resulting PDEs.

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arXiv2021

Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data

In this work we study the inhomogeneous Muskat problem, \emph{i.e.} the evolution of an internal wave between two different fluids in a porous medium with discontinuous permeability. In particular, under precise conditions on the initial datum and the physical quantities of the problem, our results ensure the decay of the solutions towards the equilibrium state in the Lipschitz norm. In addition, we establish the global existence and decay of Lipschitz solutions.

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arXiv2021

Global well-posedness and decay for viscous water wave models

The motion of the free surface of an incompressible fluid is a very active research area. Most of these works examine the case of an inviscid fluid. However, in several practical applications, there are instances where the viscous damping needs to be considered. In this paper we derive and study a new asymptotic model for the motion of unidirectional viscous water waves. In particular, we establish the global well-posedness in Sobolev spaces. Furthermore, we also establish the global well-posedness and decay of a fourth order PDE modelling bidirectional water waves with viscosity moving in deep water with or without surface tension effects.

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arXiv2021

Singularity formation for the Serre-Green-Naghdi equations and applications to abcd-Boussinesq systems

In this work we prove that the solution of the Serre-Green-Naghdi equation cannot be globally defined when the interface reaches the impervious bottom tangentially. As a consequence, our result complements the paper \emph{Camassa, R., Falqui, G., Ortenzi, G., Pedroni, M., \& Thomson, C. Hydrodynamic models and confinement effects by horizontal boundaries. Journal of Nonlinear Science, 29(4), 1445-1498, 2019.} Furthermore, we also prove that the solution to the $abcd-$Boussinesq system can change sign in finite time. Finally, we provide with a proof of a scenario of finite time singularity for the $abcd-$Boussinesq system. These latter mathematical results are related to the numerics in \emph{Bona, \& Chen, Singular solutions of a Boussinesq system for water waves. J. Math. Study, 49(3), 205-220, 2016}.

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arXiv2021

On the motion of gravity-capillary waves with odd viscosity

We develop three asymptotic models of surface waves in a non-newtonian fluid with odd viscosity. This viscosity is also known as Hall viscosity and appears in a number of applications such as quantum Hall fluids or chiral active fluids. Besides the odd viscosity effects, these models capture both gravity and capillary forces up to quadratic interactions and take the form of nonlinear and nonlocal wave equations. Two of these models describe bidirectional waves while the third PDE studies the case of unidirectional propagation. We also prove the well-posedness of these asymptotic models in spaces of analytic functions and in Sobolev spaces. Finally, we present a number of numerical simulations for the unidirectional model.

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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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arXiv2020

Well-posedness of the water-wave with viscosity problem

In this paper we study the motion of a surface gravity wave with viscosity. In particular we prove two well-posedness results. On the one hand, we establish the local solvability in Sobolev spaces for arbitrary dissipation. On the other hand, we establish the global well-posedness in Wiener spaces for a sufficiently large viscosity. These results are the first rigorous proofs of well-posedness for the Dias, Dyachenko \& Zakharov system ({\em Physics Letters A} 2008) modelling gravity waves with viscosity when surface tension is not taken into account.

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arXiv2020

On the dynamics of 3D electrified falling films

In this article, we consider a non-local variant of the Kuramoto-Sivashinsky equation in three dimensions (2D interface). Besides showing the global wellposedness of this equation we also obtain some qualitative properties of the solutions. In particular, we prove that the solutions become analytic in the spatial variable for positive time, the existence of a compact global attractor and an upper bound on the number of spatial oscillations of the solutions. We observe that such a bound is particularly interesting due to the chaotic behavior of the solutions.

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