arXiv subjects

Fernando Quirós

Publications and source records attributed to Fernando Quirós.

35 records · Page 2Linked to original sources

arXiv2019

Coupling local and nonlocal evolution equations

We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the behaviour of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffusion) in a limit procedure when we rescale the nonlocal kernel in a suitable way.

math.AP↗

arXiv2018

Anisotropic nonlocal diffusion equations with singular forcing

We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the right-hand side has very few regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.

math.AP↗

arXiv2018

A free boundary problem of Stefan type with nonlocal diffusion

We introduce and analyze a nonlocal version of the one-phase Stefan problem in which, as in the classical model, the rate of growth of the volume of the liquid phase is proportional to the rate at which energy is lost through the interphase. We prove existence and uniqueness for the problem posed on the line, and on the half-line with constant Dirichlet data, and in the radial case in several dimensions. We also describe the asymptotic behaviour of both the solution and its free boundary. The model may be of interest to describe the spreading of populations in hostile environments.

math.AP↗

arXiv2017

Regularity theory for singular nonlocal diffusion equations

We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If the nonlinearity in the equation does not oscillate too much at the origin, the solution is proved to be moreover Hölder continuous. The results are new even when the Dirichlet form is the one corresponding to the fractional Laplacian.

math.AP↗

arXiv2016

Near field asymptotics for the porous medium equation in exterior domains. The critical two-dimensional case

We consider the porous medium equation in an exterior two-dimensional domain which excludes a hole, with zero Dirichlet data on its boundary. Gilding and Goncerzewicz proved in [Gilding-Goncerzewicz-2007] that in the far field scale, $x=ξt^{\frac1{2m}}/(\log t)^{\frac{m-1}{2m}}$, $ξ\ne 0$, solutions to this problem with an integrable and compactly supported initial data behave as an instantaneous point-source solution for the equation with a variable mass that decays to 0 in a precise way, determined by the initial data and the hole. However, their result does not say much about the behavior when $|x|=o\big(t^{\frac1{2m}}/(\log t)^{\frac{m-1}{2m}}\big)$, in the so called near field scale, except that the solution is $o\big((t\log t)^{-\frac1m}\big)$ there. In particular, it does not give a sharp decay rate, neither a nontrivial asymptotic profile, on compact sets. In this paper we characterize the large time behavior in such scale, thus completing the results of [Gilding-Goncerzewicz-2007].

math.AP↗

arXiv2016

DDOTI: the deca-degree optical transient imager

DDOTI will be a wide-field robotic imager consisting of six 28-cm telescopes with prime focus CCDs mounted on a common equatorial mount. Each telescope will have a field of view of 12 square degrees, will have 2 arcsec pixels, and will reach a 10-sigma limiting magnitude in 60 seconds of r = 18.7 in dark time and r = 18.0 in bright time. The set of six will provide an instantaneous field of view of about 72 square degrees. DDOTI uses commercial components almost entirely. The first DDOTI will be installed at the Observatorio Astronómico Nacional in Sierra San Pedro Martír, Baja California, México in early 2017. The main science goals of DDOTI are the localization of the optical transients associated with GRBs detected by the GBM instrument on the Fermi satellite and with gravitational-wave transients. DDOTI will also be used for studies of AGN and YSO variability and to determine the occurrence of hot Jupiters. The principal advantage of DDOTI compared to other similar projects is cost: a single DDOTI installation costs only about US$500,000. This makes it possible to contemplate a global network of DDOTI installations. Such geographic diversity would give earlier access and a higher localization rate. We are actively exploring this option.

astro-ph.IM↗

arXiv2016

COATLI: an all-sky robotic optical imager with 0.3 arcsec image quality

COATLI will provide 0.3 arcsec FWHM images from 550 to 900 nm over a large fraction of the sky. It consists of a robotic 50-cm telescope with a diffraction-limited fast-guiding imager. Since the telescope is small, fast guiding will provide diffraction-limited image quality over a field of at least 1 arcmin and with coverage of a large fraction of the sky, even in relatively poor seeing. The COATLI telescope will be installed at the at the Observatorio Astronómico Nacional in Sierra San Pedro Mártir, México, during 2016 and the diffraction-limited imager will follow in 2017.

astro-ph.IM↗

arXiv2016

Near field asymptotic behavior for the porous medium equation on the half-line

Kamin and Vázquez proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation $u_t=(u^m)_{xx}$ on the half line with zero boundary data and nonnegative compactly supported integrable initial data behave for large times as a dipole type solution to the equation having the same first moment as the initial data, with an error which is $o(t^{-1/m})$. However, on sets of the form $0<x<g(t)$, with $g(t)=o(t^{1/(2m)})$ as $t\to\infty$, in the so called near field, the dipole solution is $o(t^{-1/m})$, and their result does not give neither the right rate of decay of the solution, nor a nontrivial asymptotic profile. In this paper we will show that the error is $o\big(t^{-(2m+1)/(2m^2)}(1+x)^{1/m}\big)$. This allows in particular to obtain a nontrivial asymptotic profile in the near field limit, which is a multiple of $x^{1/m}$, thus improving in this scale the results of Kamin and Vázquez.

math.AP↗

arXiv2016

Nonlocal filtration equations with rough kernels

We study the nonlinear and nonlocal Cauchy problem \[ \partial_{t}u+\mathcal{L}φ(u)=0 \quad\text{in }\mathbb{R}^{N}\times\mathbb{R}_+,\qquad u(\cdot,0)=u_0, \] where $\mathcal{L}$ is a Lévy-type nonlocal operator with a kernel having a singularity at the origin as that of the fractional Laplacian. The nonlinearity $φ$ is nondecreasing and continuous, and the initial datum $u_0$ is assumed to be in $L^1(\mathbb{R}^N)$. We prove existence and uniqueness of weak solutions. For a wide class of nonlinearities, including the porous media case, $φ(u)=|u|^{m-1}u$, $m>1$, these solutions turn out to be bounded and Hölder continuous for $t>0$. We also describe the large time behaviour when the nonlinearity resembles a power for $u\approx 0$ and the kernel associated to $\mathcal{L}$ is close at infinity to that of the fractional Laplacian.

math.AP↗

arXiv2015

Asymptotic behavior for a nonlocal diffusion equation in exterior domains: the critical two-dimensional case

We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, $\partial _t u=J*u-u$, where $J$ is a smooth, radially symmetric kernel with support $B_d(0)\subset\mathbb{R}^2$. The problem is set in an exterior two-dimensional domain which excludes a hole $\mathcal{H}$, and with zero Dirichlet data on $\mathcal{H}$. In the far field scale, $ξ_1\le |x|t^{-1/2}\le ξ_2$ with $ξ_1,ξ_2>0$, the scaled function $\log t\, u(x,t)$ behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by $J$. The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic \lq logarithmic momentum' of the solution, $\lim_{t\to\infty}\int_{\mathbb{R}^2}u(x,t)\log|x|\,dx$. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, $|x|\le t^{1/2}h(t)$ with $\lim_{t\to\infty} h(t)=0$, the scaled function $t(\log t)^2u(x,t)/\log |x|$ converges to a multiple of $ϕ(x)/\log |x|$, where $ϕ$ is the unique stationary solution of the problem that behaves as $\log|x|$ when $|x|\to\infty$. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, $|x|\ge t^{1/2} g(t)$ with $g(t)\to\infty$, the solution is proved to be of order $o((t\log t)^{-1})$.

math.AP↗

arXiv2014

Asymptotic behavior for a one-dimensional nonlocal diffusion equation in exterior domains

We study the long time behavior of solutions to the nonlocal diffusion equation $\partial_t u=J*u-u$ in an exterior one-dimensional domain, with zero Dirichlet data on the complement. In the far field scale, $ξ_1\le|x|t^{-1/2}\leξ_2$, $ξ_1,ξ_2>0$, this behavior is given by a multiple of the dipole solution for the local heat equation with a diffusivity determined by $J$. However, the proportionality constant is not the same on $\mathbb{R}_+$ and $\mathbb{R}_-$: it is given by the asymptotic first momentum of the solution on the corresponding half line, which can be computed in terms of the initial data. In the near field scale, $|x|\le t^{1/2}h(t)$, $\lim_{t\to\infty}h(t)=0$, the solution scaled by a factor $t^{3/2}/(|x|+1)$ converges to a stationary solution of the problem that behaves as $b^\pm{x}$ as $x\to\pm\infty$. The constants $b^\pm$ are obtained through a matching procedure with the far field limit. In the very far field, $|x|{\ge}t^{1/2} g(t)$, $g(t)\to\infty$, the solution has order $o(t^{-1})$.

math.AP↗

arXiv2014

Derivation of a Hele-Shaw type system from a cell model with active motion

We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit. Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because some kind of 'mushy region' prepares the tumor invasion.

math.AP↗

arXiv2013

Classical solutions and higher regularity for nonlinear fractional diffusion equations

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion $$ \partial_tu+(-Δ)^{σ/2}φ(u)=0, $$ posed for $x\in \mathbb{R}^N$, $t>0$, with $0<σ<2$, $N\ge1$. If the nonlinearity satisfies some not very restrictive conditions: $φ\in C^{1,γ}(\mathbb{R})$, $1+γ>σ$, and $φ'(u)>0$ for every $u\in\mathbb{R}$, we prove that bounded weak solutions are classical solutions for all positive times. We also explore sufficient conditions on the non-linearity to obtain higher regularity for the solutions, even $C^\infty$ regularity. Degenerate and singular cases, including the power nonlinearity $φ(u)=|u|^{m-1}u$, $m>0$, are also considered, and the existence of classical solutions in the power case is proved.

math.AP↗

arXiv2013

The Hele-Shaw asymptotics for mechanical models of tumor growth

Models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients through vasculature. The models can describe the tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case by means of a free boundary problem. Our first goal here is to formulate a free boundary model of Hele-Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain limit. A detailed mathematical analysis of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the asymptotic Hele-Shaw type problem. The main tools are nonlinear regularizing effects for certain porous medium type equations, regularization techniques à la Steklov, and a Hilbert duality method for uniqueness. At variance with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. A complete description requires the equation on the cell number density. Using this theory as a basis, we go on to consider the more complex model including nutrients. We obtain the equation for the limit of the coupled system; the method relies on some BV bounds and space/time a priori estimates. Here, new technical difficulties appear, and they reduce the generality of the results in terms of the initial data. Finally, we prove uniqueness for the system, a main mathematical difficulty.

math.AP↗

arXiv2012

Classical solutions for a logarithmic fractional diffusion equation

We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion $$ \partial_tu+(-Δ)^{1/2}\log(1+u)=0, $$ posed for $x\in \mathbb{R}$, with nonnegative initial data in some function space of $L \logL$ type. The solutions are shown to become bounded and $C^\infty$ smooth in $(x,t)$ for all positive times. We also reformulate this equation as a transport equation with nonlocal velocity and critical viscosity, a topic of current relevance. Interesting functional inequalities are involved.

math.AP↗

arXiv2011

A nonlocal one-phase Stefan problem that develops mushy regions

We study a nonlocal version of the one-phase Stefan problem which develops mushy regions, even if they were not present initially, a model which can be of interest at the mesoscopic scale. The equation involves a convolution with a compactly supported kernel. The created mushy regions have the size of the support of this kernel. If the kernel is suitably rescaled, such regions disappear and the solution converges to the solution of the usual local version of the one-phase Stefan problem. We prove that the model is well posed, and give several qualitative properties. In particular, the long-time behavior is identified by means of a nonlocal mesa solving an obstacle problem.

math.AP↗

arXiv2011

A general fractional porous medium equation

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \{ll} \dfrac{\partial u}{\partial t} + (-Δ)^{σ/2} (|u|^{m-1}u)=0, & \qquad x\in\mathbb{R}^N,\; t>0, [8pt] u(x,0) = f(x), & \qquad x\in\mathbb{R}^N.%. $$ We consider data $f\in L^1(\mathbb{R}^N)$ and all exponents $0<σ<2$ and $m>0$. Existence and uniqueness of a weak solution is established for $m> m_*=(N-σ)_+ /N$, giving rise to an $L^1$-contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range $0<m\le m_*$ existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above $m_*$. We also study the dependence of solutions on $f,m$ and $σ$. Moreover, we consider the above questions for the problem posed in a bounded domain.

math.AP↗