arXiv subjects

Lucas C. F. Ferreira

Publications and source records attributed to Lucas C. F. Ferreira.

At least 37 records · Page 2Linked to original sources

arXiv2019

On the two-power nonlinear Schrödinger equation with non-local terms in Sobolev-Lorentz spaces

We are concerned with the two-power nonlinear Schrödinger-type equations with non-local terms. We consider the framework of Sobolev-Lorentz spaces which contain singular functions with infinite-energy. Our results include global existence, scattering and decay properties in this singular setting with fractional regularity index. Solutions can be physically realized because they have finite local $L^2$-mass. Moreover, we analyze the asymptotic stability of solutions and, although the equation has no scaling, show the existence of a class of solutions asymptotically self-similar w.r.t. the scaling of the single-power NLS-equation. Our results extend and complement those of [F. Weissler, ADE 2001], particularly because we are working in the larger setting of Sobolev-weak-$L^p$ spaces and considering non-local terms. The two nonlinearities of power-type and the generality of the non-local terms allow us to cover in a unified way a large number of dispersive equations and systems.

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arXiv2019

Global smoothness for a 1D supercritical transport model with nonlocal velocity

We are concerned with a nonlocal transport 1D-model with supercritical dissipation $γ\in(0,1)$ in which the velocity is coupled via the Hilbert transform, namely the so-called CCF model. This model arises as a lower dimensional model for the famous 2D dissipative quasi-geostrophic equation and in connection with vortex-sheet problems. It is known that its solutions can blow up in finite time when $γ\in(0,1/2)$. On the other hand, as stated by Kiselev (2010), in the supercritical subrange $γ\in\lbrack1/2,1)$ it is an open problem to know whether its solutions are globally regular. We show global existence of non-negative $H^{3/2}$-strong solutions in a supercritical subrange (close to 1) that depends on the initial data norm. Then, for each arbitrary smooth non-negative initial data, the model has a unique global smooth solution provided that $γ\in\lbrackγ_{1},1)$ where $γ_{1}$ depends on the $H^{3/2}$-initial data norm. Our approach is inspired on that of Coti Zelati and Vicol (IUMJ, 2016).

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arXiv2019

Global well-posedness and asymptotic behavior in Besov-Morrey spaces for chemotaxis-Navier-Stokes fluids

In this work we consider the Keller-Segel system coupled with Navier-Stokes equations in $\mathbb{R}^{N}$ for $N\geq2$. We prove the global well-posedness with small initial data in Besov-Morrey spaces. Our initial data class extends previous ones found in the literature such as that obtained by Kozono-Miura-Sugiyama (J. Funct. Anal. 2016). It allows to consider initial cell density and fluid velocity concentrated on smooth curves or at points depending on the spatial dimension. Self-similar solutions are obtained depending on the homogeneity of the initial data and considering the case of chemical attractant without degradation rate. Moreover, we analyze the asymptotic stability of solutions at infinity and obtain a class of asymptotically self-similar ones.

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arXiv2017

Global well-posedness and asymptotic behavior for Navier-Stokes-Coriolis equations in homogeneous Besov spaces

We are concerned with the $3$D-Navier-Stokes equations with Coriolis force. Existence and uniqueness of global solutions in homogeneous Besov spaces are obtained for large speed of rotation. In the critical case of the regularity, we consider a suitable initial data class whose definition is based on the Stokes-Coriolis semigroup and Besov spaces. Moreover, we analyze the asymptotic behavior of solutions in that setting as the speed of rotation goes to infinity.

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arXiv2017

Global existence for an attraction-repulsion chemotaxis fluid model with logistic source

We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of RN , N = 2, 3, for small initial data in Lp-spaces.

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arXiv2017

On the 3D Euler equations with Coriolis force in borderline Besov spaces

We consider the 3D Euler equations with Coriolis force (EC) in the whole space. We show long-time solvability in Besov spaces for high speed of rotation $Ω$ and arbitrary initial data. For that, we obtain $Ω$-uniform estimates and a blow-up criterion of BKM type in our framework. Our initial data class is larger than previous ones considered for (EC) and covers borderline cases of the regularity. The uniqueness of solutions is also discussed.

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arXiv2017

Besov-weak-Herz spaces and global solutions for Navier-Stokes equations

We consider the incompressible Navier-stokes equations (NS) in $\mathbb{R}^{n}$ for $n\geq2$. Global well-posedness is proved in critical Besov-weak-Herz spaces (BWH-spaces) that consist in Besov spaces based on weak-Herz spaces. These spaces are larger than some critical spaces considered in previous works for (NS). For our purposes, we need to develop a basic theory for BWH-spaces containing properties and estimates such as heat semigroup estimates, embedding theorems, interpolation properties, among others. In particular, it is proved a characterization of Besov-weak-Herz spaces as interpolation of Sobolev-weak-Herz ones, which is key in our arguments. Self-similarity and asymptotic behavior of solutions are also discussed. Our class of spaces and its properties developed here could also be employed to study other PDEs of elliptic, parabolic and conservation-law type.

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arXiv2016

Minimizing Movement for a Fractional Porous Medium Equation in a Periodic Setting

We consider a fractional porous medium equation that extends the classical porous medium and fractional heat equations. The flow is studied in the space of periodic probability measures endowed with a non-local transportation distance constructed in the spirit of the Benamou-Brenier formula. For initial periodic probability measures, we show the existence of absolutely continuous curves that are generalized minimizing movements associated to Rényi entropy. For that, we need to obtain entropy and distance properties and to develop a subdifferential calculus in our setting.

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arXiv2015

Gradient flows of time-dependent functionals in metric spaces and applications for PDEs

We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the Wasserstein space $\mathscr{P}_{2}(\mathbb{R}^{d})$ and apply the results for a large class of PDEs with time- dependent coefficients like confinement and interaction potentials and diffusion. Our results can be seen as an extension of those in Ambrosio-Gigli-Savaré (2005)[2] to the case of time-dependent functionals. For that matter, we need to consider some residual terms, time-versions of concepts like $λ$-convexity, time-differentiability of minimizers for Moreau-Yosida approximations, and a priori estimates with explicit time-dependence for De Giorgi interpolation. Here, functionals can be unbounded from below and satisfy a type of $λ$-convexity that changes as the time evolves.

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arXiv2015

Existence and stability of global large strong solutions for the Hall-MHD system

We consider the 3D incompressible Hall-MHD system and prove a stability theorem for global large solutions under a suitable integrable hypothesis in which one of the parcels is linked to the Hall term. As a byproduct, a class of global strong solutions is obtained with large velocities and small initial magnetic fields. Moreover, we prove the local-in-time well-posedness of $H^{2}$-strong solutions which improves previous regularity conditions on initial data.

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arXiv2014

On the heat equation with nonlinearity and singular anisotropic potential on the boundary

This paper concerns with the heat equation in the half-space $\mathbb{R}_{+}^{n}$ with nonlinearity and singular potential on the boundary $\partial\mathbb{R}_{+}^{n}$. We develop a well-posedness theory (without using Kato and Hardy inequalities) that allows us to consider critical potentials with infinite many singularities and anisotropy. Motivated by potential profiles of interest, the analysis is performed in weak $L^{p}$-spaces in which we prove key linear estimates for some boundary operators arising from the Duhamel integral formulation in $\mathbb{R}_{+}^{n}$. Moreover, we investigate qualitative properties of solutions like self-similarity, positivity and symmetry around the axis $\overrightarrow{Ox_{n}}$.

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arXiv2014

Periodic solutions for a 1D-model with nonlocal velocity via mass transport

This paper concerns periodic solutions for a 1D-model with nonlocal velocity given by the periodic Hilbert transform. There is a rich literature showing that this model presents singular behavior of solutions via numerics and mathematical approaches. For instance, they can blow up by forming mass-concentration. We develop a global well-posedness theory for periodic measure initial data that allows, in particular, to analyze how the model evolves from those singularities. Our results are based on periodic mass transport theory and the abstract gradient flow theory in metric spaces developed by Ambrosio et al. [2]. A viscous version of the model is also analyzed and inviscid limit properties are obtained.

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arXiv2014

Existence and symmetry for elliptic equations in R^n with arbitrary growth in the gradient

We study the semilinear elliptic equation $Δu + g(x,u,Du) = 0$ in $\R^n$. The nonlinearities $g$ can have arbitrary growth in $u$ and $Du$, including in particular the exponential behavior. No restriction is imposed on the behavior of $g(x,z,p)$ at infinity except in the variable $x$. We obtain a solution $u$ that is locally unique and inherits many of the symmetry properties of $g$. Positivity and asymptotic behavior of the solution are also addressed. Our results can be extended to other domains like half-space and exterior domains. We give some examples.

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arXiv2014

Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces

We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of high-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier-Besov-Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large $L^{2}$-norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and couplings. Also, we show that solutions are asymptotically self-similar at infinity. Our results can be applied in a unified way for a number of active scalar PDEs like 1D models on dislocation dynamics in crystals, Burguer's equations, 2D vorticity equation, 2D generalized SQG, 3D magneto geostrophic equations, among others.

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arXiv2014

Global well-posedness and symmetries for dissipative active scalar equations with positive-order couplings

We consider a family of dissipative active scalar equations outside the $L^{2}$-space. This was introduced in [D. Chae, P. Constantin, J. Wu, to appear in IUMJ (2014)] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. We prove global well-posedness and time decay of solutions, without smallness assumptions, for initial data belonging to the critical Lebesgue space $L^{\frac{n}{γ-β}}(\mathbb{R}^{n})$ which is a class larger than that of the above reference. Symmetry properties of solutions are investigated depending on the symmetry of initial data and coupling operators.

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arXiv2013

On the Schrödinger equation with singular potentials

We study the Cauchy problem for the non-linear Schrödinger equation with singular potentials. For point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-L^{p} spaces. Specific interest is give to the point-like $δ$ and $δ'$ impurity and for two $δ$-interactions in one dimension. We also consider the periodic case which is analyzed in a functional space based on Fourier transform and local-in-time well-posedness is proved.

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arXiv2013

An approach without using Hardy inequality for the linear heat equation with singular potential

The aim of this paper is to employ a strategy known from fluid dynamics in order to provide results for the linear heat equation $u_{t}-Δu-V(x)u=0$ in $\mathbb{R}^{n}$ with singular potentials. We show well-posedness of solutions, without using Hardy inequality, in a framework based in the Fourier transform, namely $PM^{k}$-spaces. For arbitrary data $u_{0}\in PM^{k}$, the approach allows to compute an explicit smallness condition on $V$ for global existence in the case of $V$ with finitely many inverse square singularities. As a consequence, well-posedness of solutions is obtained for the case of the monopolar potential $V(x)=\fracλ{\left|x\right|^{2}}$ with $\left|λ\right|<λ_{\ast}=\frac{(n-2)^{2}}{4}$. This threshold value is the same one obtained for the global well-posedness of $L^{2}$-solutions by means of Hardy inequalities and energy estimates. Since there is no any inclusion relation between $L^{2}$ and $PM^{k}$, our results indicate that $λ_{\ast}$ is intrinsic of the PDE and independent of a particular approach. We also analyze the long time behavior of solutions and show there are infinitely many possible asymptotics characterized by the cells of a disjoint partition of the initial data class $PM^{k}$.

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arXiv2013

On nonlinear Schrödinger equations with random potentials: existence and probabilistic properties

In this paper we are concerned with nonlinear Schrödinger equations with random potentials. Our class includes continuum and discrete potentials. Conditions on the potential $V_ω$ are found for existence of solutions almost sure $ω$. We study probabilistic properties like central limit theorem and law of larger numbers for the obtained solutions by independent ensembles. We also give estimates on the expected value for the $L^{\infty}$-norm of the solution showing how it depends on the size of the potential.

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