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Maria Gualdani

Publications and source records attributed to Maria Gualdani.

16 recordsLinked to original sources
arXiv2024

A quasilinear Keller-Segel model with saturated discontinuous advection

We consider the singular limit of a chemotaxis model of bacterial collective motion recently introduced in arXiv:2009.11048 [math.AP]. The equation models aggregation-diffusion phenomena with advection that is discontinuous and depends sharply on the gradient of the density itself. The quasi-linearity of the problem poses major challenges in the construction of the solution and complications arise in the proof of regularity. Our method overcomes these obstacle by relying solely on entropy inequalities and the theory of monotone operators. We provide existence, uniqueness and smoothing estimates in any dimensional space.

math.AP↗
arXiv2024

Global smooth solutions to the Landau-Coulomb equation in $L^{3/2}$

We consider the homogeneous Landau equation in $\mathbb{R}^3$ with Coulomb potential and initial data in polynomially weighted $L^{3/2}$. We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre, yields the existence of a global-in-time smooth solution. Additionally, if the initial data belongs to $L^p$ with $p>3/2$, there is a unique solution. At the crux of the result is a new $\varepsilon$-regularity criterion in the spirit of the Caffarelli-Kohn-Nirenberg theorem: a solution which is small in weighted $L^{3/2}$ is regular. Although the $L^{3/2}$ norm is a critical quantity for the Landau-Coulomb equation, using this norm to measure the regularity of solutions presents significant complications. For instance, the $L^{3/2}$ norm alone is not enough to control the $L^\infty$ norm of the competing reaction and diffusion coefficients. These analytical challenges caused prior methods relying on the parabolic structure of the Landau-Coulomb to break down. Our new framework is general enough to handle slowly decaying and singular initial data, and provides the first proof of global well-posedness for the Landau-Coulomb equation with rough initial data.

math.AP↗
arXiv2024

Time-Fractional Porous Medium Type Equations. Sharp Time Decay and Regularization

We consider a class of porous medium type of equations with Caputo time derivative. The prototype problem reads as $\Dc u=-\A u^m$ and is posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with zero Dirichlet boundary conditions. The operator $\A$ falls within a wide class of either local or nonlocal operators, and the nonlinearity is allowed to be of degenerate or singular type, namely, $0 1$. This equation is the most general form of a variety of models used to describe anomalous diffusion processes with memory effects, and finds application in various fields, including visco-elastic materials, signal processing, biological systems and geophysical science. We show existence of unique solution and new $L^p-L^\infty$ smoothing effects. The comparison principle, which we provide in the most general setting, serves as a crucial tool in the proof and provides a novel monotonicity formula. Consequently, we establish that the regularizing effects from the diffusion are stronger than the memory effects introduced by the fractional time derivative. Moreover, the solution attains the boundary conditions pointwise. Finally, we prove that the solution does not vanish in finite time if $0 0$. Our findings indicate that memory effects weaken the spatial diffusion and mitigate the difference between slow and fast diffusion.

math.AP↗
arXiv2023

Regularization estimates of the Landau-Coulomb diffusion

The Landau-Coulomb equation is an important model in plasma physics featuring both nonlinear diffusion and reaction terms. In this manuscript we focus on the diffusion operator within the equation by dropping the potentially nefarious reaction term altogether. We show that the diffusion operator in the Landau-Coulomb equation provides a much stronger L^1 to L^\infty rate of regularization than its linear counterpart, the Laplace operator. The result is made possible by a nonlinear functional inequality of Gressman, Krieger, and Strain together with a De Giorgi iteration. This stronger regularization rate illustrates the importance of the nonlinear nature of the diffusion in the analysis of the Landau equation and raises the question of determining whether this rate also happens for the Landau-Coulomb equation itself.

math.AP↗
arXiv2023

Nonlinear regularization estimates and global well-posedness for the Landau-Coulomb equation near equilibrium

We consider the Landau equation with Coulomb potential in the spatially homogeneous case. We show short time propagation of smallness in $L^p$ norms for $p>3/2$ and instantaneous regularization in Sobolev spaces. This yields new short time quantitative a priori estimates that are unconditional near equilibrium. We combine these estimates with existing literature on global well-posedness for regular data to extend the well-posedness theory to small $L^p$ data with $p$ arbitrarily close to $3/2$. The threshold $p = 3/2$ agrees with previous work on conditional regularity for the Landau equation in the far from equilibrium regime. In light of the monotonicity of the Fisher information shown in the recent preprint [arXiv:2311.09420], our primary nonlinear regularization estimate holds even in the far-from-equilibrium regime. As a consequence, we obtain exponential convergence to equilibrium for suitably localized solutions in every Sobolev norm.

math.AP↗
arXiv2021

Hardy's inequality and (almost) the Landau equation

In this manuscript we establish an $L^\infty$ estimate for the isotropic analogue of the homogeneous Landau equation. This is done for values of the interaction exponent $γ$ in (a part of) the range of very soft potentials. The main observation in our proof is that the classical weighted Hardy inequality leads to a weighted Poincaré inequality, which in turn implies the propagation of some $L^p$ norms of solutions. From here, the $L^\infty$ estimate follows from certain weighted Sobolev inequalities and De Giorgi-Nash-Moser theory.

math.AP↗
arXiv2020

Existence of weak solutions to a continuity equation with space time nonlocal Darcy law

In this manuscript we consider a porous medium equation with non-local diffusion effects given by a fractional heat operator $\partial_t + (-Δ)^s$ in two space dimensions. Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality, a higher order integral estimate, and a generalized version of the Div-Curl lemma.

math.AP↗
arXiv2018

Longtime behavior and weak-strong uniqueness for a nonlocal porous media equation

In this manuscript we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator \begin{equation*} \partial_t u = \mbox{div}(u\nabla p),\qquad \partial_t p = -(-Δ)^s p + u^2, \end{equation*} in three space dimensions for $3/4\le s < 1$ and analyze the long time asymptotics. The proof is based on energy methods and leads to algebraic decay towards the stationary solution $u=0$ and $\nabla p=0$ in the $L^2(\mathbb{R}^3)$-norm. The decay rate depends on the exponent $s$. We also show weak-strong uniqueness of solutions and continuous dependence from the initial data. As a side product of our analysis we also show that existence of weak solutions, previously shown in [Caffarelli, Gualdani, Zamponi 2018] for $3/4\le s \le 1$, holds for $1/2 < s\le 1$ if we consider our problem in the torus.

math.AP↗
arXiv2018

On $A_p$ weights and the Landau equation

In this manuscript we investigate the regularization of solutions for the spatially homogeneous Landau equation. For moderately soft potentials, it is shown that weak solutions become smooth instantaneously and stay so over all times, and the estimates depend only on the initial mass, energy, and entropy. For very soft potentials we obtain a conditional regularity result, hinging on what may be described as a nonlinear Morrey space bound, assumed to hold uniformly over time. This bound always holds in the case of moderately soft potentials, and nearly holds for general potentials, including Coulomb. This latter phenomenon captures the intuition that for moderately soft potentials, the dissipative term in the equation is of the same order as the quadratic term driving the growth (and potentially, singularities). In particular, for the Coulomb case, the conditional regularity result shows a rate of regularization much stronger than what is usually expected for regular parabolic equations. The main feature of our proofs is the analysis of the linearized Landau operator around an arbitrary and possibly irregular distribution. This linear operator is shown to be a degenerate elliptic Schrödinger operator whose coefficients are controlled by $A_p$-weights.

math.AP↗
arXiv2017

Global existence of weak even solutions for an isotropic Landau equation with Coulomb potential

In this manuscript we consider an isotropic modification for the Landau equation with Coulomb potential in three space dimensions. Global in time existence of weak solutions for even initial data is shown by employing a time semi-discretization of the equation, an entropy inequality and a uniform estimate for the second moment of the solution to the discretized problem. Moreover, under an additional condition that has to be satisfied uniformly over time, uniform boundedness of the solution is proved, with bounds depending solely on the mass, second moment and entropy of the solution. A byproduct of our analysis is a proof of improved regularity for weak solutions to the Landau equation with Coulomb potential.

math.AP↗
arXiv2017

A review for an isotropic Landau model

We consider the equation $u_t = \mbox{Div}(a[u]\nabla u - u\nabla a[u])$, $-Δa = u$. This model has attracted some attention in the recents years and several results are available in the literature. We review recent results on existence and smoothness of solutions and explain the open problems.

math.AP↗
arXiv2017

Spectral gap and exponential convergence to equilibrium for a multi-species Landau system

In this paper we prove new constructive coercivity estimates and convergence to equilibrium for a spatially non-homogeneous system of Landau equations with soft potentials. We show that the nonlinear collision operator conserves each species' mass, total momentum, total energy and that the Boltzmann entropy is nonincreasing along solutions of the system. The entropy decay vanishes if and only if the Boltzmann distributions of the single species are Maxwellians with the same momentum and energy. A linearization of the collision operator is computed, which has the same conservation properties as its nonlinear counterpart. We show that the linearized system dissipates a quadratic entropy, and prove existence of spectral gap and exponential decay of the solution towards the global equilibrium.

math-ph↗
arXiv2014

Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential

Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow up in the $L^\infty$-norm at a finite time $T$ can occur only if the $L^{3/2}$-norm of the solution concentrates for times close to $T$. The bounds are obtained using the comparison principle for the Landau equation and for the associated mass function. This method provides long-time existence results for the isotropic version of the Landau equation with Coulomb potential, recently introduced by Krieger and Strain.

math.AP↗
arXiv2014

Global well-posedness for the homogeneous Landau equation

Global well-posedness and exponential decay to equilibrium are proved for the homogeneous Landau equation from kinetic theory. The initial distribution is only assumed to be bounded and decaying sufficiently fast at infinity. In particular, discontinuous initial configurations that might be far from equilibrium are covered. Despite the lack of a comparison principle for the equation, the proof of existence relies on barrier arguments and parabolic regularity theory. Uniqueness and decay to equilibrium are then obtained through weighted integral inequalities. Although the focus is on the spatially homogeneous case with Coulomb potential, the methods introduced here may be applied elsewhere in nonlinear kinetic theory.

math.AP↗
arXiv2012

A discrete Bernoulli free boundary problem

We consider a free boundary problem for the $p$-Laplace operator which is related to the so-called Bernoulli free boundary problem. In this formulation, the classical boundary gradient condition is replaced by a condition on the distance between two different level surfaces of the solution. For suitable scalings our model converges to the classical Bernoulli problem; one of the advantages in this new formulation lies in the simplicity of the arguments, since one does not need to consider the boundary gradient. We shall study this problem in convex and other regimes, and establish existence and qualitative theory.

math.AP↗