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Matteo Muratori

Publications and source records attributed to Matteo Muratori.

At least 37 records · Page 2Linked to original sources
arXiv2020

Uniqueness of very weak solutions for a fractional filtration equation

We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in ${\mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in [19] that if two bounded solutions $u,w$ of (1.1) satisfy $u-w\in L^1({\mathbb R}^d\times(0,T))$, then $u=w$. We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions, provided they are nonnegative. Indeed, we show that a minimal solution exists and that any other solution must coincide with it. As a consequence, distributional solutions have locally-finite energy.

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arXiv2018

Gradient flows and Evolution Variational Inequalities in metric spaces. I: structural properties

This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space $(X,\mathsf{d})$ that can be characterized by Evolution Variational Inequalities. We present new results concerning the structural properties of solutions to the $\mathrm{EVI}$ formulation, such as contraction, regularity, asymptotic expansion, precise energy identity, stability, asymptotic behaviour and their link with the geodesic convexity of the driving functional. Under the crucial assumption of the existence of an $\mathrm{EVI}$ gradient flow, we will also prove two main results: the equivalence with the De Giorgi variational characterization of curves of maximal slope and the convergence of the Minimizing Movement-JKO scheme to the $\mathrm{EVI}$ gradient flow, with an explicit and uniform error estimate of order $1/2$ with respect to the step size, independent of any geometric hypothesis (such as upper or lower curvature bounds) on $\mathsf{d}$. In order to avoid any compactness assumption, we will also introduce a suitable relaxation of the Minimizing Movement algorithm obtained by the Ekeland variational principle, and we will prove its uniform convergence as well.

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arXiv2018

The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case

We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan-Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an unexpected separate-variable behaviour that reminds one of Dirichlet problems on bounded Euclidean domains. As a crucial step, we prove existence of solutions to a related sublinear elliptic problem, a result of independent interest. Uniqueness of solutions vanishing at infinity is also shown, along with comparison principles, both in the parabolic and in the elliptic case. Our results complete previous analyses of the Porous Medium Equation flow on negatively curved Riemannian manifolds, which were carried out first for the hyperbolic space and then for general Cartan-Hadamard manifolds with a negative curvature having at most quadratic growth. We point out that no similar analysis seems to exist for the linear heat flow. We also translate such results into some weighted Porous Medium Equations in the Euclidean space having special weights.

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arXiv2018

Blow-up and global existence for the porous medium equation with reaction on a class of Cartan-Hadamard manifolds

We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite time. If $p<m$, large data blow up at worst in infinite time, and under the stronger restriction $p\in(1,(1+m)/2]$ all data give rise to solutions existing globally in time, whereas solutions corresponding to large data blow up in infinite time. The results are in several aspects significantly different from the Euclidean ones, as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation.

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arXiv2017

Porous medium equations on manifolds with critical negative curvature: unbounded initial data

We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan-Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also prove a blow-up theorem in a suitable weighted space, for initial data that grow sufficiently fast at infinity.

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arXiv2017

Smoothing effects for the filtration equation with different powers

We study the nonlinear diffusion equation $ u_t=Δϕ(u) $ on general Euclidean domains, with homogeneous Neumann boundary conditions. We assume that $ ϕ^\prime(u) $ is bounded from below by $ |u|^{m_1-1} $ for small $ |u| $ and by $ |u|^{m_2-1} $ for large $|u|$, the two exponents $ m_1,m_2 $ being possibly different and larger than one. The equality case corresponds to the well-known porous medium equation. We establish sharp short- and long-time $ L^{q_0} $-$ L^\infty $ smoothing estimates: similar issues have widely been investigated in the literature in the last few years, but the Neumann problem with different powers had not been addressed yet. This work extends some previous results in many directions.

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arXiv2016

The porous medium equation with measure data on negatively curved Riemannian manifolds

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive. We then establish uniqueness in the class of nonnegative solutions. On the other hand, we prove that any weak solution of the porous medium equation necessarily takes on as initial datum a finite Radon measure. In addition, we obtain some results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions, and properties of potentials of positive Radon measures. Such results are new and of independent interest, and are crucial for our approach.

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arXiv2016

The porous medium equation with large initial data on negatively curved Riemannian manifolds

We show existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at infinity at a prescribed rate, that depends crucially on the curvature bounds. The curvature conditions we require are sharp for uniqueness in the sense that if they are not satisfied then, in general, there can be infinitely many solutions of the Cauchy problem even for bounded data. Furthermore, under matching upper bounds on sectional curvatures, we give a precise estimate for the maximal existence time, and we show that in general solutions do not exist if the initial data grow at infinity too fast. This proves in particular that the growth rate of the data we consider is optimal for existence. Pointwise blow-up is also shown for a particular class of manifolds and of initial data.

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arXiv2016

Weighted interpolation inequalities: a perturbation approach

We study optimal functions in a family of Caffarelli-Kohn-Nirenberg inequalities with a power-law weight, in a regime for which standard symmetrization techniques fail. We establish the existence of optimal functions, study their properties and prove that they are radial when the power in the weight is small enough. Radial symmetry up to translations is true for the limiting case where the weight vanishes, a case which corresponds to a well-known subfamily of Gagliardo-Nirenberg inequalities. Our approach is based on a concentration-compactness analysis and on a perturbation method which uses a spectral gap inequality. As a consequence, we prove that optimal functions are explicit and given by Barenblatt-type profiles in the perturbative regime.

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arXiv2016

The fractional Laplacian in power-weighted $L^p$ spaces: integration-by-parts formulas and self-adjointness

We consider the fractional Laplacian operator $(-Δ)^s$ (let $ s \in (0,1) $) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the $ L^2(\mathbb{R}^d) $ scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space $ \dot{H}^s(\mathbb{R}^d) $. More precisely, we focus on functions belonging to some weighted $ L^2 $ space whose fractional Laplacian belongs to another weighted $ L^2 $ space: we prove and disprove the validity of the integration-by-parts formula depending on the behaviour of the weight $ ρ(x) $ at infinity. The latter is assumed to be like a power both near the origin and at infinity (the two powers being possibly different). Our results have direct consequences for the self-adjointness of the linear operator formally given by $ ρ^{-1}(-Δ)^s $. The generality of the techniques developed allows us to deal with weighted $ L^p $ spaces as well.

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arXiv2016

Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods

This paper is the second part of the study. In Part~I, self-similar solutions of a weighted fast diffusion equation (WFD) were related to optimal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied to radially symmetric functions. For these inequalities, the linear instability (symmetry breaking) of the optimal radial solutions relies on the spectral properties of the linearized evolution operator. Symmetry breaking in (CKN) was also related to large-time asymptotics of (WFD), at formal level. A first purpose of Part~II is to give a rigorous justification of this point, that is, to determine the asymptotic rates of convergence of the solutions to (WFD) in the symmetry range of (CKN) as well as in the symmetry breaking range, and even in regimes beyond the supercritical exponent in (CKN). Global rates of convergence with respect to a free energy (or entropy) functional are also investigated, as well as uniform convergence to self-similar solutions in the strong sense of the relative error. Differences with large-time asymptotics of fast diffusion equations without weights will be emphasized.

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arXiv2016

Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincar{é} inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.

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arXiv2016

Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities

We use the formalism of the R{é}nyi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition which determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by R{é}nyi entropy powers, and since some invariances are lost, the estimates based on the Emden-Fowler transformation have to be modified.

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arXiv2016

The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour

We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such manifolds. We also estimate the location of the free boundary. A global Harnack principle follows. We also present a change of variables that allows to transform radially symmetric solutions of the PME on model manifolds into radially symmetric solutions of a corresponding weighted PME on Euclidean space and back. This equivalence turns out to be an important tool of the theory.

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arXiv2015

Smoothing effects for the porous medium equation on Cartan-Hadamard manifolds

We prove three sharp bounds for solutions to the porous medium equation posed on Riemannian manifolds, or for weighted versions of such equation. Firstly we prove a smoothing effect for solutions which is valid on any Cartan-Hadamard manifold whose sectional curvatures are bounded above by a strictly negative constant. This bound includes as a special case the sharp smoothing effect recently proved by Vázquez on the hyperbolic space in \cite{V}, which is similar to the absolute bound valid in the case of bounded Euclidean domains but has a logarithmic correction. Secondly we prove a bound which interpolates between such smoothing effect and the Euclidean one, supposing a suitable quantitative Sobolev inequality holds, showing that it is sharp by means of explicit examples. Finally, assuming a stronger functional inequality of sub-Poincaré type, we prove that the above mentioned (sharp) absolute bound holds, and provide examples of weighted porous media equations on manifolds of infinite volume in which it holds, in contrast with the non-weighted Euclidean situation. It is also shown that sub-Poincaré inequalities cannot hold on Cartan-Hadamard manifolds.

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arXiv2015

Fractional porous media equations: existence and uniqueness of weak solutions with measure data

We prove existence and uniqueness of solutions to a class of porous media equations driven by the fractional Laplacian when the initial data are positive finite Radon measures on the Euclidean space. For given solutions without a prescribed initial condition, the problem of existence and uniqueness of the initial trace is also addressed. By the same methods we can also treat weighted fractional porous media equations, with a weight that can be singular at the origin, and must have a sufficiently slow decay at infinity (power-like). In particular, we show that the Barenblatt-type solutions exist and are unique. Such a result has a crucial role in [24], where the asymptotic behavior of solutions is investigated. Our uniqueness result solves a problem left open, even in the non-weighted case, in [42]

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arXiv2014

On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation.

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