arXiv subjects
Publications and source records attributed to Nikita Simonov.
We study weighted Carlson-Levin inequalities on cones, computing the optimal constant and classifying optimizers via an Entropy method. The limiting case of a logarithmic Carlson-Levin inequality is also treated. As a Corollary, we obtain a rigidity result for a weighted Gagliardo-Nirenberg inequality. We conclude with some examples involving monomial weights.
In this paper, we review recent results on stability and instability in logarithmic Sobolev inequalities, with a particular emphasis on strong norms. We consider several versions of these inequalities on the Euclidean space, for the Lebesgue and the Gaussian measures, and discuss their differences in terms of moments and stability. We give new and direct proofs, as well as examples and discuss the stability of a logarithmic uncertainty principle. Although we do not cover all aspects of the topic, we hope to contribute to establishing the state of the art.
Our focus is on the fast diffusion equation driven by the $p$-Laplacian operator, that is $\partial_t u=Δ_p u$ with $1<p<2$, posed in the whole space $\mathbb{R}^N$, $N\geq 2$. The nonnegative solutions are expected to converge in time toward a stationary profile. While such convergence had been previously established for $p$ close to $2$, no quantitative rates were known, and the asymptotic behaviour remained poorly understood across the full fast diffusion range. In fact, the long time behaviour of solutions to the $p$-Laplace Cauchy problem drastically change in different subranges of the $p$. Some of them are analysed here for the first time. In this work, we provide the convergence rates for nonnegative, integrable solutions in the so-called good fast diffusion range, $p_c=\tfrac{2N}{N+1} <p<2$, where mass is conserved. We prove that solutions converge to a self-similar profile with matching mass, with explicit rates measured in relative error. Our constructive proof is based on a new entropy method that remains effective even when the entropy is not displacement convex -- where optimal transport techniques fail. In the very fast diffusion range $1<p<p_c$, we give the first asymptotic analysis near the extinction time. We uncover new critical exponents -- especially in high dimensions -- that give rise to markedly different qualitative behaviour depending on the value of $p$. We also establish convergence rates for the gradients of radial solutions in the good fast diffusion range, again measured in relative error. Finally, we analyze the structural properties required for the entropy method to apply, thereby opening a broader investigation into the basin of attraction of Barenblatt-type profiles, particularly in the singular case of $p$ close to $1$.
We establish an optimal \emph{Widder theory} for a weighted porous medium equation with rough and inhomogeneous density that may be singular at a point and tends to zero at spatial infinity. Specifically, for this equation, we identify a class $X$ of initial measure data that give rise to very weak solutions, we show that non-negative very weak solutions necessarily admit an initial trace in $X$ at time $t=0$, and we prove that any two non-negative solutions having the same initial trace are equal. The corresponding theory for the classical (unweighted) equation was established by exploiting various properties that are not available in our weighted setting, such as the continuity of solutions, the explicit scale invariance of the equation, Aleksandrov's reflection principle, and the Aronson--Bénilan inequality. Therefore, to complete the Widder theory, we must devise several proofs by means of entirely new methods. We also establish an optimal quantitative \emph{a priori} smoothing estimate for unsigned local solutions without resorting to scale invariance, which seems to be new in this form even for the classical porous medium equation. Finally, we show that non-negative very weak solutions are always locally bounded, and in particular that they have locally finite energy.
The cutoff phenomenon, conceptualized at the origin for finite Markov chains, states that for a parametric family of evolution equations, started from a point, the distance towards a long time equilibrium may become more and more abrupt for certain choices of initial conditions, when the parameter tends to infinity. This threshold phenomenon can be seen as a critical competition between trend to equilibrium and worst initial condition. In this note, we investigate this phenomenon beyond stochastic processes, in the context of the analysis of nonlinear partial differential equations, by proving cutoff for the fast diffusion and porous medium Fokker-Planck equations on the Euclidean space, when the dimension tends to infinity. We formulate the phenomenon using quadratic Wasserstein distance, as well as using specific relative entropy and Fisher information. Our high dimensional asymptotic analysis uses the exact solvability of the model involving Barenblatt profiles. It includes the Ornstein-Uhlenbeck dynamics as a special linear case.
The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods.
This paper is devoted to stability results for the Gaussian logarithmic Sobolev inequality, with explicit stability constants.
We consider Gagliardo-Nirenberg inequalities on the sphere which interpolate between the Poincaré inequality and the Sobolev inequality, and include the logarithmic Sobolev inequality as a special case. We establish explicit stability results in the subcritical regime using spectral decomposition techniques, and entropy and carré du champ methods applied to nonlinear diffusion flows.
This paper is devoted to Gaussian interpolation inequalities with endpoint cases corresponding to the Gaussian Poincaré and the logarithmic Sobolev inequalities, seen as limits in large dimensions of Gagliardo-Nirenberg-Sobolev inequalities on spheres. Entropy methods are investigated using not only heat flow techniques but also nonlinear diffusion equations as on spheres. A new stability result is established for the Gaussian measure, which is directly inspired by recent results for spheres.
We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy - entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case.
We study weighted inequalities of Hardy and Hardy-Poincaré type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile.
We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=Δ_p u$ on the whole Euclidean space, in the so-called "good fast diffusion range" $\tfrac{2N}{N+1}<p<2$. It is well-known that non-negative solutions behave for large times as $\mathcal{B}$, the Barenblatt (or fundamental) solution, which has an explicit expression. We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of $\mathcal{B}$. This can be considered the nonlinear counterpart of the celebrated Gaussian estimates for the linear heat equation. We characterize the maximal (hence optimal) class of initial data such that the GHP holds, by means of an integral tail condition, easy to check. The GHP is then used as a tool to analyze the fine asymptotic behavior for large times. For initial data that satisfy the same integral condition, we prove that the corresponding solutions behave like the Barenblatt with the same mass, uniformly in relative error. When the integral tail condition is not satisfied we show that both the GHP and the uniform convergence in relative error, do not hold anymore, and we provide also explicit counterexamples. We then prove a "generalized GHP", that is, pointwise upper and lower bounds in terms of explicit profiles with a tail different from $\mathcal{B}$. Finally, we derive sharp global quantitative upper bounds of the modulus of the gradient of the solution, and, when data are radially decreasing, we show uniform convergence in relative error for the gradients. To the best of our knowledge, analogous issues for the linear heat equation $p=2$, do not possess such clear answers, only partial results are known.
This document comes as supplementary material of the paper Stability in Gagliardo-Nirenberg inequalities by the same authors. It is intended to state a number of classical or elementary statements concerning constants and inequalities for which we are not aware of existing published material or expressions detailed enough for our purpose. We claim no originality on the theoretical results and rely on standard methods in most cases, except that we keep track of the constants and provide constructive estimates.
We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the Fast Diffusion Equation with weights (WFDE) $u_t=|x|^γ\mathrm{div}\left(|x|^{-β}\nabla u^m\right)$ posed on $(0,+\infty)\times\mathbb{R}^d$, with $d\ge 3$, in the so-called good fast diffusion range $m_c<m<1$, within the range of parameters $γ, β$, optimal for the validity of the so-called Caffarelli-Kohn-Nirenberg inequalities. It is a natural question to ask in which sense such solutions behave like the Barenblatt $\mathfrak{B}$ (fundamental solution): for instance, asymptotic convergence, i.e. $\|u(t)-\mathfrak{B}(t)\|_{{\rm L}^p(\mathbb{R}^d)}\xrightarrow[]{t\to\infty}0$, is well known for all $1\le p\le \infty$, while only few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data $\mathcal{X}\subset{\rm L}^1_+(\mathbb{R}^d)$ that produces solutions which are pointwise trapped between two Barenblatt (Global Harnack Principle), and uniformly converge in relative error (UREC), i.e. ${\rm d}_\infty(u(t))=\|u(t)/\mathcal{B}(t)-1\|_{{\rm L}^\infty(\mathbb{R}^d)}\xrightarrow[]{t\to\infty}0$. Such characterization is in terms of an integral condition on $u(t=0)$. To the best of our knowledge, analogous issues for the linear heat equation $m=1$, do not possess such clear answers. Our characterization is also new for the classical, non-weighted, FDE. We are able to provide minimal rates of convergence to $\mathcal{B}$ in different norms. Such rates are almost optimal in the non weighted case, and become optimal for radial solutions. To complete the panorama, we show that solutions with data in ${\rm L}^1_+(\mathbb{R}^d)\setminus\mathcal{X}$, preserve the same "fat" spatial tail for all times, hence UREC fails.
We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation $u_t = |x|^γ \nabla\cdot (|x|^{-β} \nabla u^m)$, with $0 < m <1$ posed on cylinders of $(0,T)\times{\mathbb R}^N$. The weights $|x|^γ$ and $|x|^{-β}$, with $γ< N$ and $γ-2 < β\leq γ(N-2)/N$ can be both degenerate and singular and need not belong to the class $\mathcal{A}_2$, a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting. The weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. We therefore prove quantitative - with computable constants - upper and lower estimates for local weak solutions, focussing our attention where a change of geometry appears. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, we obtain Hölder continuity of the solutions, with a quantitative (even if non-optimal) exponent. Our results apply to a quite large variety of solutions and problems. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper. Our techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, $m=1$, we also prove quantitative estimates, recovering known results in some cases and extending such results to a wider class of weights.