arXiv subjects
Publications and source records attributed to Rico Zacher.
We establish the local boundedness of (sub-)solutions to nonlinear kinetic diffusion equations with $p$-growth, where the kinetic p-Laplace equation is a prototypical example. A key ingredient is the derivation of kinetic Gagliardo-Nirenberg inequalities, where the Lebesgue norm of a function is estimated in terms of its transport and diffusive directions controlled in different Lebesgue spaces.
We introduce and analyse a mathematical model describing the dynamics of particles generated by charge-exchange interactions. The model extends the well-established exchange-driven growth model, previously studied in several works, by allowing for particle densities defined on the entire integer lattice. Despite the many similarities between the two models, substantial differences arise both in their qualitative behaviour and in their mathematical analysis. Under suitable assumptions on the kernel in the collision operator, we establish global well-posedness in the class of nonnegative densities with finite first moment. Moreover, under a detailed balance condition, we investigate the structure of equilibria and analyse their stability by means of entropy methods.
These notes give a brief introduction to differential Harnack inequalities and summarise the main results of the mini-course ``Li-Yau and Harnack estimates for nonlocal diffusion problems'', presented by the author at the Seasonal School on PDEs ``Oscillation Phenomena, PDEs, and Applications: A Comprehensive School in Mathematical Analysis'', held at Ghent University in October 2025.
We show that the nonlocal Fisher information - defined as the entropy dissipation of the Boltzmann entropy for nonlocal heat equations - admits a natural lifting in the sense of Guillen and Silvestre (2025). Important examples include the discrete Fisher information arising in Markov chains and the fractional Fisher information $i_s$ associated with the fractional Laplacian $(-\Delta)^{s}$ on $\mathbb{R}^d$, $s\in (0,1)$. We further establish a Blachman-Stam inequality (BSI) for the fractional Fisher information $i_s$, and prove that, for a large class of functions, $i_s$ converges to the classical Fisher information as $s\to 1$. Through this nonlocal-to-local limit, we recover the classical BSI and the lifting property of the classical Fisher information.
In this paper we establish the Harnack inequality for globally positive local solutions to a general class of nonlocal in time subdiffusion equations in one space dimension, which includes time-fractional diffusion equations with time order less than one. It is already known that for these equations the classical Harnack inequality does not hold if the space dimension is greater than or equal to two. Here, we complete the analysis, by providing a positive result in one space dimension.
We prove duality estimates for time-fractional and more general subdiffusion problems. An important example is given by subdiffusive porous medium type equations. Our estimates can be used to prove uniqueness of weak solutions to such problems, and they allow to extend a key estimate from classical reaction-diffusion systems to the subdiffusive case. Besides concrete equations involving a Laplacian, we also consider abstract problems in a Hilbert space setting.
We construct critical trajectories in kinetic geometry, i.e. curves in $\mathbb{R}^{1+2n}$ that are: tangential to the vector fields $\partial_t+v\cdot \nabla_x$ and $\nabla_v$, connecting any two given points, respecting the underlying kinetic scaling, and with the property, that the singularity of the $v$-tangent vector near the starting point equates the degeneracy of the dependency of the curve velocity in terms of the endpoint velocity. The construction is based on Newton's laws of motion, where the ansatz for the forcing of the kinetic trajectory is the superposition of functions combining the correct power scaling with desynchronised logarithmic oscillations. These critical trajectories provide a robust and versatile ''almost exponential map'' that allows to prove several functional analytic estimates. We introduce a notion of kinetic mollification and, as an application, deduce the kinetic Sobolev inequality with optimal exponent without relying on the fundamental solution. Moreover, we establish a universal estimate for the logarithm of positive supersolutions to the Kolmogorov equation with rough coefficients inspired by the work of Moser (1961, 1964) on elliptic and parabolic problems. Combining this estimate with De Giorgi-Moser iterations and a lemma due to Bombieri and Giusti, we give an alternative proof of the (weak) Harnack inequality for the Kolmogorov equation with rough coefficients, following the ideas of Moser (1971). Our result gives the optimal range of exponents in the weak Harnack inequality and the optimal (geometric) dependency of the Harnack constant on the bounds of the diffusion matrix.
We study the regularity of weak solutions to nonlocal in time subdiffusion equations for a wide class of weakly singular kernels appearing in the generalised fractional derivative operator. We prove a weak Harnack inequality for nonnegative weak supersolutions and Holder continuity of weak solutions to such problems. Our results substantially extend the results from our previous work [12] by leaving the framework of distributed order fractional time derivatives and considering a general PC kernel and by also allowing for an inhomogeneity in the PDE from a Lebesgue space of mixed type.
We prove Li-Yau and Harnack inequalities for systems of linear reaction-diffusion equations. By introducing an additional discrete spatial variable, the system is rewritten as a scalar diffusion equation with an operator sum. For such operators in a mixed continuous and discrete setting, we introduce the hybrid curvature-dimension condition $CD_{hyb} (κ,d)$, which is a combination of the Bakry-Émery condition $CD(κ,d)$ and one of its discrete analogues, the condition $CD_Υ(κ,d)$. We establish a hybrid tensorisation principle and prove that under $CD_{hyb} (0,d)$ with $d<\infty$ a differential Harnack estimate of Li-Yau type holds, from which a Harnack inequality can be deduced by an integration argument.
We study the regularity of weak solutions to evolution equations with distributed order fractional time derivative. We prove a weak Harnack inequality for nonnegative weak supersolutions and Hölder continuity of weak solutions to this problem. Our results substantially generalise analogous known results for the problem with single order fractional time derivative.
We consider the porous medium equation (PME) on a locally finite graph and identify suitable curvature-dimension (CD) conditions under which a discrete version of the fundamental Aronson-Bénilan estimate holds true for positive solutions of the PME. We also show that these estimates allow to prove Harnack inequalities which are structurally similar to the continuous case. The new CD conditions are illustrated with several concrete examples, e.g.\ complete and chain-like graphs.
In 1971 Moser published a simplified version of his proof of the parabolic Harnack inequality. The core new ingredient is a fundamental lemma due to Bombieri and Giusti, which combines an $L^p-L^\infty$-estimate with a weak $L^1$-estimate for the logarithm of supersolutions. In this note, we give a novel proof of this weak $L^1$-estimate. The presented argument uses parabolic trajectories and does not use any Poincar\'e inequality. Moreover, the proposed argument gives a geometric interpretation of Moser's result and could allow transferring Moser's method to other equations.
In this article, we give a trajectorial proof of a kinetic Poincar\'e inequality which plays an important role in the De Giorgi-Nash-Moser theory for kinetic equations. The present work improves a result due to J. Guerand and C. Mouhot [10] in several directions. We use kinetic trajectories along the vector fields $\partial_t + v \cdot \nabla_x$ and $\partial_{v_i}$, $i = 1,\dots, d$ and do not rely on higher-order commutators such as $[\partial_{v_i},\partial_t + v \cdot \nabla_x] = \partial_{x_i}$ or on the fundamental solution. The presented method also applies to more general hypoelliptic equations. We illustrate this by studying a Kolmogorov equation with $k$ steps.
We consider operators of the form $L u(x) = \sum_{y \in \mathbb{Z}} k(x-y) \big( u(y) - u(x)\big)$ on the one-dimensional lattice with symmetric, integrable kernel $k$. We prove several results stating that under certain conditions on the kernel the operator $L$ satisfies the curvature-dimension condition $CD_Υ(0,F)$ (recently introduced by two of the authors) with some $CD$-function $F$, where attention is also paid to the asymptotic properties of $F$ (exponential growth at infinity and power-type behaviour near zero). We show that $CD_Υ(0,F)$ implies a Li-Yau inequality for positive solutions of the heat equation associated with the operator $L$. The Li-Yau estimate in turn leads to a Harnack inequality, from which we also derive heat kernel bounds. Our results apply to a wide class of operators including the fractional discrete Laplacian.
We establish a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li-Yau inequality for positive solutions $u$ to the fractional (in space) heat equation of the form $(-Δ)^{β/2}(\log u)\leq C/t$, where $β\in (0,2)$. We also illustrate our general result with an example in the discrete setting by proving a sharp Li-Yau inequality for diffusion on a complete graph.
We introduce the concept of kinetic maximal $L^p$-regularity with temporal weights and prove that this property is satisfied for the (fractional) Kolmogorov equation. We show that solutions are continuous with values in the trace space and prove, in particular, that the trace space can be characterized in terms of anisotropic Besov spaces. We further extend the property of kinetic maximal $L^p_\mu$-regularity to the Kolmogorov equation with variable coefficients. Finally, we show how kinetic maximal $L^p_\mu$-regularity can be used to obtain local existence of solutions to a class of quasilinear kinetic equations and illustrate our result with a quasilinear kinetic diffusion equation.
We prove existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator w.r.t. time belongs to the large class of ${\cal PC}$ kernels. In particular, the case of a fractional time derivative of order less than 1 is included. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type which involves function spaces defined in terms of the integro-differential operator in time. Boundedness of the solution is obtained by the De Giorgi iteration technique. Sufficiently regular solutions are shown to be unique by means of an $L_1$-contraction estimate.
We consider continuous-time (not necessarily finite) Markov chains on discrete spaces and identify a curvature-dimension inequality, the condition $CD_Υ(κ,\infty)$, which serves as a natural analogue of the classical Bakry-Émery condition $CD(κ,\infty)$ in several respects. In particular, it is tailor-made to the classical approach of proofing the modified logarithmic Sobolev inequality via computing and estimating the second time derivative of the entropy along the heat flow generated by the generator of the Markov chain. We prove that curvature bounds in the sense of $CD_Υ$ are preserved under tensorization, discuss links to other notions of discrete curvature and consider a variety of examples including complete graphs, the hypercube and birth-death processes. We further consider power type entropies and determine, in the same spirit, a natural CD condition which leads to Beckner inequalities. The $CD_Υ$ condition is also shown to be compatible with the diffusive setting, in the sense that corresponding hybrid processes enjoy a tensorization property.