Papers

The aim of the book is to present some recent results in the theory of stochastic Itô equations with singular deterministic part (drift) and its applications to second-order elliptic and parabolic equations with singular first-order coefficients. The singularity is characterized by means of Morrey spaces and this allows for much more singular coefficients than those from Lebesgue spaces. For instance, first-order coefficients having behavior like $1/|x|$ near the origin are allowed. In the first part of the book we are dealing with equations having just measurable coefficients and treat the Markov diffusion time-inhomogeneous processes $X$ corresponding to parabolic operators. In particular, mixed-norm parabolic Aleksandrov estimates, Harnack inequality and Hölder continuity of $X$-caloric functions are investigated. This produces the corresponding results in PDEs such as extended Aleksandrov maximum principle, Harnack inequality and Hölder continuity of PDE-caloric functions. In two remaining chapters we concentrate on weak and strong solutions of Itô equations which requires some regularity restrictions on the diffusion matrix (or second-order coefficients in the PDE language). We give the best to date conditions in terms of Morrey spaces for the existence and uniqueness of weak and strong solutions of Itô equations with singular drift. The majority of our main results are new even if the drift part is zero.

We show that the leading-order term in the late-time asymptotics of solutions to the linear wave equation on radially symmetric stationary perturbations of $(2 + 1)$-dimensional Minkowski space is proportional to $u^{-1/2}v^{-1/2}$ (which solves the wave equation on Minkowski space), where $u$ and $v$ are double null coordinates. Our proof adapts the physical space techniques in the work of Gajic (arXiv:2203.15838) on the wave equation with an inverse-square potential on the Schwarzschild spacetime. In particular, we extend the $r^p$-weighted energy estimates of Dafermos–Rodnianski (arXiv:0910.4957) to two space dimensions.

The aim of this work is to establish an existence and uniqueness solution for spatiocharacteristic second-order quasilinear hyperbolic problems in Sobolev type spaces with weights to clarify and complete the previous work done by H. Muller Zum Hagen and H.J. Seifert, Gen. Rel. and Gravit. 1977. We use this result in P. G. Louokdom tamto, PhD thesis ongoing, 2026 to establish a semi-global existence and uniqueness result for second-order quasilinear Goursat problems where the coefficients of the second derivatives depend linearly on the unknown in weighted Sobolev-type spaces, which we will apply to the harmonic gauge vacuum Einstein equations

We consider, on a connected metric graph $\mathcal{G}$, a family of nonlinear Schrödinger equations $$ -u'' + W_n(x) u + \lambda_n u = \rho_n(x)|u|^{p-2}u, \quad n \in \mathbb{N}. \qquad (*) $$ We assume that $p > 2$, $(W_n)$, $(\rho_n) \subseteq L^{\infty}(\mathcal{G})$ with $\rho_n \geq 0$, $|W_n|_{L^\infty(\mathcal{G})}$ and $|\rho_n|_{L^\infty(\mathcal{G})}$ are bounded and $\lambda_n \to +\infty$. Given $n \in \mathbb{N}$, we call "solution" a function $u_n \in H^1(\mathcal{G})$ which satisfies (*) for that $n\in \mathbb{N}$ together with the Kirchhoff conditions at the vertices. Focusing on the limiting behavior of sequences $(u_n) \subseteq H^1(\mathcal{G})$ of solutions as $\lambda_n \to + \infty$ and assuming that the Morse index $m(u_n)$ of $u_n$ is uniformly bounded, we establish, the existence of a finite subset of blow-up points away from which, up to a subsequence, $|u_n|$ has a global exponential decay. These points are generally a strict subset of the blow-up points, and their number is estimated by the bound on the Morse index of $(u_n)$. It is the first time that this global exponential decay property is established on graphs even if one consider only signed solutions. In the last part of the paper we derive various results of a priori bounds on the solutions in $L^\infty$ and $L^2$. Our blow-up analysis, combined with ODE arguments allows, for frequently considered classes of graphs, to obtain a fairly complete picture of the relationships between the number of nodal regions, Morse index, $L^\infty$ and $L^2$ norms of solutions.

We prove optimal boundary $C^{1,\alpha}$ regularity for viscosity solutions of degenerate fully nonlinear uniformly elliptic equations with oblique boundary conditions and Hamiltonian terms of the form \[ \begin{cases} |Du|^{\gamma}F(D^2 u) + \varrho(x)|Du|^{\sigma} = f(x) & \text{in } \Omega,\\ \beta(x)\cdot Du+\zeta(x)u = g(x) & \text{on } \partial \Omega, \end{cases} \] where $\gamma>0$ and $0<\sigma\le 1+\gamma$. We develop a compactness framework for affine translations, linking the size of the translation to the Hamiltonian structure. This is combined with a boundary improvement-of-flatness argument adapted to oblique boundary data, yielding the optimal boundary regularity.

We develop a unified analytical and computational framework for the generalized Abel ordinary differential equation $y^{\prime }(x)=a_n(x)\bigl(% y^n+\lambda_{n-1}(x)y^{n-1}+\dots+\lambda_0(x)\bigr)$ of arbitrary degree $% n\ge1$ on the unbounded interval $[x_0,\infty)$. Under mild structural hypotheses on the coefficients and on the existence of a stable moving equilibrium branch $E(x)$, we prove a new Asymptotic Plateau Theorem establishing that the solution issued from $y(x_0)=0$ is globally defined, strictly monotone, trapped between zero and $E(x)$, and converges to a finite positive limit $L=\lim_{x\to\infty}E(x)$. We further obtain an explicit, computable rate of convergence and a degree-reduction principle that generalizes the classical Liouville substitution. The theory is complemented by a high-order Radau IIA implementation whose output reproduces the predicted plateaus to nine significant digits. A detailed application to a generalized Merton structural credit-risk model derives an Abel-type equation for the long-maturity state profile of the credit spread and illustrates the economic relevance of the framework.

In this paper, we establish a quantitative weak unique continuation theorem on an annular domain for a backward degenerate parabolic equation with a degenerate interior point. Our methodology hinges on approximating the solution of the degenerate parabolic equation through solutions of non-degenerate parabolic counterparts. Subsequently, we establish Carleman estimates for the non-degenerate parabolic equation across two separate domains. By virtue of these estimates, we deduce a quantitative weak unique continuation property for the degenerate parabolic equation, thereby substantiating the weak unique continuation result for the original degenerate parabolic equation.

In this paper, we focus on two types of degenerate partial differential equations: a degenerate elliptic equation and a degenerate parabolic equation. Significantly, both categories are characterized by the same principal operator. To obtain solutions for these equations, we introduce a novel approximation approach, termed the shape design approximation. As a practical application of this method, we derive a Carleman estimate for the backward degenerate parabolic equation. This estimate plays a pivotal role in establishing the null controllability of the degenerate parabolic equation. A notable advantage of employing the shape design approximation in deriving the Carleman estimate is that it enables us to bypass the requirement for second order derivatives in the degenerate equation. Usually, this has been a significant obstacle in the derivation of Carleman estimates for degenerate parabolic equations.

This paper addresses the inverse source problem for a mixed-type fractional wave-diffusion-wave equation posed in a cylindrical domain. The governing equation involves a time-dependent variable-order fractional derivative, which enables the model to effectively capture temporal transitions between wave-like and diffusive behaviors. The solution is constructed in the form of a Fourier-Bessel series. By employing the method of separation of variables together with fundamental properties of Bessel functions, we analyze the uniform convergence of the resulting infinite series. This analysis ultimately leads to a rigorous proof of the existence of a solution.

We propose a new approach for crowd motion models where the density constraint can only slow down the motion of each agent, with no effect on those agents who are not in a saturated area or who have no saturated density ”in front” of them. This is done by means of a limit of conservation laws inspired by the equations used for traffic as in Follow the leader-type models. We study the asymptotics of the solutions of these conservation laws in a certain asymptotic regime, and obtain a PDE at the limit of a whole new type. One of the main goals of the paper is to prove uniform BV estimates on the density, and thus strong compactness to prove the existence of solutions to this limit equation. We also discuss the qualitative behavior of solutions, provide numerical illustrations both in dimension 1 and 2, and establish the new entropy inequalities associated with this limit equation.

Existence of solutions to an obstacle $p$-Laplacian problem exhibiting a singular, discontinuous reaction is proved. The reaction term may be discontinuous in a Lebesgue-negligible set. Moreover, solutions are shown to be locally $C^{1,\alpha}$ far away from the contact set. Under a differentiability hypothesis on the obstacle, solutions belong to $C^{1,\alpha}(\overline{\Omega})$.

In this paper, we develop a general homogenization theory for elliptic equations with coefficients that oscillate periodically at infinitely many scales $\varepsilon = (\varepsilon_1, \varepsilon_2, \cdots) \in (0,1)^\infty$, with $\varepsilon_1>\varepsilon_2>\cdots$ and $\varepsilon_n \to 0$ as $n \to \infty$. Such problems arise naturally in the study of fractal materials and diffusion in fluids. Under suitable scale-separation assumptions, we prove a qualitative homogenization theorem and obtain optimal $L^2$ convergence rates. We also establish interior and boundary Lipschitz estimates that are uniform in $\varepsilon$.

In this paper, we prove a spectral multiplier theorem for sub-Laplacians with drift on Métivier groups. We improve the result of [Martini, Ottazzi and Vallarino, Rev. Mat. Iberoam, 2019] in case of Métivier groups, by reducing the required smoothness condition on the multiplier function from homogeneous dimension to the topological dimension of the underlying group.

In this paper we describe in a formal way how the derivation of the turbulent wave equation for the Schrödinger equation breaks down for times close to the self similar blow up of the wave turbulence kinetic equation. To this end, we study how the derivation of the cumulants hierarchy can not be approximated using solutions of the wave turbulence kinetic equation near the blow up time. It tuns out that near the blow up time the kinetic equation has to be replaced by a hierarchy of equations which is equivalent to a random field, defined for times $t\in (-\infty, \infty)$ and satisfying a nonlinear non autonomous Schrödinger equation.

We study the observability of the Schrödinger equation on the $d$-dimensional torus $\T^d=\R^d/(2\pi\Z)^d$, $d \geq 1$, from an open subset $\omega \subset \T^d$. Our main result establishes a quantitative observability estimate for the free Schrödinger equation in the regime of small times $T$ and small observation sets of the form $\omega = \prod_{j=1}^{d}(a_j,b_j)$. A central ingredient in the proof is a cluster decomposition method combined with an induction scheme introduced by Bourgain and further developed by Burq and Zhu.

We study nonlinear elliptic equations modeled on \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = \mu, \] where $2\le p<q<\infty$, $a(\cdot) \ge 0$, and $\mu$ is a signed Borel measure with finite total mass. We prove local Calderón–Zygmund type gradient estimates for SOLA (Solutions Obtained as Limits of Approximations) by finding new and natural assumptions on $p$, $q$ and $a(\cdot)$.

We develop local discontinuous Galerkin (LDG) methods for conservation laws with heterogeneous stochastic fluxes, where the Stratonovich-driven transport terms may be linear or nonlinear. Such equations arise, for example, in simplified turbulence models, mean field games, and fluctuating hydrodynamics. Starting from the Itô formulation, we construct semi-discretizations that build the cancellation mechanism of transport noise into the numerical method. At the discrete energy level, the second-order Stratonovich-Itô correction is balanced by the quadratic variation, up to numerical flux terms, so that the hyperbolic stability structure is retained. Suitable numerical fluxes yield discrete energy conservation or energy dissipation, valid either pathwise or in expectation. The resulting high-order schemes are proved well posed through stability estimates combined with a Khasminskii-type argument, without imposing linear growth assumptions. Numerical experiments confirm stability and high-order accuracy.

The interplay of chemotaxis and diffusion of nutrients or signaling chemicals in bacterial suspensions can produce a variety of structures with locally high concentrations of cells, including phyllotactic patterns, filaments, and concentrations in fabricated microstructures, which is described by the chemotaxis-Navier-Stokes flow by Tuval et al. in 2005. Dombrowski et al. also observed that Bacterial flow in a sessile drop related to those in the Boycott effect of sedimentation can carry bioconvective plumes, viewed from below through the bottom of a petri dish, and the horizontal "turbulence" white line near the top is the air-water-plastic contact line. It's interesting to verify these turbulent phenomena mathematically. For varying chemotactic and velocity viscosities, we derive the boundary layer equations of the chemotaxis-Navier-Stokes system rigorously in a two-dimensional half-space under the Navier-slip boundary condition and obtain the vanishing viscosity limit of the 2D chemotaxis-fluid coupled system in the anisotropic conormal Sobolev spaces.

We investigate the regularity of the viscosity solutions to a class of degenerate/singular fully nonlinear elliptic equations with Hamiltonian terms. To overcome the difficulty caused by the simultaneous presence of the general degenerate/singular gradient terms and Hamiltonian terms, we analyze the coupled interplay between the degeneracy/singularity law and the growth of Hamiltonian terms and establish lower regularity results. Finally, we obtain sharp interior $C^{1,\alpha}$ regularity estimates via a geometric tangential method.

The stationary Navier–Stokes–Cahn–Hilliard equations are considered, governing the motion of a compressible, two-phase fluid mixture with a diffuse interface. The free energy density in this paper has a singular logarithmic (Flory-uggins) form, ensuring that the mass fraction remains in the physical range and allowing for vacuum states. We prove the existence of weak solutions in a three-dimensional bounded domain under structural assumptions on the adiabatic exponent. The stationary setting poses two main mathematical challenges: the absence of an energy inequality driven by the evolution process to control the singular potential, and the degeneracy of the density near the vacuum. To address these issues, we introduce a specialized regularization of the logarithmic term that eliminates the quadratic growth induced by anti-diffusion, thereby restoring compactness. Uniform estimates are obtained through a special choice of artificial pressure and an interpolation argument that controls the desired norm of the density. A two-level limiting process then yields a weak solution that satisfies the physical bounds almost everywhere on the support of the density.To our knowledge, this is the first existence result for the steady compressible Navier–Stokes–Cahn–Hilliard system that incorporates both a singular free energy and vacuum regions.

We prove weighted $L^2$ and refined $L^p$ decoupling estimates for functions whose Fourier transforms are supported in a small neighborhood of the unit sphere or the truncated paraboloid with an additional lower-dimensional frequency localization property. As a special case, we recover the fractal $L^2$ restriction estimate of Du and Zhang, with a sharper dependence on the density of the weight. We also derive weighted refined decoupling estimates related to the Falconer distance set problem, improving earlier results under the stronger assumption that the underlying weight is $\alpha$-dimensional at every scale.

We consider the normal operator of the X-ray transform, weighted with Gaussian weights, in Euclidean space with dimension at least 3. We show the eigenfunctions of the normal operator are joint eigenfunctions of the harmonic oscillator and the spherical Laplacian, and we relate the spectrum to that of elliptic operators in the 1-cusp pseudodifferential calculus.

In this article, we prove that viscosity subsolutions to nonlocal mean curvature-type equations satisfy universal volumetric estimates at all scales. Our results hold for general symmetric kernels that are comparable to the fractional Laplacian. Furthermore, we prove that subsolutions with low density (with respect to a universal constant) necessarily have `fat boundary', that is, have topological boundary with positive Lebesgue measure.

The article is dedicated to the memory and enduring legacy of Professor Robert V. Kohn, Courant Institute, NYU. In this memorial article, we record thoughts and reminiscences of his exemplary life.

We investigate the mean-field limit for interacting particle systems through a duality-based framework and obtain quantitative estimates on the convergence of marginals as well as on correlation functions. The analysis applies to second-order Vlasov systems with pairwise interactions under a mean-field scaling and relies on a hierarchy of dual cumulants associated with the particle dynamics. In particular, for merely square–integrable interaction forces, we derive the natural fluctuation–scale rate $\mathcal{O}(N^{-1/2})$. By introducing an iterative argument on the hierarchy of dual cumulants, we leverage this bound to recover the optimal mean-field rate $\mathcal{O}(N^{-1})$ and to obtain robust estimates on the dual cumulants, at the expense of corresponding regularity assumptions on the interaction kernel. Finally, using the relation between dual and direct correlations, we transfer these bounds to direct cumulants, yielding refined information on correlations and deviations from chaos. The approach provides a unified framework for simultaneously controlling the mean-field limit and the higher-order correlation structure of the particle system.

We study distributional Bernoulli-type conditions in geometrically irregular domains $\Omega$. Here the zero extension of $u$ to $\mathbb{R}^n$ satisfies $$ \Delta u \;=\; c\,\mathcal{H}^{n-1}\!\lfloor_{\partial^*\Omega}\;-\;f(u)\,\mathbf{1}_\Omega\,dx $$ in the distributional sense. This is a weak version of the one-phase Bernoulli free boundary condition, which avoids the uniform Lipschitz/density assumptions of classical Alt–Caffarelli theory. We prove that for every $n \ge 2$ and every $f \in C^2(\mathbb{R})$ with $f(0)>0$, there exist bounded, non-spherical, finite-perimeter domains $\Omega\subset \mathbb R^n$ satisfying this distributional Bernoulli law with $$ 0<\mathcal H^{n-1}(\partial^*\Omega)<\infty, \qquad \mathcal H^{n-1}(\partial\Omega\setminus\partial^*\Omega)=0, $$ yet $$ {\rm ess}\sup_{x\in\partial^*\Omega} \sup_{0<r<1} \frac{\mathcal H^{n-1}(B_r(x)\cap\partial^*\Omega)}{r^{n-1}} =\infty. $$ This shows the key constraint is not absence of a reduced boundary, but failure of uniform all-scale surface density bounds. For $f \equiv 1$, these results yield counterexamples to the weak Serrin-type overdetermined problems in all dimensions, proving the distributional Bernoulli law alone cannot replace the uniform growth/density conditions core to Alt–Caffarelli theory. On the other hand, we prove a planar rigidity result: Within the Smirnov class, the associated harmonic quadrature identity forces $\Omega$ to be a disk. Thus, for the constant-source Serrin/Bernoulli law, Smirnov regularity is the threshold for weak Bernoulli rigidity in $\mathbb R^2$, while uniform upper density bounds form the threshold for $n\ge 3$ according to [23].

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on } \partial \Omega, \end{aligned} \end{cases} \end{equation} where $\Delta_{\gamma}$ denotes the Grushin operator defined by \begin{equation} \Delta_{\gamma} := \Delta_x + (1+\gamma)^2 |x|^{2\gamma}\Delta_y, \end{equation} with $\gamma>0$, $z=(x,y)\in \mathbb{R}^N$, $N=n+m$, $n \geq 1$, $m\geq 1$, $\Omega \subset \mathbb{R}^N$ a smooth bounded domain, $\lambda>0$, $1<p<\infty$, and $\delta>0$. The analysis depends on the exponent $p$, which may be subcritical, critical, or supercritical, that is, $p<2_\gamma^*-1$, $p=2_\gamma^*-1$, or $p>2_\gamma^*-1$, respectively, where $2_\gamma^*=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the Grushin operator, and $Q=m+(1+\gamma)n$ is the corresponding homogeneous dimension.

We prove a heat semigroup characterization of the total variation for compactly supported ${\rm BV}$ on arbitrary smooth complete weighted Riemannian manifolds, extending the main result in \cite{GP15}. We then provide an example of a weighted manifold where such equivalence does not hold for a large class of sets of finite perimeter.

This is a memorial article for Yvonne Choquet-Bruhat, who was one of the great pioneers of mathematical general relativity and of partial differential equations. Starting with her 1952 result on local existence of solutions of the vacuum Einstein field equations, she obtained many results on the Einstein evolution equations, the Einstein constraint equations, and the equations of supergravity. Her methods have also been important for numerical relativity. She also wrote several textbooks and a memoir. An abridged version of this article has been submitted to AMS Notices.

We study the underdamped Langevin dynamics with invariant measure $\mu(\,\mathrm{d}x\,\mathrm{d}v)\propto \mathrm{e}^{-U(x)-\lvert v\rvert^2/2}\,\mathrm{d}x\,\mathrm{d}v$. Assume that the position marginal $\mu_x(\,\mathrm{d}x)\propto \mathrm{e}^{-U(x)}\,\mathrm{d}x$ satisfies a logarithmic Sobolev inequality with constant $\rho>0$, and that $U$ is convex on $\mathbb{R}^d$ and satisfies some growth conditions. We introduce a modified entropy approach with a Wasserstein entropy-current corrector \begin{equation*} \mathcal H_\epsilon(g)=\operatorname{Ent}_\mu(g) +\epsilon\int \Pi_v(v\,g)\cdot\bigl(x-T_q(x)\bigr)\,\mu_x(\mathrm{d}x), \end{equation*} where $\Pi_v$ denotes averaging over the velocity variable against the standard Gaussian $\kappa(\mathrm{d}v)=(2\pi)^{-d/2}\mathrm{e}^{-\lvert v\rvert^2/2}\,\mathrm{d}v$, $q=\Pi_v g$ is the position marginal density of $g$, and $T_q$ is the Brenier optimal transport map from $q\mu_x$ to $\mu_x$. For friction $\gamma=\Gamma\sqrt\rho$ with $\Gamma>0$, and for any initial law $p_0$ with finite relative entropy, if $p_t$ denotes the law of underdamped Langevin dynamics at time $t$, we establish the explicit entropy decay \begin{equation*} \operatorname{Ent}(p_t\mid\mu) \leq \frac{1+\theta}{1-\theta}\,\mathrm{e}^{-\Lambda t}\,\operatorname{Ent}(p_0\mid\mu), \qquad t\ge0, \end{equation*} with rate \begin{equation*} \Lambda=\frac{\theta}{2(1+\theta)}\sqrt\rho, \qquad \theta=\min\Bigl\{\tfrac{\Gamma}{12},\tfrac{1}{4\Gamma}\Bigr\}. \end{equation*} In particular, the entropy convergence rate has optimal $\sqrt\rho$ order.