Papers

In this note we show the existence of a residual set (in the sense of Baire) of divergence free initial data $u_0\in L^2(D)$, $D=\mathbb{R}^2$ or $\mathbb{T}^2$, for which global existence and uniqueness of weak solutions to the incompressible 2D Euler equations holds. The associated solutions $u$ satisfy the energy balance and are recovered in the vanishing viscosity limit from solutions to 2D Navier-Stokes, which as a consequence cannot display anomalous dissipation of energy. Additionally, there exists a unique regular Lagrangian flow associated to such $u$, and the associated transport equation is well-posed. Finally, when $D=\mathbb{T}^2$, the solution $u$ is recovered as the limit of Galerkin approximations. The proof relies on global existence of smooth solutions and weak-strong uniqueness arguments.

We address Gromov's Quantification of $C^0$ Convergence Conjecture in dimension three. Let $B$ be the unit ball in $\mathbb R^3$. Let $g$ and $g_0$ be smooth metrics on $B$. We prove there are constants $C$ and $\epsilon_0$ depending only on $g_0$ so that \[ \inf_{x\in B} R_g(x) \leq R_{g_0}(0) + C \|g-g_0\|_{C^0}^{1/2} \] provided $\|g-g_0\|_{C^0}\leq \epsilon_0$. We also construct examples to show that the exponent $1/2$ is sharp. This explicitly quantifies the fact that scalar curvature lower bounds are preserved under $C^0$ convergence of metrics. When $g_0$ is merely $C^2$ we prove a related estimate with a slightly weaker rate, and when $g_0$ has rotational symmetry we prove a related estimate with a stronger linear rate. To prove these results, we use harmonic functions to define a local quantity that detects the scalar curvature. Then we use classical elliptic PDE estimates to show that this quantity is stable under $C^0$ perturbations of the metric. As a further application of this method, we give a partial answer to a question of Gromov on the preservation of scalar curvature lower bounds for metrics that are converging in measure.

We present a model for sticky particles in which cluster sizes after a reaction have $\ell$ fewer total particles than the sum of their reactants. The finite particle system is modeled as a Markov process under a mean-field assumption for selecting reactants. The limiting kinetic equations form an infinite system of nonlinear differential equations similar to the Smoluchowski coagulation equations with multiplicative kernel. We show existence and uniqueness for systems whose cluster sizes are either bounded above or below by the emission size $\ell$. When clusters have at most $\ell$ particles, well-posedness can be extended until an exhaustion time in which certain cluster fractions vanish. For clusters with more than $\ell$ particles, we prove short-time well-posedness, along with explicit formulas for cluster sizes and moments. We also conduct numerical experiments which suggest these formulas hold until a gelation time, at which an infinite-sized cluster forms.

We introduce a family of (nonlinear) pairing measures that ensure the validity of the divergence rule for composite functions $\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a bounded divergence-measure vector field, and $u$ is a scalar function of bounded variation. The elements of the family depend on the choice of the pointwise representative of $u$ on its jump set. Beyond the standard properties, such as the Coarea and Gauss-Green formulas on sets of finite perimeter, this flexibility allows us to characterize the pairings that ensure the lower semicontinuity of the corresponding functionals along sequences converging in $L^1$ with controlled precise values. We show that these lower semicontinuous pairings arise as the relaxation of integral functionals defined in Sobolev spaces.

In this paper, given a convex, bounded, open set $\Omega \subset \mathbb{R}^n$ we prove a sharp inequality involving the Laplacian torsional rigidity and both the perimeter and the measure of the domain. Our result generalizes to arbitrary dimensions the inequality established by Makai in the plane which, as conjectured in arXiv:2007.02549. Furthermore, we establish quantitative estimates that provide key insights into the geometric structure and the thickness of the underlying optimizing sequences.

We consider the simplicial de Rham complex and the Čech-de Rham complex, two bigraded Hilbert complexes whose Hodge-Laplace problems govern spatially coupled problems in mixed dimension and homogeneous dimension, respectively. The former complex can be realized as a subcomplex of the latter. In this paper, we quantify how close these complexes are to each other by constructing bounded cochain complexes between them, and thus we quantify how close a mixed-dimensional formulation of a problem is to an equidimensionally coupled formulation of the same problem. From this construction, we derive a priori- and a posteriori error estimates between the associated Hodge-Laplace problems on the two complexes. These estimates represent the error which is introduced by treating a spatially coupled problem as mixed-dimensional, rather than an equidimensional problem with thin overlaps.

For any $\beta_0<1/3$ we construct divergence free vector fields in $ C_{x,t}^{\beta_0}$ and a sequence of diffusivities $\kappa_q \searrow 0$ such that, for an arbitrary initial datum from a low regularity class, the classical solution $\rho_q$ to the advection-diffusion equation exhibits anomalous dissipation along the sequence $\kappa_q$. At the same time $\rho_q$ remains uniformly bounded in $C_t^{0} C_x^{\alpha_0}$, where $\beta_0 + 2\alpha_0<1$. Our result confirms a conjecture of Armstrong and Vicol \cite{ArmstrongVicol} and shows sharpness of the Obukhov-Corrsin threshold within the context of iterated homogenization. Our construction confirms time-homogeneity of the dissipation anomaly, as required in turbulence theory, and as a consequence we also obtain better time regularity for the scalar $\rho_q$ than the classical prediction of Yaglom.

We consider a shape optimization problem for a hybrid energy combining local confinement and nonlocal Coulomb repulsion. Specifically, for any open set $\Omega \subseteq \mathbb{R}^3$ of prescribed volume, we consider the ground state energy of an $L^2$-normalized function supported in $\Omega$, defined as a linear combination of its homogeneous $\dot{H}^1$ and $\dot{H}^{-1}$ seminorms. We show that in the small mass regime, volume-constrained minimizers of this geometric functional exist and are $C^{2,\alpha}$ perturbations of a ball. The proof relies on a combination of surgery techniques, $\Gamma$-convergence, elliptic PDE theory, and one-phase free boundary regularity. A key novelty of this paper lies in the treatment of the Coulombic repulsive term: unlike standard competitive models, the lack of (a priori) sign constraints on the optimal functions forces the nonlocal term to exhibit two natures: it acts both as a scattering and an homogenizing force.

We build solutions to Kac's particle system and show that their empirical measures converge to the solution of the space-homogeneous Boltzmann equation in the regime of very soft potentials. This proves propagation of chaos for the last class of kernels for which it was still open. The proof relies on new estimates on the dissipation of the Fisher information along the Boltzmann equation, which allow us to control the strong singularities of the system. These estimates are obtained thanks to a new inequality related to the fractional heat flow on the sphere, that might be of independent interest.

This paper studies local existence and the singularity formation of the solutions of the one-dimensional hyperbolic Navier-Stokes equations, in particular proving the gradient blow-up of the derivatives of the solutions. The underlying model introduces a relaxation mechanism that leads to hyperbolization, achieved both through a nonlinear Cattaneo law for heat conduction and through Maxwell-type constitutive relations for the stress tensor. Our main approach is to prove that the hyperbolic Navier-Stokes equations are indeed hyperbolic, and to prove that they possess two genuinely nonlinear eigenvalues, thereby establishing the blow-up of the gradient of the solution. In addition, we provide a derivation of the equation of state for the hyperbolic Navier-Stokes equations in the appendix.

We determine the explicit value of the optimal constant in the trace inequality for functions of bounded variations in the case the domain has a particular class of singularities.

We investigate both qualitative and quantitative issues related to the classification of non-negative energy solutions to the anisotropic critical $p$-Laplace equation in $\mathbb{R}^n$, for $1<p<n$. Specifically, we establish an anisotropic version of Struwe's decomposition, along with the interaction estimate for the family of bubbles in this decomposition. Moreover, we provide a short proof of the classification result as well as a quantitative stability result, proving that every energy solution to a perturbation of the anisotropic critical equation must be closed to a bubble, in the absence of bubbling.

In this paper we consider the volume-constrained minimization of a variant of the Ohta-Kawasaki functional with an anisotropic surface energy replacing the standard perimeter. Following and suitably adapting the second variation approach devised in arXiv:1211.0164, we prove local minimality results for the horizontal lamellar configuration, in analogy with the isotropic case, under the assumption that the anisotropy is uniformly elliptic. If instead the Wulff shape of the anisotropy has upper and lower horizontal facets, we prove that the lamella exhibits a rigid behavior and is an isolated local minimizer for all parameter values. We conclude by showing some global minimality results, mostly focusing on the planar case.

We study area- and length-preserving curvature flows for embedded closed curves on pinched Hadamard surfaces. In the variable-curvature setting, the evolution equations contain additional lower-order terms, so the PDE analysis requires refined comparison arguments and delicate curvature estimates. For smooth convex initial curves, we prove preservation and instantaneous strictness of convexity, long-time existence, and uniform bounds for the curvature and its higher derivatives. Under additional geometric assumptions, we obtain convergence of the curvature to a constant. In the rotationally symmetric case, the area-preserving flow exhibits a dichotomy: either the evolving curves converge exponentially to a geodesic circle, or they drift off to infinity and approach a constant-curvature limit curve. We also identify a geometric condition on the initial curve that prevents escape to infinity and guarantees convergence to a geodesic circle.

We study Poincare-Sobolev type inequalities for compactly supported smooth functions which are defined in the Euclidean $n$-space and whose absolute value of gradient are Choquet $\delta /n$-integrable with respect to the $\delta$-dimensional Hausdorff content, $n\geq 2$, $\delta\in (0,n]$. In particular, our results imply a new Sobolev inequality for quasicontinuous functions defined in the Sobolev space $W^{1,1}_0(\mathbb{R}^n)$. As an application we extend a recently introduced superlevel Sobolev inequality into a context of the Hausdorff content.

The pointwise space-time behavior of the Green's function of the three-dimensional modified Vlasov-Poisson-Boltzmann system is studied in this paper. It is shown that the Green's function has a decomposition of the macroscopic diffusive waves and Huygens waves with the speed $\sqrt{\frac{8}{3}}$ at low-frequency, the singular kinetic wave and the remainder term decaying exponentially in space and time. In addition, we establish the pointwise space-time estimate of the global solution to the nonlinear modified Vlasov-Poisson-Boltzmann system based on the Green's function.

In this work we study the large-time behaviour of solutions of the Heat Equation in the hyperbolic space $\mathbb{H}^d$, providing precise speeds of convergence in $L^1$ and $L^\infty$ to their asymptotic profiles by means of an adaptation of entropy estimates. For $L^1$ initial conditions we are able to identify the asymptotic profile in $L^1$, which is not universal but contains a certain memory of the initial distribution of the mass of the solution. We improve thus on previous results, where speed of convergence was absent and asymptotic profiles where not known in the general case, and show a way to adapt entropy estimates employed in the study of diffusion processes to non-compact Riemannian manifolds. The main strategy to prove this is to consider transient profiles as minimizers of the entropy functional. These profiles are time-dependent and encompass the geometric information of the Riemannian manifold.

In the present paper, we study the diffusion limit of the classical solution to the modified Vlasov-Poisson-Boltzmann (mVPB) System with initial data near a global Maxwellian. Based on the spectral analysis, weprove the convergence and establish the convergence rate of the global strong solution to the mVPB system towards the solution to an incompressible Navier-Stokes-Poisson-Fourier system with the precise estimation on the initial layer.

In this paper, we consider the stochastic reaction-diffusion equation $\mathrm{d}u = (\mathcal{A} u + f(u))\mathrm{d}t + \sigma(u)\mathrm{d}W$ on a smooth bounded domain $\mathcal{O}$ with homogeneous Dirichlet boundary conditions. We investigate the long-time behavior of solutions with a strongly dissipative drift nonlinearity and superlinear multiplicative noise in the Bochner space $L^q(\Omega; C_0(\overline{\mathcal{O}}))$, $q \ge 2$. Here $\mathcal{A}$ is a second-order self-adjoint elliptic operator and $W$ is a two-sided trace-class Wiener process. The standard Galerkin method fails to yield energy estimates in $L^q(\Omega; L^q(\mathcal{O}))$ via the Itô formula for $q > 2$, owing to the interference of projection operators when dealing with nonlinear terms; meanwhile, the classical theory of mild solutions lacks sufficient spatial regularity to apply the Itô formula directly. To overcome these difficulties, we consider mild solutions and establish a critical regularity estimate for the corresponding stopped process $u_n(t)$ in $W_0^{1,q}(\mathcal{O})$, which rigorously justifies the use of the Itô formula in the non-Hilbert space $L^q(\Omega; L^q(\mathcal{O}))$. As a result, we derive explicit moment energy estimates and quantitative dissipativity bounds, yielding global existence, uniqueness, and exponential asymptotic decay of solutions in $L^q(\Omega; C_0(\overline{\mathcal{O}}))$. Unlike previous qualitative results in continuous function spaces, our framework provides a fully quantitative theory of global dissipativity.

We investigate standing waves for the energy critical Schrödinger system with three waves interaction arising as a model for the Raman amplification in a plasma. Several results are proved: simultaneous existence of stable and unstable standing waves, existence of global solutions, and absence of small data scattering. Our main results show some specific features arising from the three waves interaction differently from the classical energy critical Schrödinger equation, and they support some experimental observations on Raman amplification.

In this paper, we investigate a class of critical Ambrosetti-Prodi type problems involving the sub-Laplacian on a Carnot group. Specifically, we consider \[ \left\{ \begin{aligned} -\Delta_{\mathbb{G}} u &= \lambda u + u_{+}^{2_{Q}^{*}-1} + f(\xi) \quad &&\text{in } \Omega,\\[2mm] u &= 0 \quad &&\text{on } \partial\Omega, \end{aligned} \right. \] where $\Delta_{\mathbb{G}}$ is the sub-Laplacian on a Carnot group $\mathbb{G}$, $\Omega \subset \mathbb{G}$ is an open bounded domain with smooth boundary, $\lambda>0$ is a real parameter, $f\in L^{\infty}(\Omega)$, $u_{+}$ denotes the positive part of $u$, and $2_{Q}^{*}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q$. Motivated by the classical Ambrosetti-Prodi problem, we establish existence and multiplicity results for the cases $\lambda<\lambda_{1}$ and $\lambda>\lambda_{1}$, where $\lambda_{k}$ denotes the $k$-th Dirichlet eigenvalue of $-\Delta_{\mathbb{G}}$. We also prove the existence of solutions at resonance when $\lambda=\lambda_{1}$ and show that bifurcation occurs from each eigenvalue $\lambda_{k}, k >1$.

This article investigates the existence of closed, star-shaped hypersurfaces for a class of Hessian quotient type curvature equations, in which the operator $\frac{\sigma_k}{\sigma_l}(\Lambda)$ arising in these equations can be viewed as a generalization of the classical Hessian quotient operator. By combining a priori estimates with the continuity method, we establish the existence and uniqueness of $(\mathbf{p}, k)$-convex hypersurfaces for both nonhomogeneous and homogeneous equations of this type. Furthermore, by exploiting the recently discovered “inverse convexity” property of the operator $\frac{\sigma_k}{\sigma_l}(\Lambda)$, we prove a constant rank theorem and thereby obtain the existence and uniqueness of strictly convex solutions to these curvature equations.

In this paper, we study the Boltzmann equation in a close to the hydrodynamic limit regime, set in bounded spatial domains with non-isothermal Maxwell boundary conditions. We establish the existence, uniqueness, and asymptotic stability of a non-equilibrium steady state under suitable smallness assumption on the temperature fluctuations at the boundary.

We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains.

This paper aims to extend to Orlicz-Sobolev spaces some results of integral representation for the simultaneous homogenization and dimensional reduction of integral energies defined on fields taking values on a differentiable manifold. Since our functional framework goes beyond the classical Sobolev's spaces, we also prove, via $\Gamma$-convergence, a general integral representation results in the unconstrained Orlicz setting. Due to $\Delta_2$ and $\nabla_2$ conditions verified by the Young function $\Phi$ (which modulated the growth behaviour), we prove that the density of the $\Gamma$-limit is a tangential quasiconvex integrand represented by a cell formula.

In this paper, we investigate the higher weak differentiability of solutions to a class of mixed local and nonlocal degenerate elliptic equations in the Heisenberg group $\mathbb{H}^n$. Owing to the non-commutative property and two-step nilpotent Lie algebra structure of $\mathbb{H}^n$, we first employ an iterative scheme involving fractional difference quotients to establish the weak differentiability of solutions in the vertical direction. This is subsequently extended to the horizontal and vertical gradients. Then, by coupling a truncation argument with the difference quotient method, we prove the higher weak differentiability of the gradients of solutions.

We consider the Dirichlet problem for the complex Monge–Ampère equation on strongly pseudoconvex Kähler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth solutions in a number of natural circumstances, following work of Chou-Wang.

In this investigation, we conduct a systematic computational search for potential singularities in 3D Navier-Stokes flows on a periodic domain $\Omega$ based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that for a solution $\mathbf{u}(t)$ of the Navier-Stokes system to be regular on an interval $[0,T]$, the integral $\int_{0}^T \|\mathbf{u}(t)\|_{L^q}^p\,dt$, where $2/p+3/q=1,\;q>3$, and the expression $\sup_{t \in [0,T]} \|\mathbf{u}(t)\|_{L^3}$ must be bounded. Flows which might become singular and violate these conditions are sought by solving a family of variational PDE optimization problems where we identify initial conditions $\mathbf{u}_{0}$ with the corresponding flows $\mathbf{u}(t)$ locally maximizing the integral $\int_{0}^T \|\mathbf{u}(t)\|_{L^q}^p\,dt$ for a range of different values of $q$ and $p$ or the norm $\|\mathbf{u}(T)\|_{L^3}$ for different time windows $T$ and increasing sizes $\| \mathbf{u}_0 \|_{L^q}$ of the initial data. We consider two formulations where these expressions are maximized over appropriate Lebesgue spaces $L^q(\Omega)$ or the largest Hilbert-Sobolev spaces $H^s(\Omega)$ embedded in them. The lack of Hilbert-space structure in the first case necessitates development of a novel computational approach to solve the problem. While no evidence of unbounded growth of the quantities of interest, and hence also for singularity formation, was detected, we were able to quantify how "close" the flows realizing such worst-case scenarios come to forming a singularity. A comparison of these results with estimates on the rate of growth of the norms $||\mathbf{u}(t)||_{L^q}$ and of the enstrophy $\mathcal{E}(t)$ indicates that the extreme flows do enter a regime where these quantities are amplified at a rate consistent with singularity formation in finite time, but this growth is not sustained long enough for singularities to form.

These lecture notes are devoted to solutions of hyperbolic-parabolic systems with persistent oscillations. We consider two examples both from mechanics: (i) The system of viscoelasticity of Kelvin-Voigt type with strain energies involving double well potentials, as employed in phase transitions. (ii) The compressible Navier-Stokes equations for a barotropic gas. For each system we construct solutions with persistent oscillations. In a later part we consider the nonlinear homogenization problem. For the systems of viscoelasticity in one-space dimension in Lagrangian coordinates, and for the compressible Navier-Stokes system for barotropic fluids we show how ideas from the kinetic formulation of conservation laws can be used to derive effective equations. The effective equation consists by a kinetic equation coupled with the macroscopic flow.

In this article, we show that for one-phase free boundary problems in noncollapsed limits of $n$-dimensional manifolds with two-sided Ricci curvature bounds, the Hausdorff dimension of the singular set of the free boundary can be bounded by $n-5$, which is sharp in this context.