Papers

We are concerned with the properties of Ornstein-Uhlenbeck evolution operators acting on functions defined in a Hilbert space and p-integrable with respect to a suitable Gaussian measure. These operators provide solutions to the infinite dimensional, non-autonomous backward Kolmogorov equation. The first part of the paper focuses on some general regularization properties, while the second part carries on a deep analysis of the asymptotic behaviour in the periodic case. In particular, we identify the optimal convergence rate of the Ornstein-Uhlenbeck operator and we give an optimality criterion depending only on the drift term of the Kolmogorov equation.

This paper concerns the $H$-convergence of nonlinear nonlocal monotone operators defined through the Riesz fractional gradient and divergence. We show that the $H$-convergence in this nonlocal framework is equivalent to the $H$-convergence of the corresponding local one. As a consequence, we obtain a $H$-compactness result for a suitable class of nonlocal monotone operators. We then study the $\Gamma$-convergence of nonlocal energy functionals associated with the subclass of conservative monotone operators, proving that it is equivalent to the $\Gamma$-convergence of the corresponding local energies. A key ingredient is a new uniqueness result for the integral representation of both local and nonlocal functionals. As a by-product, we obtain the $\Gamma$-compactness of the class of nonlocal energies under consideration. Finally, we show the equivalence between the $H$-convergence of nonlocal conservative monotone operators and the $\Gamma$-convergence of the associated energy functionals.

An $\infty$-harmonic function is a viscosity solution of $\nabla^2 u(\nabla u,\nabla u)=0$, or equivalently, an absolute minimizer of $\|\nabla u\|_{L^\infty}$. We prove a variety of new structural and regularity results in two dimensions, including: 1. $\infty$-harmonic functions in domains of $\mathbb{R}^2$ are $C^{1,1/3}$. 2. Critical points are isolated, and at each critical point, the solution has a unique quasiradial blow-up. 3. Entire solutions with polynomial growth have unique quasiradial blow-downs, and are determined by their Fourier modes at infinity. These results are consequences of a new theory relating $\infty$-harmonic functions to inverse mean curvature flow (IMCF) clusters – which are piecewise weak solutions of IMCF with common obstacle-type boundary conditions on the interfaces (a simple example is an embedded family of cuspidal curves evolving by inverse curvature). This connection arises as the $p\to\infty$ limit of the classical duality between $p$-harmonic and $q$-harmonic functions in $\mathbb{R}^2$, where $\frac1p+\frac1q=1$.

We construct Hilbert-space isomorphisms identifying the spaces of radiation fields on the event horizon and null infinity that are induced by the forward scattering map on the Schwarzschild exterior for fixed spherical harmonic mode solutions to the linear scalar wave equation arising from Cauchy data sets at $t=0$ which are regular at the event horizon, in the sense that the local energy at the event horizon is finite. We show that fixed spherical harmonic mode solutions with no radiation on the event horizon must decay exponentially in time, at a rate that is determined by the surface gravity. On the other hand, we construct examples of fixed spherical harmonic mode solutions with no radiation on null infinity and which have polynomial decay along the event horizon. Finally, we construct examples of polynomially decaying solutions to the linear scalar wave equation which are regular at the event horizon, have unbounded support in spherical harmonic modes, and induce no radiation on the event horizon.

We prove stability for the affine Sobolev inequality for exponents $p\geq 2$ with best possible norm and best possible stability exponent. We also show a corresponding result for critical points of the functional in the absence of bubbling. An important ingredient in our proof is the classification of positive energy solutions to the critical affine $p$-Laplace equation.

We study the solvability of the divergence equation $$ \operatorname{div} \u = f $$ in bounded $C^2$ domains under homogeneous Dirichlet boundary conditions for data $f\in C^{0,\alpha}(\Omega)$ satisfying the compatibility condition $ \int_\Omega f =0. $ We construct a solution $\u$ such that for every $0<\beta<\alpha$ $$ \u\in C^{1,\beta}(\Omega)^n $$ satisfies $$ \|\u\|_{C^{1,\beta}(\Omega)} \le C\|f\|_{C^{0,\alpha}(\Omega)}. $$ The proof combines localization techniques with a boundary flattening procedure reducing the problem to a model half-cube.

We study local well-posedness of the initial value problem for a class of fourth-order nonlinear dispersive systems on the one-dimensional torus. The main difficulty comes from the loss of derivatives in the nonlinear terms. By introducing suitable structural conditions that compensate for derivative loss through cancellation mechanisms, and constructing modified energies via gauge-type transformations combined with a diagonalization procedure, we establish local well-posedness in high-regularity Sobolev spaces. The assumptions in the present result relax those in previous works, at the expense of requiring higher regularity. In particular, our structural conditions extend the range of admissible nonlinearities beyond the scalar case and provide a unified framework for treating multi-component systems.

We study the Cauchy problem for the three-dimensional compressible Navier–Stokes equations with eddy diffusion, an anisotropic dissipative mechanism that arises naturally in geophysical fluid dynamics (cf.~\cite{Jabin-Bresch-2018,Temam-Ziane-2004}). In contrast to the classical compressible Navier–Stokes system, the momentum equation here carries no full vertical Laplacian: the velocity is diffused only in the horizontal directions, and the sole vertical regularization it receives is the partial one transmitted through the compressible mode $\operatorname{div}\mathbf{u}$. This degeneracy invalidates the standard parabolic energy framework as well as the classical high–low frequency Green-function bounds. We prove that the constant non-vacuum equilibrium $(\bar{\rho},0)$ is globally nonlinearly stable against small Sobolev perturbations: global classical solutions exist in $H^{N}(\mathbb{R}^{3})$ for every $N\ge 3$, and the density and velocity relax to equilibrium with explicit, genuinely anisotropic decay rates. The mechanism behind the result is a hidden dissipation produced by the pressure–divergence coupling between $\nabla\rho$ and $\operatorname{div}\mathbf{u}$, which compensates for the missing vertical smoothing of the density and the compressible part of the velocity; the solenoidal part of the velocity, by contrast, is governed by a purely horizontal heat flow and therefore decays only at the two-dimensional rate. The analysis rests on a refined anisotropic spectral decomposition of the Green matrix, a div–curl treatment of the velocity, and time-weighted nonlinear energy estimates tailored to the degenerate dissipation. To the best of our knowledge, this is the first global stability and large-time behavior result for the three-dimensional compressible Navier–Stokes equations with eddy diffusion in the whole space.

This paper studies the long time statistics and small noise asymptotics of Markov cocycles associated with Markov processes in random environments modeled by measure preserving dynamical systems on a standard Borel probability space. Our first result provides an abstract criterion for exponential mixing of stationary measures for such cocycles, formulated toward SPDE applications with assumptions that can be verified directly from a priori estimates. To overcome the nonuniformity from the environment, we combine generalized coupling arguments with ergodic theoretic methods. This allows us to convert nonuniform estimates along the environment into contraction on a positive density set of times, and then upgrade this to all time contraction by introducing a block gap-counting argument. Our second result establishes a Freidlin–Wentzell large deviation principle(LDP) for the unique stationary measure in the small noise limit with a good rate function. For the upper bound, the noise is allowed to be degenerate, while the deterministic pullback attractor may have nontrivial dynamics. The abstract theory applies to nonautonomous SPDEs. We illustrate it with two examples: the two-dimensional Navier–Stokes equations on bounded domains and damped Sine–Gordon equations, where both the deterministic forcing and the degenerate additive noise depend on the random environment.

This article established the existence results for the $m$-harmonic equation involving critical Choquard nonlinearity and subcritical perturbation. We first explore the minimizers of the $m$-harmonic operator with the critical Choquard equation. Then, using these minimizers, we establish delicate estimates to show the energy below the threshold level, which helps to recover the compactness. Further, we prove the existence of a nontrivial solution for our problem with different kinds of local and nonlocal subcritical perturbations. To the best of our knowledge, this is the first article dealing with the polyharmonic equation and critical Choquard type nonlinearity. The results obtained are even new for $m\geq 2$.

We derive and analyze a time-domain pressure–interface evolution model for a compressible gas bubble in a compressible liquid with surface tension. Starting from the nonlinear two-phase Euler free-boundary problem and linearizing about a spherical Young–Laplace equilibrium, we obtain a coupled bulk–surface hyperbolic system for the liquid pressure, the gas pressure, and the normal displacement of the interface. The time-domain analysis is complicated by the fact that the surface-tension quadratic form is indefinite: the \(Y_0^0\) component is the volume-changing breathing mode, the \(Y_1^m (m = \{-1,0,1\})\) components are neutral translations, and only the higher spherical harmonics give a coercive shape-mode energy. We exploit this decomposition. The coercive sector is treated by a \(C^0\)-semigroup argument, while the breathing mode is analyzed separately by Fourier–Laplace methods. This yields well-posedness for admissible finite-energy data and classical solutions under the natural compatibility conditions. We then justify two limiting descriptions with quantitative error estimates. In an acoustic quasi-static regime, the model reduces to the linearized Rayleigh–Plesset equation for the breathing mode and to the linearized Rayleigh–Lamb equations for the shape modes. In a different regime, it reduces to the frozen-interface acoustic transmission model. Finally, a frequency-domain analysis identifies the corresponding Minnaert, Rayleigh–Lamb, and Fabry–Pérot-type resonance mechanisms as different components and limits of the same pressure–interface formulation.

We construct smooth, uniformly convex planar domains that admit minimal, strictly stable solutions of a semilinear Dirichlet problem whose superlevel sets are nonetheless nonconvex. The class of admissible nonlinearities includes, in particular, two prototypical cases: the Gelfand-type nonlinearity $e^u$ and the family of shifted power-type nonlinearities $(a+u)^p$, where $a>0$ and $p>1$. By applying the elementary scaling properties of the Dirichlet problem, we also show that the same lack of convexity of superlevel sets holds for the corresponding parameter-dependent equations. These results provide a negative answer to a question posed by Brezis, who inquired whether the stability of a solution necessarily entails quasiconcavity for these prototypical stable configurations.

In 2008, Beckner (Proc. Amer. Math. Soc. 136(5), 1871-1885) proved two inequalities of the Herbst type, which are the critical forms of the Stein-Weiss inequality. In 2018, Chen et al. (Tran. Amer. Math. Soc. 370(12), 8429-8450) established the reversed Stein-Weiss inequality. In this paper, we are concerned about its critical case and give a reversed Herbst inequality. Namely, $$ \left|\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|x-y|^{\alpha/q'-n}|y|^{\alpha/q'}g(x)h(y)dxdy\right| \geq C_{n,\alpha,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n)}\|h\|_{L^p(\mathbb{R}^n)} $$ holds for any nonnegative functions $g \in L^{q'}(\mathbb{R}^n)$ and $h \in L^p(\mathbb{R}^n)$, where $n\geq 1$, $p, q' \in (0,1)$, $\alpha>n$ satisfying ${1}/{p}+{1}/{q'}-{2\alpha}/(q'n)=1$. Such an inequality is not covered by the reversed Stein-Weiss inequality. Meanwhile, we prove the existence of extremal functions of this inequality. Finally, we study the Euler-Lagrange system satisfied by those extremal functions $$ \left\{\begin{matrix} u(x)=\int_{\mathbb{R}^n}|x-y|^{\beta-n}v^{-p_2}(y)|y|^{\beta}dy, v(x)=\int_{\mathbb{R}^n}|x-y|^{\beta-n}u^{-p_1}(y)|x|^{\beta}dy. \end{matrix}\right. $$ We obtain necessary conditions for the existence of positive solutions, and investigate their integrability and asymptotic behavior when $|x| \to 0$ and $|x| \to \infty$.

Whether the solutions of conformal equations in the whole space are radially symmetric is an interesting topic. Chen-Li-Ou proved the radial symmetry for integral systems of the Hardy-Littlewood-Sobolev type and the Stein-Weiss type by the method of moving planes in integral form. In 2015, Dou-Zhu obtained the radial symmetry of extremal functions of the reversed Hardy-Littlewood-Sobolev inequality by the method of moving spheres, and Liu proved the radial symmetry of solutions of the Euler-Lagrange system by the method of moving planes developed by Dou-Guo-Zhu. In this paper, we also use the method of moving planes to prove the radial symmetry of positive solutions of the Euler-Lagrange system satisfied by the extremal functions of the reversed Stein-Weiss inequality established by Chen-Liu-Lu-Tao in 2018.

In this paper, we study qualitative properties of distribution solutions of a fourth order equation $$ -\Delta u(x)+a^2\Delta^2u(x)=u^q(x), \quad u(x)>0 \ \ in \ \ \mathbb{R}^3, $$ where $a>0$ and $q>0$. It is the static equation of a mixed dispersion Schrodinger equation, and also the Euler-Lagrange equation satisfied by extremal functions of an embedding inequality. We obtain some Liouville theorems and the corresponding related critical exponents, which imply the best constant of the embedding inequality cannot be attainable. We also obtain some regularity results (involving differentiability, integrability, radial symmetry) and asymptotics at infinity of distribution solutions. Here an equivalent integral equation with the Coulomb potential $|x|^{-1}(1-e^{-|x|/a})$ plays a key role. In addition, we also use the Pohozaev identity in integral form to obtain the Liouville theorem of this integral equation. Such the Pohozaev identity still works to handle the Allen-Cahn-type integral equation.

We introduce a compensated Piola graph method for Lagrangian stability estimates in critical homogeneous Besov spaces below the usual product threshold. For $d\ge2$, $2d\le p<\infty$, and $s=d/p$, we establish local Hadamard well-posedness and a continuation criterion in the scaling-critical Besov phase space $\dot B^{s-1}_{p,1}(\mathbb R^d)^d\times \dot B^s_{p,1}(\mathbb R^d)$, for a class of incompressible parabolic systems coupled to nondiffusive internal variables. The main obstruction is that the formal Piola product between the inverse deformation gradient and the Lagrangian velocity is only borderline at $p=2d$ and is not continuous in general for $p>2d$. We replace this product by a closed solenoidal Piola graph, using a compensated divergence structure that survives in the high-integrability range. The principle applies to viscous non-resistive MHD, Hookean incompressible viscoelasticity, and nondiffusive Oldroyd–B systems with affine objective terms. In particular, it closes the previously untreated high-$p$ Hadamard well-posedness range for critical non-resistive MHD and for the nondiffusive Oldroyd–B systems considered here; combined with the known low-$p$ MHD theory, it gives the finite-$p$, $q=1$ critical Besov picture for non-resistive MHD.

In this paper, we consider the following critical nonlinear elliptic equation: \[ - \Delta u = a(x) |u|^{2^*-2}u \quad \text{in } \mathbb{R}^N, \quad u \in \mathcal{D}^{1,2}(\mathbb{R}^N) \] where $N \ge 3$, $2^* = \frac{2N}{N - 2}$, $a(x) \in C(\mathbb{R}^N, \mathbb{R})$ is a positive function that is invariant under the map $x \to -\frac{x}{|x|^2}$. Under some assumptions on $a(x)$, we show the existence of a positive solution to the equation that is invariant under the Kelvin transform. The symmetry condition imposed here is substantially weaker than the invariance under a noncompact symmetry group that is typically assumed in the literature. The key to the proof is a classification of the Palais–Smale sequences of the associated energy functional. To this end, we establish a new abstract profile decomposition theorem incorporating symmetries such as the Kelvin transform.

We study topological relaxation in ordered viscoelastic conformation flows at finite epsilon. In an ordered region, a positive spectral gap selects an oriented principal axis and hence an S^2-valued director with a Hopf class. We show that a change of this class must be accompanied, before the sharp-interface limit, by one of a finite list of costs: exterior gap concentration, ordered-core mass, boundary flux, FENE/collar loss, or a topology exit. The result is proved for a concrete Landau-de Gennes ordered-core closure coupled to an Oldroyd/FENE-type transport law. The structural hypotheses used in the argument are verified up to the first typed exit time: the Morse-Bott ordered well, tubular soft coordinates, massive-mode coercivity, a projected transported Ginzburg-Landau equation, exterior gap control, and tame FENE/collar coefficients. The projected Ginzburg-Landau equation separates translation modes from the remaining residuals. The translation modes give the normal line force, while the orthogonal soft, massive, geometric, and collar terms are absorbed by coercivity or charged to the corresponding exit. A modulated-energy argument propagates a nonempty class of vortex-tube data on regular intervals. On each such interval, the normalized core measures converge to an integral one-varifold satisfying a transported forced Brakke inequality with the computed force. The theorem therefore derives the force projection, open-basin propagation, and Brakke compactness estimates before invoking any limiting Brakke flow, and it records the finite-epsilon cost when the regular ordered-core description breaks down.

We study the abstract damped wave equation on a Banach space, allowing the damping coefficient to be unbounded. By recasting the equation as a first-order system and identifying conditions under which the associated block operator generates a $C_0$-group, we construct generalized cosine and sine families that represent mild and classical solutions, extending the classical undamped theory. We establish existence, uniqueness, regularity, invariant subspaces, growth rate, and trigonometric type identities for these families. Our setup applies to a broad class of damped wave, Klein–Gordon, and higher-order PDE examples, including cases where damping restores well-posedness that fails in the undamped equation.

We prove that in a non-trivial conformal circle-preserving transformation (CPT for short), $e^{\sigma}F$ on a (forward or backward) complete Finslerian manifold $(M,F)$, the conformal factor has at most two critical points. Then a diffeomorphism classification based on the number of critical points as well as some curvature rigidity properties - of Finslerian manifolds that admit nontrivial conformal CPTs - is presented.

We consider a derivative nonlinear Schrödinger system called the Chen-Lee-Liu type system on the torus. This system is known as a completely integrable system. We prove the flow map fails to be continuous at every point in the Sobolev space $H^s(\mathbb{T}) \times H^s(\mathbb{T})$. Moreover, we establish an additional condition required for the flow map to be continuous. For the discontinuity, we take a sequence converging to the initial data for which the corresponding solutions do not exist.

This paper is concerned with the decay of solutions and the analysis of the local and global exact controllability to trajectories for a class of one-dimensional nonlinear parabolic systems with weakly degenerate diffusion coefficients. The system is controlled through the coefficient of the reaction term. Our approach relies on a well-known local inversion method combined with suitable a priori estimates specifically adapted to the degenerate setting.

We study the controllability and reachability of sphere-constrained evolution equations under degenerate (low-mode) forcing, with the harmonic map heat flow as the principal application. Exploiting the underlying geometric structure, we reformulate the problem as an infinite-dimensional control-affine system in Fourier variables and analyze the Lie algebra generated by the controlled vector fields. We prove that iterated Lie brackets generate new admissible directions, providing a mechanism through which finitely many control modes propagate their influence across infinitely many Fourier components. The results provide a Lie-algebraic framework for controlling manifold-valued evolution equations.

We discuss a free boundary model for tumor invasion that describes a cloud of cells that diffuse and, at the same time, are drifted along the vector field of the chemotactic direction. The model captures the evolution of a solid tumor, including the process of angiogenesis, which consists in the formation of new blood vessels that supply the tumor with oxygen and other nutrients, thereby promoting its spread and growth. We prove that, once formed, the tumor survives through time, maintaining strictly positive thickness. An explicit expression in terms of the initial data is derived. Moreover, we distinguish two regimes depending on the ratio $\kappa$ between the spreading of tumor cells and the growth of the tumor mass. If $\kappa$ is sufficiently large, then the tumor grows exponentially in time and invades the entire host tissue. In contrast, if $\kappa$ is small enough, then either the tumor remains bounded in size over time or may experience a fast contraction.

We study the Cauchy problem in $\mathbb{R}^{3}$ for the repulsive compressible Navier-Stokes-Riesz system with Riesz exponent $0<s<1$ and viscosity $0<\varepsilon\leq1$, where the Riesz interaction $\nabla(-\Delta)^{-s}(\rho-\bar{\rho})$ is a generalization of the Coulomb interaction for electrons. For small perturbations of a constant equilibrium, with the solenoidal component of the initial velocity of order $\mathcal{O}(\varepsilon)$, we prove the global existence and uniqueness of smooth solutions. We derive time-decay estimates in $L^{2}$ norms and $L^{\infty}$ norms that capture both uniform-in-$\varepsilon$ dispersive behavior and viscosity-dependent dissipation. We further establish a global-in-time inviscid limit to the irrotational global solution of the compressible Euler-Riesz system whose initial data consist of the same density and the curl-free component of the velocity, with an explicit convergence rate in $W^{k,p}$ norms. The proof combines viscosity-adapted dispersive estimates, normal-form analysis and nonlinear energy estimates with control of both negative and positive Sobolev norms.

Let $\Omega\subset\mathbb{R}^N$ be a bounded open convex set, and let $0=\mu_0(\Omega)<\mu_1(\Omega)\le \mu_2(\Omega)\le\cdots$ be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds $$ \mu_2(\Omega)\le 4\mu_1(\Omega),\qquad \mu_3(\Omega)\le 9\mu_1(\Omega). $$ The first estimate resolves a problem attributed to Henrot, while the second gives the next sharp case predicted by the one-dimensional model. The constants are optimal in every dimension.

The goal of this note is to investigate quantitative stability properties of the critical Sobolev inequality in ${\sf CD}(N-1,N)$ metric measure spaces. Assuming that the optimal constant for the inequality is almost the same as the one of the round sphere, we show that the cumulative distribution of any almost extremal function is close, in Wasserstein distance, to the one of an Aubin-Talenti bubble on the round sphere. We obtain similar results for the log Sobolev inequality and the spectral gap under various curvature and dimension assumptions. In all cases we obtain a quantitative stability with sharp exponent.

In this paper, we consider the three-dimensional incompressible rotating Navier–Stokes equations and establish the sharp $L^p$ decay estimates of global solutions. We reveal that the optimal $L^p$ decay rates for $2<p<\infty$ are strictly faster than those obtained in existing results by interpolation between the $L^2$ unitary identity and $L^\infty$ dispersive estimates, although the endpoint cases were known to be sharp. Moreover, the optimality of decay rates is also proved by the lower bound estimate for a specific initial datum. The underlying mechanism lies in the anisotropic degeneracy of the oscillatory integrals arising from the Coriolis force.

We study quantitative convergence rates for vanishing viscosity approximations of possibly degenerate viscous Hamilton–Jacobi equations on the flat torus. The limiting equation contains a spatially dependent diffusion coefficient a(x) >= 0, which is allowed to vanish. Under standard structural assumptions on the Hamiltonian, we first prove a pointwise convergence rate of order O(epsilon |log epsilon|). We then show that, when the error is tested against a smooth probability density, the logarithmic loss can be removed and an averaged O(epsilon) rate holds. The proof is based on the nonlinear adjoint method, weighted Hessian estimates, and entropy estimates for the adjoint density.

We establish the first results on $\mathrm{L}^p$ bounds for Riesz transforms associated with non-autonomous second order parabolic differential operators in divergence form with bounded coefficients that depend measurably on all variables. In the case of complex coefficients, we identify the maximal open range of exponents $1<p \leq2$ through the availability of $\mathrm{L}^p$ resolvent bounds. This open range always contains the lower parabolic Sobolev conjugate of $2$ and the result is sharp in spatial dimension $n \geq 2$. For real coefficients, we prove extrapolation to the full range. Our argument relies on novel space-time off-diagonal bounds based on two complementary geometries: parabolic cubes on small scales and regions modeled after the half-order time derivative of a parabolic Bessel potential on large scales.