Papers

In this article, we investigate the existence and properties of time-periodic solutions for damped evolutionary partial differential equations subject to periodic forcing. Particular emphasis is placed on configurations where the energy decay of the corresponding free equation is non-uniform. We link the speed of decay and the existence of periodic solutions for the forced equation. Furthermore, we characterize the relationship between the resolvent growth and the associated loss of regularity. The theoretical framework is illustrated through several examples, including linear and nonlinear damped wave equations and coupled hyperbolic-parabolic systems. Finally, we provide a counterexample demonstrating the occurrence of resonance, in which regular but unbounded solutions emerge despite damping.

We consider elliptic measure data problems of the type \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = \mu \] in a bounded domain in $\mathbb{R}^n$, where $p<q$ and $a(\cdot) \ge 0$. We prove local Calderón–Zygmund estimates in the singular case $2-1/n < p < 2$, under natural assumptions on $p$, $q$ and $a(\cdot)$.

This article mainly proves the existence of stationary correctors under space-time spectral gap conditions, which exhibit different properties from those of elliptic operator correctors. Additionally, new flux correctors and their fluctuation estimates are introduced.Based on this, we obtain the optimal homogenization error in the sense of strong and weak norms on C1 cylinders by using the duality and distance-weighted arguments, in which the (weighted) annealed Calderon-Zygmund estimates coupled with a novel form of the minimal radius are developed. Throughout the paper, no small-scale smoothness of the coefficients is used.

The energy dissipation in the inviscid limit is a central problem in turbulence theory. Kolmogorov's K41 theory predicts a positive dissipation rate independent of viscosity – a phenomenon known as anomalous dissipation. Brué and De Lellis gave the first rigorous construction, but it relies on extremely precise geometric conditions. Based on quasi-self-similar mixing, we prove structural stability under pure normal perturbations of the central curves. We establish C^2 stability of the maps and C^1 stability of the local fields, and obtain Hölder estimates and high-frequency energy concentration. A contradiction gives a positive dissipation lower bound independent of the perturbation, and embedding into the (2+1/2)-dimensional framework shows C^6 structural stability. The main novelty is that the Brué–De Lellis construction remains stable under such perturbations, so anomalous dissipation occurs in an open neighbourhood of function spaces, providing a rigorous foundation for K41 theory.

We first prove that all the functions in L 2 whose directional derivative is in L 2 have a directional trace on the boundary of any open bounded domain, without assumptions on its regularity. This enables us to define the omnidirectional trace of the elements of the Sobolev space H 1 ($\Omega$) for which there exists a function on the boundary that is almost everywhere equal, with respect to the directional measure, to the directional trace, regardless of the direction. The set of all these elements of H 1 ($\Omega$), denoted by H 1 tr ($\Omega$), is shown to be closed, and to always contain the closure in H 1 ($\Omega$) of the set C 0 ($\Omega$)$\cap$ H 1 ($\Omega$) (it is always equal to this set in the 1D case, and can be strictly greater in higher dimensions). The omnidirectional trace always satisfies an integration-by-parts formula, which combines the values of the trace on opposite points of the boundary. Examples show that this notion enables the resolution of variational problems involving the values at the boundary of the domain.

In this paper, we establish unique continuation inequalities at two time points for the Dunkl–Schrödinger equation. The proof is based on quantitative uncertainty principles for the Dunkl transform. In particular, we prove that pairs of (\varepsilon,k)-thin sets form strong annihilating pairs for the Dunkl transform, which yields quantitative unique continuation properties for solutions to the Dunkl–Schrödinger equation.

We investigate the global regularity problem for the three-dimensional incompressible Navier-Stokes equations restricted to axisymmetric flows in a finite cylinder $D = \{(r,\theta,x_3): 0 \le r \le 1, 0 \le \theta < 2\pi, 0 \le x_3 \le 1\}$, subject to the Navier-Hodge-Lions (NHL) boundary condition. While global existence of smooth solutions is known in the swirl-free case, the presence of swirl ($v_\theta \neq 0$) introduces vortex stretching that may potentially lead to finite-time singularity formation. In this work, we prove that if the initial data satisfy a scaling-invariant smallness condition of the form \[ \frac{9C_1C_3^{1/2}}{4}\left(\frac{1}{2}\|V_0\|_{L^4}^4 + \|\Omega_0\|_{L^2}^2\right)^{1/4}\|\Gamma_0\|_{L^4} \le \frac{1}{4}, \] where $V = v_\theta/\sqrt{r}$, $\Omega = \omega_\theta/r$, $\Gamma = r v_\theta$, and $C_1, C_3$ are explicit constants given in this paper, then the solution remains globally regular for all time. The proof proceeds via a transformed system for $\Omega$ and $V$, leveraging a maximum principle for $\Gamma$, refined Agmon-type inequalities to control $\|v_r/r\|_{L^\infty}$, and delicate boundary analysis of the finite cylinder geometry. Key energy estimates yield $L^\infty_T L^4_x$ bounds for all velocity components, which fall within the regularity class, thereby precluding finite-time blow-up. The result extends the known criticality theory for axisymmetric Navier-Stokes flows to the setting of NHL boundary conditions, which are physically relevant for flows with stress-free or slip-type constraints on lateral and horizontal boundaries.

In this paper, we address the existence of Fredholm backstepping transformations for self-adjoint and skew-adjoint operators $A$. Under suitable assumptions on the operator $A$ and the possibly unbounded control operator $B$, we prove the existence of a Fredholm backstepping transformation for operators of order strictly greater than $1$. This work overcomes two major limitations of the classical Fredholm backstepping framework. One of the main contributions is the explicit identification of the underlying isomorphism used in the construction of the transformation $T$, thereby bypassing the compactness arguments and Riesz basis mechanisms traditionally used in the literature. This explicit structure enables us to derive quantitative and sharp estimates for $\|T\|_{\mathcal{L}(H;H)}$ and $\|T^{-1}\|_{\mathcal{L}(H;H)}$ with respect to the decay rate $\lambda$. As a consequence, we obtain quantitative rapid stabilization and small-time null controllability results for a broad class of operators.

n this paper, we revisit the existence of global weak solutions of wave maps from $\R^n$ into the sphere $\mathbb{S}^{L-1}$, $\Box u\perp T_u \mathbb{S}^{L-1}$, by establishing it as a singular limit of maps from $\R^n\times \R_+$ to $\mathbb S^{L-1}$ that minimize elliptic regularized variational functionals that contain an exponential weight in the time direction with a small parameter $\varepsilon$, where the initial data of the Cauchy problem serve as the boundary condition. The idea went back to De Giorgi \cite{Giorgi1996}, which has been implemented by Serra and Tilli \cite{Serra-Tilli2012, Serra-Tilli2016} for certain class of nonlinear wave equations. This approach is also applicable to the $SO(m)$-target manifold.

We present a comprehensive framework for gravity-driven, surfactant-laden thin films flowing over slippery substrates, elucidating how wall slip modifies the coupled hydrodynamics and interfacial transport. A long-wave model is formulated with a conservative bulk-surface mass balance and a Navier slip condition. The Orr-Sommerfeld eigenvalue problem governs the linear regime, while a weighted-residual model captures the nonlinear evolution over a range of equilibrium surfactant coverages, Marangoni strengths, and adsorption kinetics. The analysis predicts a non-monotonic variation of the critical Reynolds number with equilibrium coverage, exhibiting a maximum at intermediate $\Gamma_e$, and a slip-induced transition from single- to double-hump solitary structures with increasing Marangoni number, accompanied by attenuated capillary ripples. Under fast adsorption kinetics, the surface field homogenizes, preserving the mean film shape and flux while flattening both the surface concentration $\Gamma$ and the bulk inventory $\chi + h\phi$. A spurious interfacial mass growth reported by Pascal et al.(PRF, 2019) and D'Alessio et al.(JFM, 2020) is resolved through a revised surface balance ensuring strict conservation. Wall slip thus emerges as a key control parameter, reducing viscous resistance and mitigating Marangoni back-stress. The slip parameter $\beta$ is a useful control knob for surfactant-laden films. Slip prevents fragile multi-hump bound states, promoting a single broad crest or an almost flat, uniform sheet by carefully bonding $\beta$ to wave selection, ripple damping, and the bulk-surface surfactant balance.

On a general Riemannian manifold the Navier-Stokes equations admit several inequivalent formulations, differing in the choice of viscous operator: the Hodge Laplacian, the Bochner Laplacian, or the deformation Laplacian. We show that a Lagrangian kinematic construction, in which the strain rate is built from the rate of change of inner products of Lie-dragged connecting vectors, uniquely selects the deformation Laplacian for fluids whose configuration space is intrinsically the manifold. The Hodge Laplacian is excluded at the kinematic step (before introducing constitutive assumptions) because the strain rate constructed from inner-product geometry is symmetric and has no antisymmetric part. We further show that when the fluid arises as a thin-shell limit of an ambient three-dimensional flow, the operator that emerges depends on the boundary condition imposed in the normal direction: stress-free (Navier slip) conditions recover the deformation Laplacian, while Hodge boundary conditions recover the Hodge Laplacian, via an explicit decomposition of the ambient Bochner Laplacian into intrinsic and extrinsic pieces. The intrinsic piece is the deformation Laplacian regardless of the boundary condition. As an analytical confirmation, we show that the kinematic selection is consistent with the known failure of the energy inequality for the Hodge Laplacian on the hyperbolic plane $\HH^2$: the deformation Laplacian is coercive on $\HH^2$ while the Hodge Laplacian is not, because the Ricci term has the opposite sign in the two operators. We further prove that on any complete two-dimensional manifold with Gaussian curvature bounded above by a negative constant, the incompressible Navier-Stokes equation with the deformation Laplacian admits a unique global weak solution with exponential energy decay, resolving the analytical obstruction preventing the corresponding result for the Hodge Laplacian.

We study a new fully averaged poroelastic Kirchhoff plate model coupled with the flow of an incompressible, viscous fluid governed by the time-dependent Stokes equations. The fully averaged formulation offers several advantages over the classical Biot poroelastic plate model: both elastodynamic and pressure equations are posed on a codimension-one interface, the resulting numerical schemes are simpler to implement and computationally more efficient, and the fluid-structure coupling is more natural. We analyze a linearly coupled fluid-structure interaction problem with kinematic and dynamic interface conditions enforcing continuity of normal velocities, the Beavers-Joseph-Saffman slip in the tangential velocities, and balance of forces between the fluid and the poroelastic structure. We establish the existence of weak solutions using energy methods, and then prove global-in-time existence of a unique strong solution to a regularized version of the problem using sectoriality of the associated spatial operator and maximal $\mathrm{L}^p$-regularity of the resulting Cauchy problem. For data with exponential decay, we prove exponential decay of solutions. Finally, we develop a finite element method for the numerical approximation of the coupled system and show that it provides an excellent approximation of the full Biot-Stokes system in the thin-structure regime. The main advantage of this model lies in the remarkably simple implementation, as the poroelastic plate equations constitute a surface model bounding a bulk fluid domain. These results provide a rigorous analytical and computational framework for the study of coupled fluid-poroelastic structure interactions involving thin poroelastic interfaces modeled by the fully averaged Kirchhoff poroelastic plate equations.

Concentrated forces acting at the tip of a two-dimensional wedge give rise to the classical Flamant solution to linear elasticity, whose displacement and strain are singular at the tip of the wedge. Starting from nonlinear elasticity, we prove that the Flamant solution gives the leading order response of a slightly truncated wedge to small boundary displacements or loads. This asymptotic result holds for general hyperelastic energies with super-quadratic growth at infinity; it also holds in the borderline case of quadratic growth at infinity, so long as the tip of the wedge is subjected to small enough displacements or loads. A main point of the proof is to restore compactness to low-energy sequences. We do so by applying a logarithmic change of variables sufficiently far from the tip. To justify this change of variables, we prove a geometric rigidity inequality in $L^p$ for truncated wedge domains with a constant that is uniform in the truncation length. This follows from the bi-Lipschitz invariance of the constant in the $L^p$ Friesecke–James–Müller inequality. Using this change of variables, we derive an asymptotic variational principle characterizing the Flamant solution in the singular limit of an ideal wedge.

In this paper, we study the stability of the inverse conductivity problem of determining a convex polyhedral inclusion embedded in a homogeneous isotropic medium from a single boundary measurement. The main tools in our analysis are singularity decomposition for elliptic equations in non-smooth domains, propagation of smallness, and microlocal analysis. Combining these tools, we establish a logarithmic stability estimate for the Hausdorff distance between inclusions in terms of the measurement error.

We investigate one-dimensional scalar balance laws with singular convolution-type source terms. Under appropriate convexity and kernel assumptions, we establish the global existence of entropy weak solutions in ${\bf L}^2(\mathbb{R})$, together with two partial uniqueness results, in the ${\bf L}^2$-periodic setting and non-periodic setting with ${\bf L}^1(\mathbb{R})$ kernel. In the ${\bf L}^1$-kernel case, the characteristic speed satisfies an Oleinik-type estimate, and entropy weak solutions possess locally bounded fractional variation for all positive times. Furthermore, we derive a simple criterion characterizing local smoothness and wave breaking of solutions, which, in particular, includes both the Burger-Poisson and the Burgers-Hilbert equation as special cases.

We prove a threshold-sharp stability theory for the conformal scalar-curvature sector on zero-curvature Carter backgrounds. The main result is a fully closed bounded-slab theorem: the reflecting evolution is constructed, the conserved energy is proved positive, the complete affine threshold obstruction is identified, and all remaining finite-energy dynamics are shown to be uniformly stable with no unstable modes. This is the sharp statement for compact reflecting slabs, where genuine time decay is false in general. We then extend the same threshold philosophy to black-hole exteriors, separating the intrinsic conformal mechanism from the exterior scalar-wave inputs needed for red-shift, local energy, limiting absorption, and zero-frequency control. The framework gives main applications to Kerr, Reissner-Nordström, slowly rotating weakly charged Kerr-Newman wall exteriors, and extremal horizon-charge obstructions. Our precise result is that it proves stability only for the conformal scalar-curvature sector, not tensorial or nonlinear gravitational stability, and it distinguishes boundedness, qualitative local decay, polynomial decay, and extremal Aretakis-type obstruction without conflating them.

We construct an approximation of the kernel of the solution of the time dependent Schrödinger equation whose Hamiltonian is a 2D harmonic oscillator in Aharonov-Bohm magnetic field. The main tools used here were established in the paper of A. Laptev and I.M. Sigal, where the authors considered a class of Fourier Integral Operators with global complex phases approximating the fundamental solutions (propagators) for time-dependent Schrödinger equations. For the example considered in this paper we are able to find the main term in the approximation of the kernel that equals a version of the Mehler formula.

We introduce a Rayleigh-quotient minimax method for locating maximal one-sided saddle-node bifurcations in nonlinear equations, including non-variational ones. The method avoids branch continuation and instead selects the critical point directly through the minimax bifurcation formula \[ \lambda^* := \sup_{u\in\mathcal U^o} \inf_{v\in\mathcal S\setminus\{0\}} \mathcal R(u,v), \] where \(\mathcal R\) is a two-variable extended Rayleigh quotient on fixed cones. A saddle point of this quotient simultaneously determines the critical parameter, the bifurcation solution, and the adjoint singularity relation. This gives a direct characterization of the bifurcation threshold and leads to Galerkin minimax approximations, a posteriori parameter-margin estimates, and perturbation bounds for the critical value. The abstract assumptions are verified for nonlinear elliptic systems, including non-potential systems.

In this paper, we study the uncertainty principle for Schrödinger equations with a bounded time-independent potentials on certain Cartan-Hadamard manifolds endowed with an asymptotic hyperbolic metric in dimensions $n\geq2$. The classical Hardy uncertainty principle in Euclidean space, as developed in the works of Escauriaza-Kenig-Ponce-Vega (JEMS, 2008; Duke Math. J., 2010), reveals a rigidity phenomenon for solution $u$ to Schrödinger equations: sufficiently strong Gaussian decay at two distinct times yields $u\equiv0$. In this work, we show that a similar rigidity persists in the setting of hyperbolic geometry, despite the absence of translation invariance and Fourier representation. Our approach follows a general strategy of Escauriaza-Kenig-Ponce-Vega, where the underlying geometry brings an essential change. This enables us to establish new Carleman estimates and logarithmic convexity. Unlike the Euclidean setting, the hyperbolic geometry exhibits exponential volume growth and nontrivial geodesic escape at infinity, which fundamentally alters the propagation mechanism of Schrödinger evolutions. Based on the newly-built virial identities and an approximation argument, we derive the logarithmic convexity. The main difficulty in proving the logarithmic convexity is the lack of convolution structure on general manifolds. By making use of the exponential map and Jacobi field, we define a new mollifier on curved geometry. Meanwhile, to establish the Carleman estimate adapted to hyperbolic space, we introduce a new weight function adapted to the curved manifold. Our results highlight the role of curvature in shaping quantitative uniqueness properties for dispersive equations.

We establish a simple and explicit criterion for wave breaking for a general class of perturbed Burgers equations that cover several Burgers-type models, including the Fractional KdV equation, the Whitham equation, and the Fornberg-Whitham equation. The proof is both rigorous and straightforward.

For the physically important case in which the viscosity coefficients depend on the density $\rho$ through a power law (i.e., $\rho^\delta$ with some exponent $\delta \in (\frac{1}{2},1)$), we establish the global well-posedness of regular solutions of the compressible Navier-Stokes equations for barotropic flow with large initial data of spherical symmetry in two and three spatial dimensions. The initial density considered here is positive everywhere but vanishes in the far field, ensuring that the resulting solutions satisfy the conservation laws of total mass and momentum. The most crucial step in our analysis is to obtain a uniform upper bound for the density, which is challenging due to the combined difficulties of degeneracy near the far-field vacuum, coordinate singularity at the origin, and nonlinearity of viscosity coefficients. Furthermore, the methodology developed here can also be applied to the corresponding problem in which the density remains strictly away from the vacuum.

We consider a stochastic Keller-Segel-Navier-Stokes system in $R^2$ describing the collective motion of cells in an ambient stochastic fluid flow, where the cells are attracted by a chemical substance and transported by the ambient fluid velocity, and the fluid motion is self-consistently driven by forces induced by the cells. We prove the existence of a unique mild solution globally-in-time to the two-dimensional stochastic Keller-Segel-Navier-Stokes system with subcritical mass.

We extend Newton's problem of minimal resistance to the Lorentz-Minkowski space. We derive the functional energy and determine the Euler-Lagrange equation. In contrast to the Euclidean case, this equation is quasilinear elliptic, and thus, a maximum principle holds in this context. We obtain the solutions of separable variables of this equation via separation of variables. Furthermore, we find all radial solutions to the problem, which present conical singularities at the origin. We also analyze the Single Shock Condition.

We prove existence of a weak solution to the three-dimensional Beris–Edwards system in the whole space under the stable bulk assumption $c>0$. The solution satisfies the natural bounds $Q\in L^\infty_tH^1_x\cap L^2_tH^2_x$ and $u\in L^\infty_tL^2_x\cap L^2_tH^1_x$, the distributional form of the equations, and the expanded Leray–Hopf type energy inequality used in weak–strong uniqueness arguments. The proof does not pass directly to the limit in that expanded inequality, where the non-corotational terms contain products of the form $|Q^n|^4Q^n:\nabla u^n$. It first obtains the physical free-energy inequality through a hyperviscous approximation and a localized tail estimate, and then derives the expanded inequality from a low-order chain rule for the bulk part of the energy. The last section records the elementary uniaxial reduction which explains why the present argument is restricted to stable bulk potentials.

Building on Willis' homogenization framework, recent work has revealed that heterogeneous conductors exhibit macroscopic thermal bianisotropy, in which the macroscopic heat flux and entropy are nonlocally coupled to both temperature and temperature gradient. Existing numerical examples, however, are limited to the subwavelength regime. Here, we provide the first explicit demonstration of this spatial nonlocality by computing the effective kernels of a periodic laminate using three independent homogenization methods. The three approaches yield consistent nonlocal cross-coupling terms, clarifying the roles of spatial asymmetry and averaging choice. We also calculate the corresponding thermal impedance and show that it is direction-dependent, highlighting a physical signature of thermal bianisotropy relevant to thermal metamaterials.

We study stochastic Euler equations in compressible and incompressible regimes, on the whole space and the torus, driven by mixed multiplicative noise: continuous Stratonovich/Itô components and a discontinuous Marcus component. The noise amplitudes are pseudo-differential operators including the transport operator. We develop a local-in-time theory of classical solutions, establishing existence, uniqueness, and a blow-up criterion. Discontinuous Marcus noise requires new analytical tools to control interactions between jump discontinuities and nonlocal operators. For compressible equations, we formulate a transformation-and-extension principle recovering the Makino variable and its symmetric quasilinear formulation. Using transformed variables, compatible nonlinear Sobolev estimates are developed to close high-order stochastic energy bounds. This accommodates broad physically relevant state equations, including piecewise $\gamma$-laws, Chaplygin laws, and the white dwarf pressure law. Many equations remain unexplored in multidimensional stochastic compressible settings, even under pure Itô forcing. For the incompressible damped case, we identify damping–noise regimes guaranteeing global existence, uniform bounds, and decay. To study statistical behavior, we establish a novel existence criterion for invariant probability measures tailored to Markov semigroups satisfying a restricted Feller property under mismatched metrics. Bypassing single-topology Feller continuity robustly extends the Krylov–Bogoliubov theory. We use this to construct invariant measures for singular stochastic evolution systems in Hilbert spaces. For mixed multiplicative noise and $d\ge 2$, we prove existence of invariant measures for stochastic damped Euler equations on $\mathbb{T}^d$ under moderate damping–noise, and uniqueness under strong damping–noise on $\mathbb{T}^d$ and $\mathbb{R}^d$.

We study the well-posedness of a Cauchy problem associated with the general form of the Laguerre operator and relate it to the corresponding global problem for the harmonic oscillator. To this end, we carry out a detailed analysis of the continuity properties of the associated propagator. Furthermore, we establish connections between several integral transforms, including the fractional Fourier transform and the fractional Hankel transform. Our results highlight the role of Pilipović spaces on positive orthants when studying problems involving the Laguerre operator.

Motivated by PDE-learning, we give a classifying space for nonlinear partial differential operators equivariant under the action of isometries. This classifying space is for operators defined on Riemannian model spaces. The nonlinear operators we are considering are those that can be written as a polynomial in linear operators. We show that the classifying space for such operators can be realized as the vector space spanned by equivalence-classes of multigraphs. We also illustrate how this realization can help us discover non-trivial linear dependence relations between nonlinear differential operators relative to the dimension of the manifold. Finally, we give some comments on operators equivariant under the identity component of the isometry group and under isometry groups of sub-Riemannian model spaces.

We consider an SDE system for signed Coulomb particles moving in $\mathbb R^2$. Due to the singular Coulomb interaction force, collisions between particles of opposite sign will happen in finite time. Upon collision, the colliding particles are removed from the system. Our main results are the existence and uniqueness of strong global solutions and the characterization of all possible collisions. The challenge of the proofs is to deal with the singularity of the interactions. We overcome this by using scaling invariance of the process and by putting together several tools from [FT25] developed for the similar Keller–Segel particle system.

We develop a quantitative statistical theory of transformers in the large-context regime by adopting the abstraction of contextual flow maps (CFMs): dynamical systems that evolve a distinguished token in the presence of a contextual measure across a stack of attention blocks. Within this framework, the finite-context model approximates an idealized infinite-context system in which the contextual measure is replaced by its underlying population, so that the context length $n$ becomes a statistical resource. Exploiting the McKean–Vlasov structure of the dynamics and the classical machinery of propagation of chaos, we establish a forward bound controlling the deviation between the finite- and infinite-context CFMs uniformly along depth, and a backward bound controlling the deviation between the corresponding training trajectories uniformly across iterations of online gradient descent. Both bounds achieve the optimal Wasserstein rate $n^{-1/d}$ for general CFMs and parametric rate $n^{-1/2}$ for a restricted class of CFMs that includes transformers as a special case. The analysis rests on a new Eulerian adjoint formulation of the loss gradient and stability estimates for the resulting forward–adjoint system, both of which may be of independent interest.