Papers

We study a one-dimensional active-line equation motivated by thin stress-sheet dynamics in the high-Weissenberg Oldroyd-B regime. A positive periodic line density $\rho=m+\eta$ satisfies $\rho_t+cP_0\{\rho\Lambda\rho-(\mathcal H\rho)\rho_s\}+\gamma(\rho-m)=0$, where $\Lambda=\mathcal H\partial_s$ and $P_0f=f-\langle f\rangle$. We prove that every strictly positive smooth initial density of arbitrary size generates a unique global smooth solution. The key is the pointwise cancellation obtained after one differentiation: for $w=\rho_s$, $w_t+c\rho\Lambda w-c(\mathcal H\rho)w_s+\gamma w=0$. Its maximum principle controls the slope globally, while critical-drift Hölder and Schauder estimates close all higher derivatives. We also prove quantitative small-oscillation stability. Independently, an exact fourth-difference sum-of-squares identity gives $\int_{\mathbb T}\rho^2\Lambda^3\rho\,ds\geq0$ for every smooth nonnegative density. The stronger derivative-energy sign leads, in its sharp phase-opposed form, to the cubic convolution inequality $\mathcal C(x)\leq2AE_3$, where $A=\sum_{n\geq1}x_n$, $E_3=\sum_{n\geq1}n^3x_n^2$, and $\mathcal C(x)=\sum_{a,b\geq1}(a+b)(a^2+ab+b^2)x_ax_bx_{a+b}$. The constant $2$ is sharp along critical $n^{-3/2}$ plateaux. We prove the inequality for several cutoff-uniform and infinite-support classes, including selected Schur, dyadic-layer, Mellin, and moment families. The unrestricted inequality remains open, but it is not needed for the global regularity theorem.

We prove a finite-scale estimate for vortex stretching in spatially filtered three-dimensional Navier–Stokes flow. The positive near-field part of the filtered stretching is bounded by a pairwise defect of filtered vorticity directions. A magnitude-weighted direction inequality converts this angular defect into a first-order difference quotient of filtered vorticity, and the resulting term is absorbed by filtered diffusion up to a lower-order enstrophy reservoir. In the localized filtered enstrophy balance, the remaining positive surplus is assigned to far-field strain, commutator forcing, and localization residuals. The far-field term is reduced to weighted packing and conditional annular Carleson embedding. The differentiated commutator stress is controlled by a scale-invariant increment defect adapted to the filter and its derivative. At the critical exponent, bounded increment defects generate cylindrical generalized Young-measure profiles.

We consider the cubic nonlinear Schrödinger (NLS) equation on two-dimensional flat tori with varying aspect ratios. In this formulation, the choice of aspect ratio governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors. We present a geometry-conditioned Fourier neural operator (FNO) for the cubic defocusing NLS equation, where the input consists of the real and imaginary parts of the solution together with the aspect-ratio parameter \(\omega^2\). The model is trained to approximate the one-step solution operator and is evaluated on unseen trajectories generated from random-phase initial data using Fourier pseudospectral method. Our numerical experiments show that the learned operator captures the main solution dynamics on both tori and reproduces the distinct Sobolev norm behavior of the two geometries, with stronger \(H^2\)-growth on the rational torus and more constrained behavior on the irrational torus, consistent with the findings of \cite{hrabski2021energy}. We perform ablation studies to examine the roles of retained Fourier modes, activation functions, Fourier-layer depth, and explicit geometry conditioning. The results indicate that including $\omega^2$ improves long-time predictive accuracy, especially for the rational geometry, and supports the use of geometry-aware neural operators for learning spectral-transfer phenomena in nonlinear dispersive partial differential equations.

We invariantly define Gelfand-Shilov spaces on classes of non-compact manifolds with a certain “structure at infinity”. We also construct and study a global calculus of pseudodifferential operators on such manifolds, locally defined by symbols satisfying estimates associated with weight sequences. The operators so obtained act naturally on the previously defined Gelfand-Shilov spaces. These generalise analogous functional spaces and operators defined on Euclidean spaces.

This paper is devoted to the analysis of a quasi-variational–hemivariational inequality associated with the convective Brinkman–Forchheimer extended Darcy (CBFeD) equations for Bingham fluids in both two and three spatial dimensions. The considered model describes incompressible fluid flow through porous media while incorporating convection, nonlinear damping effects, and Forchheimer-type resistance, together with non-smooth and non-convex slip boundary conditions. We first derive an appropriate weak formulation of the problem, which leads naturally to a Bingham-type quasi-variational–hemivariational inequality with a velocity-dependent constraint set. By employing the Kakutani–Ky Fan fixed point theorem, we establish the existence of weak solutions for the resulting multivalued quasi-variational inequality formulation of the CBFeD system. Furthermore, we prove that every weak solution of the associated quasi-variational inequality is also a solution of the corresponding quasi-variational–hemivariational inequality. The analysis presented in this work provides a rigorous mathematical framework for CBFeD models with Bingham fluids under non-monotone and non-smooth boundary interactions.

We study periodic traveling waves in a general class of nonlinear dispersive equations with Burgers dissipation and a monostable Fisher-KPP reaction source. Under natural assumptions on the dispersion symbol, we prove the existence of small-amplitude periodic traveling waves bifurcating from a constant equilibrium by means of a Lyapunov-Schmidt reduction. We then investigate the spectral stability of these waves by combining Floquet-Bloch decomposition with perturbation theory for linear operators. It is shown that the Floquet spectrum of the linearized operator intersects the unstable complex half-plane, implying spectral instability with respect to localized perturbations. The results recover the recently established instability theory for the Korteweg-de Vries-Burgers-Fisher equation as a special case and provide a unified framework for a broad class of local and nonlocal dispersive systems. These results reveal a common instability mechanism generated by the interplay of dispersion, Burgers dissipation, and monostable reaction dynamics.

We establish a local stability result for a class of Saint-Venant type inequalities. Given the solution $u$ of the Dirichlet torsion problem in a domain $\Omega$, we consider shape functionals $\mathcal{J}(\Omega)$ involving the integral of $j(u)$, where $j$ is convex and satisfies suitable structural assumptions. By Talenti's comparison principle, balls maximize $\mathcal{J}$ among sets of prescribed measure. We prove that this extremal property is stable in the class of nearly spherical sets: the deficit from the optimal value controls the square of the $H^{1/2}$-norm of the boundary perturbation. The argument relies on shape derivative techniques, including the computation of the second variation and the introduction of an adjoint state. As applications, the result covers several relevant examples, including the torsional rigidity, $L^p$-norms of the torsion function for $p\ge 2$, and Moser-Trudinger functional in dimension two.

In this paper, we mainly study the interior $C^{2}$ estimates for a class of sum Hessian quotient equations on Riemannian manifolds. For $0\leq l<k< n$, we establish interior $C^{2}$ estimates at the center of a geodesic ball. Let $\Omega$ be a bounded domain (with smooth boundary) on Riemannian manifold. For $0\leq l<k\leq n$, we establish Pogorelov type estimates on $\Omega$ with the vanishing Dirichlet boundary condition.

We consider the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with step-like initial data. Using the nonlinear steepest descent method, we derive the long-time asymptotic expansion of the solution to the Cauchy problem in three distinct transition regions. In the first two transition regions, the leading-order asymptotics are characterized by Painlevé \uppercase\expandafter{\romannumeral34\relax}-type formula, while in the third one is a collisionless shock region, the leading-order asymptotics is describedin terms of Riemann theta functions. Our analysis is based on the Riemann-Hilbert formulation associated with the Cauchy problem of the defocusing NLS equation.

We study the local well-posedness of the Cauchy problem for a class of hyperbolic quasilinear evolution systems with fractional dissipation and commutators of the fractional Laplacian $(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$. The analysis is motivated by the compressible isentropic Navier-Stokes equations with fractional diffusion and the compressible Euler alignment system with singular communication weights. We show that the contribution of the commutators to the energy is absorbed by the dissipation. Our approach does not rely on the cancellation condition previously used to recover the coercivity of the highest-order non-local terms. We further show that the commutator term induces a loss of regularity in the stability estimates, preventing the standard contraction argument from being applied. Therefore, we instead establish the relative compactness of the solution operator and apply Schauder's fixed point theorem. The resulting theory provides local existence for arbitrary orders $\alpha\in(0,1)$ and identifies the obstacles to local uniqueness in the presence of fractional commutators.

Quantum neural networks have emerged as powerful models for approximating nonlinear functions. Yet their use in solving integro-differential equations (IDEs) and fractional integro-partial differential equations (FIPDEs), which involve inherently nonlocal operators, remains unexplored. This work introduces a quantum physics-informed neural network (QPINN) framework that combines a quantum neural network with the governing equations of general nonlinear IDEs and FIPDEs. The proposed quantum network uses an affine feature map and variational quantum circuits to produce trial solutions with explicit trigonometric structure. We prove a quantitative $L^{2}(\mu)$ universal approximation theorem for this architecture, achieving a convergence rate of $\mathcal{O}(n^{-1/2})$. This extends classical Fourier approximation theory to quantum circuits for physics-informed learning. We propose two QPINN variants: the numerical-quadrature QPINN (N-QPINN), which handles nonlocal integrals and fractional operators via high-order numerical quadrature while computing local derivatives through automatic differentiation of quantum trial solutions; and the auxiliary-function QPINN (A-QPINN), which eliminates numerical quadrature by introducing auxiliary variables that reformulate each integro-differential equation as an equivalent coupled system of partial differential equations, enabling a multi-output quantum neural network to simultaneously represent the solution and its associated variables. A series of numerical experiments demonstrates that the proposed QPINN framework accurately captures the behavior of nonlinear IDEs and FIPDEs and outperforms classical physics-informed neural networks.

We introduce a particle method for the numerical approximation of time-dependent first-order Mean Field Games (MFGs) systems with non-separable, displacement monotone Hamiltonians and terminal costs, for arbitrary time-horizons and (possibly) singular initial player distributions in $\mathcal{P}_2(\mathbb{R}^d)$. The numerical scheme is based on an implicit Euler discretization in time and sampling in space of the characteristic Hamiltonian/Pontryagin system associated with the continuous MFGs system. We prove convergence of the approximations of the player distribution in the $L^{\infty}(\mathcal{W}_2)$-metric and the approximations for the gradient of the value function along optimal trajectories in the $L^{\infty}{(L^2)}$-norm as the number of spatial samples tends to infinity jointly with the temporal time-step vanishing. The error bound that we establish for this convergence further implies rates of convergence of the scheme for a range of spatial sampling techniques. Provided that the Lagrangian and terminal costs are additionally locally Lipschitz continuous, we also establish an asymptotic error bound in the $L^{\infty}(L^1)$-norm for the approximations of the value function along optimal trajectories. This is the first work in the literature on rigorous numerical approximation and analysis of first-order MFG systems that handles non-separable Hamiltonians and potentially singular initial agent distributions for arbitrary long time horizons. We illustrate the performance of the scheme in numerical experiments for a range of initial agent distributions, time horizons and space dimensions.

We derive quantitative uniform resolvent estimates for Schrödinger operators on the half-line with inverse-square potentials, which provide a sharp behaviour in the limit of large coupling. Our approach is based on a matrix representation of the boundary value of a weighted resolvent. The partial wave decomposition then turns these one-dimensional channel estimates into explicit weighted resolvent estimates for the Laplacian, its inverse-square potential perturbations and for the magnetic Laplacian with an Aharonov–Bohm potential. We also obtain exact Simon-type identities for the imaginary parts of the weighted resolvents of these operators.

We study dual-cascade flux laws for the stochastic forced–dissipative surface quasi-geostrophic (SQG) equation on a large periodic box. For statistically stationary solutions, under a weak anomalous dissipation assumption, we derive rigorous mixed third-order structure-function laws for the dual cascade: a Yaglom-type law for the direct cascade of surface potential energy (SPE) and an antisymmetrized mixed flux law for the inverse cascade of the Hamiltonian. In particular, the inverse Hamiltonian law appears to be new even as an explicit third-order structure-function relation. We also prove Onsager-type obstruction results showing that sufficiently regular stationary families cannot sustain the corresponding non-zero fluxes: $B^s_{3,\infty}$-regularity above the Onsager threshold $1/3$ rules out the direct SPE flux, while sufficient low-frequency Besov regularity rules out the inverse Hamiltonian flux. These results provide a rigorous formulation of the SQG dual-cascade phenomenology in a stochastic stationary setting.

In this paper, we first propose two different quantities on graphs, namely local entropy and cohesion, then design two corresponding flows for edge weights: the local entropy flow and the cohesion flow. We establish the global existence and uniqueness of solutions for both flows and investigate their asymptotic behaviors, including the case that the limit goes to positive infinity. Moreover, they can be applied to fundamental network analysis tasks, including community detection and node classification. Empirical evaluations demonstrate that our method achieves performance competitive with Ollivier Ricci flow and Lin-Lu-Yau Ricci flow on benchmark network analysis tasks. In experimental scenarios, we first apply the cohesion flow to evolve the edge weights of the graph, and then apply the local entropy flow to further update the resulting weighted graph. Both flows are computationally efficient, leading to a significant reduction in overall computational cost and improved scalability.

For every \(0<\kappa<\sqrt{5}-2\), we prove global existence for the two-dimensional generalized parabolic Anderson model on the whole plane $\mathbb R^2$ with nonlinearity $F\in C_b^2(\mathbb R)$, driven by an enhanced noise $(\eta,\Psi)$. The noise $\eta$ has polynomially weighted spatial Besov–Hölder regularity $-1-\kappa$, and $\Psi$ is the corresponding renormalized second-order object. If $F''$ is globally Lipschitz, the solution is unique. The proof combines a weight-compatible annular high–low decomposition with a paracontrolled transport representation. The final remainder is estimated simultaneously in a weighted $L^\infty$ norm and in a higher-order weighted parabolic Hölder norm, using two strictly different polynomial weights. This weight gap absorbs the polynomial losses generated by the enhanced noise, the localization procedure, and the transport coefficient. Several refinements of earlier work allow the maximum-principle and Schauder estimates to yield a global a priori bound for a larger range of $\kappa$. Uniqueness is proved in a time-dependent exponentially weighted topology.

We study self-similar singularity formation for the one-dimensional weak-advection Hou–Li model, a reduced model motivated by the axisymmetric Euler equations. In the periodic setting, we construct exact finite-time self-similar blowup solutions for $2/3<a<1$, with profiles that are neither focusing nor expanding. In the whole-space setting with a Neumann condition, we construct exact finite-time self-similar blowup solutions for the full range $0<a\leq1$, with profiles of focusing, non-expanding/non-focusing, or expanding form depending on the sign of the self-similar scaling parameter. The construction is based on a fixed-point formulation near the origin, followed by an ODE extension argument. We also establish regularity, asymptotic behavior, monotonicity properties of the profiles, and uniqueness up to the natural scaling invariance.

We are interested in the local higher integrability of solutions to elliptic equations with linear growth of the form $$-\text{ div}A(x,Du)=f(x). $$ Under a Besov regularity on the partial map $x \mapsto A(x,\xi)$ and a summability condition on the datum $f$ in the scale of Lebesgue spaces, we prove that the solutions are almost Lipschitz continuous, i.e. their gradients belong locally to $L^q$, for any finite exponent $q$. In turn, solutions are locally $\gamma$-Hölder continuous, for every $\gamma \in (0,1)$. The difficulty arising from the lack of an explicit second variation for the problem is overcome by testing the equation with a function proportional to a power of the finite difference quotient of the solution. To the best of our knowledge, this technique is used in this context for the first time. We also provide an example showing the sharpness of our result in the scale of Lebesgue spaces.

We investigate a class of nonlocal equations whose leading operator is modeled on either the fractional $p$-Laplacian or the regional fractional $p$-Laplacian, $p \in (1,\infty)$. We prove local self-improving properties of weak solutions to the fractional $p$-Laplacian in the case $p\in(1,\infty)$ with non-integrable right-hand side, as well as to the regional fractional $p$-Laplacian in the subquadratic case $1 < p < 2$, by extending the nonlinear commutator estimates developed by Schikorra (Math. Ann. 366 (1-2):695–720, 2016).

For a time dependent Schrödinger equation, the scattering map is the map sending the asymptotic profile of a solution as $t \to-\infty$ to its asymptotic profile as $t\to+\infty$. In this paper we show that, for a certain class of metrics, the scattering maps and Poisson operators associated to two Schrödinger operators on the same curved space only differ by a compact operator on a critical level if and only if the two potentials are equal.

We study the lifespan of mild solutions to the fractional semilinear parabolic Cauchy problem with a Hardy–Hénon-type weight \[ u_t + (-\Delta)^s u = |x|^{-\gamma}\,|u|^p, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0,x)=\varepsilon\,u_0(x), \] where $0<s<1$, $0\le\gamma<\min(2s,N)$, $p>1$ and $u_0\in L^1\cap L^\infty$ with $\int_{\mathbb{R}^N}u_0(x)\,dx>0$. Setting \[ p_F \;:=\; 1+\frac{2s-\gamma}{N}, \] we prove that the lifespan $T_\varepsilon$ obeys, for every sufficiently small $\varepsilon>0$, \[ T_\varepsilon \;\approx\; \begin{cases} \varepsilon^{-\,\beta^{-1}},& 1<p<p_F,\\[1mm] \exp\!\big(C\,\varepsilon^{-(p-1)}\big),& p=p_F,\\[1mm] +\infty,& p>p_F, \end{cases} \qquad \beta \;=\;\frac{(2s-\gamma)-N(p-1)}{2s(p-1)}. \] The lower bound rests on fractional heat-kernel estimates and an $L^1$–$L^\infty$ Hardy-type interpolation inequality; the upper bound is obtained by testing the equation against the backward fractional heat kernel, a globally defined positive weight for which $(-\Delta)^s$ is controlled everywhere and the linear terms cancel identically by self-adjointness. This circumvents the compactly supported cutoffs of the classical test-function method, which are incompatible with a nonlocal operator. The exponent $\beta$ is sharp; for $\gamma=0$ it reduces to the fractional Lee–Ni exponent $\frac{1}{p-1}-\frac{N}{2s}$. To the best of our knowledge, these results are new even for $\gamma=0.$ We also establish a large-data lifespan law, sharp lower bounds on the blow-up rate together with a conditional Type-I upper bound, a conditional self-similar profile result.

We prove sharp $k$-linear $L^p$ estimates for Carleson–Sjölin oscillatory integral operators with arbitrary separated frequency scales for all $k\ge 2$ and $1\le p\le \infty$. The estimates are sharp, including the endpoint logarithmic behavior for general Carleson–Sjölin phases. Moreover, we obtain log-free endpoint bilinear spectral cluster estimates on every closed three-dimensional Riemannian manifold, resolving a problem of Burq–Gérard–Tzvetkov. As a consequence, we establish sharp $k$-linear $L^p$ spectral cluster estimates for all $k\ge 2$ and $1\le p\le \infty$.

In this article, we consider the anisotropic Calderón problem of determining the potential from boundary measurements of the time-independent Schrödinger equation on compact Riemannian manifolds with boundary. We prove a global uniqueness theorem for small perturbations of the Euclidean metric.

We study qualitative properties of positive singular solutions to a weakly coupled elliptic system with critical exponents in a punctured ball. We give a sharp criterion on the removablity of the isolated singularity. We prove that semi-singular solutions (i.e., solutions with one component bounded near the singularity and the other component unbounded near the singularity) do not exist for the dimensions $N\geq 4$ but do exist for $N=3$. Asymptotic symmetry and sharp pointwise estimates are also proved for singular solutions. These generalizes some classical results of (Caffarelli, Gidas and Spruck, Comm. Pure Appl. Math, 1989) to the weakly coupled system. Moreover, for the weakly coupled system, we demonstrate a novel phenomenon that does not arise in the scalar equation.

This paper investigates the long-time behavior of a semilinear beam equation in a domain $\Omega \subset R^{n}$, with memory and fractional damping of the form $(-\Delta)^{\alpha}u_{t}$ ($\alpha \in [0,2]$ the dissipation index). Two critical growth indices of the nonlinear term are determined for smooth and $C^2$ boundaries respectively, concerning the existence of the associated semigroup. We prove the existence of global attractor for the semigroup by showing that it possesses a bounded absorbing set and asymptotic compactness. Furthermore, we find out a new way to obtain, for all $\alpha$, higher regularities than anticipated for the attractors, and the regularity result indicates an interesting phenomenon that even much weaker damping can produce regularity that is infinitely close to that in the case of strong damping ($\alpha =2$). As a consequence, our regularity result deepens and extends the existing related ones for the case when the memory is absent.

In this article, a non-homogeneous $Tb$ type theorem for arbitrary dimensional Calderón-Zygmund singular integral operators is proved. This is an extension of an analogous non-homogeneous $Tb$ theorem for the Cauchy transform, in the planar setting, due to Nazarov, Treil and Volberg. The novelties of the present work are the change of dimension and the fact that the operators to which the theorem applies are not necessarily antisymmetric. The techniques used in the proof include, among others, suppressed kernels, decompositions in $L^2(\mu)$, where $\mu$ is a Radon measure in $\mathbb{R}^d$, and a probabilistic argument resulting from taking averages of the operators involved.

Given a closed, real, non-singular 1-form on a compact manifold $\Omega$, global properties of the associated differential complex are studied. We completely characterize global solvability in the first and last levels of the complex. Furthermore, in the particular case where the $1$-form is rational, we prove global solvability for every degree and give a complete description of the cohomology spaces. Finally, a complete characterization for global hypoellipticity is obtained, building on the work of A. Meziani (Comm. PDE., 2002). In all cases, it is shown that the conditions depend exclusively on the arithmetic nature of the form's periods.

For piecewise deterministic samplers such as Randomized Hamiltonian Monte Carlo (RHMC), Bouncy Particle Sampler (BPS) or Zig-Zag Process (ZZP), long-time exponential convergence rates have been established in previous works using Harris or $L^2$ hypocoercivity approaches. In particular, in the $L^2$ framework, a so-called diffusive-to-ballistic speedup was known for log-concave targets, according to which the convergence rates of these samplers, with suitable parameters, are quadratically improved with respect to the standard overdamped Langevin diffusion process. A recent work by Jianfeng Lu showed that this speedup also holds for the kinetic Langevin diffusion process when the convergence is stated in terms of relative entropy, raising the question whether this also holds for piecewise deterministic samplers. The present work provides a positive and a negative answer to this: first, we show that the speedup holds in entropy for RHMC; second, we show that for BPS or ZZS, even for a standard Gaussian target, a similar result cannot hold, and even that exponential convergence (at any rate) in entropy fails.

The paper deals with initial boundary value problem for the Magnetic Zakharov system arising in plasma physics in two-dimensional domains. It was proved the global well-posedness of this problem in some Sobolev type class.

In this paper, we investigate the regularity theory of local minimizers of multi-phase energy functionals. As a key feature of our work, instead of the classical Hölder continuity assumptions on the modulating coefficients and interaction between the growth exponents, we assume that these coefficients belong to a suitable class of Muckenhoupt weights. The presence of multiple growth phases with the degenerate or singular nature of Muckenhoupt weights poses substantial analytical difficulties that prevent a direct application of classical theory. Our approach requires a refinement of localized energy estimates and an adapted iteration scheme that exploits the reverse Hölder properties of the Muckenhoupt weights. As our main results, we establish the higher integrability, local boundedness, and Hölder continuity of local minimizers. Most notably, we prove the Harnack inequality for non-negative local minimizers, which, to the best of our knowledge, stands as the first result of its kind in the multi-phase setting involving Muckenhoupt modulating coefficients. This paper is a contribution toward a better understanding of the qualitative behavior of minimizers in non-uniformly elliptic variational problems, and offers a new framework that complements the existing literature beyond the classical Hölder continuity of modulating coefficients.