Papers

Let $X$ be a compact Kähler manifold. In prior work, we constructed diffeological moduli stacks of Higgs and flat bundles on $X$, related by extension completion of smooth harmonic families. Here, we develop the relative analytic theory. On Sobolev completions over arbitrary plots, we prove that every smooth stable Higgs family satisfying the numerical conditions admits a global smooth harmonic metric. Fixing a Hermitian–Einstein determinant metric removes scalar freedom, and then elliptic regularity and normalized gluing yield plotwise smoothness. The theorem holds at every finite parameter regularity $C^d$ and on reduced singular parameter spaces with ambient extensions. For a Higgs deformation $\eta$, the normalized metric variation satisfies $L_hs=-\mathcal S_h(\eta)$ and $s=-G_h\mathcal S_h(\eta)$ up to an independent rank-one determinant term for $\mathrm{GL}_r$. This computes the plotwise differential and recovers the classical comparison. Locally split, constant-type polystable families admit smooth harmonic metrics. Real-analytic examples show general polystable families may have neither continuous harmonic metrics nor relative harmonic filtrations and may lie outside every $C^d$ extension-generated locus. In one example a singular harmonic reduction produces a continuous adjoint Higgs field and a flat family with semisimple slices. This defines a weak $C^0$ operator-level harmonic mediator, strictly larger than the metric-regular one, whose endpoint images after finite extension completion and stackification satisfy $\mathscr M_{\mathrm{Dol},0}^{\mathrm{wk}\mathcal H}(X)\simeq\mathscr M_{\mathrm{dR},0}^{\mathrm{wk}\mathcal H}(X)$. We characterize the extension-generated stack by relative harmonic filtrations, develop their obstruction theory, analyze the loss of extension data under heat flow, and construct the smooth Hodge $\lambda$-family on the stable locus.

We study Serrin-type overdetermined problems for a class of possibly degenerate elliptic operators under a vertical boundary condition forcing gradient blow-up. We identify the precise regime in which the problem admits a solution on a ball. In this regime, we prove rigidity: every solution domain is a ball, and the solution is uniquely determined by an explicit radial profile.

We consider collisional kinetic equations whose collision operator is not necessarily bilinear and prove quantitative convergence to the Navier-Stokes-Fourier system in weighted Sobolev spaces, together with a description of the initial layers. The aim of this paper is to conciliate the conditional convergence result of Bardos-Golse-Levermore for abstract kinetic equations conserving macroscopic quantities and dissipating entropy with the spectral strategy initiated by Bardos-Ukai for the Boltzmann equation. This work extends the abstract approach of Gervais-Lods which was restricted to bilinear collisions (Boltzmann, Landau or quadratic approximation of other models) to non-bilinear models such as the Boltzmann-Fermi-Dirac equation, the BGK equation and the nonlinear Fokker-Planck equation.

The content of the following pages was part of a special topics lecture given at the Australian National University during the first semester of 2026. That was a very nice expericence, I really enjoyed giving this 12 weeks (2 hours a week) lecture. It is meant to be self contained, starting with results on Fourier transform and Sobolev spaces. As a toy model, before treating the Navier-Stokes system, we focus on the non linear heat equation where the non linearity is polynomial. Ultimately, we prove existence and uniqueness of mild solutions of the Navier-Stokes equations in critical spaces. This script contains some exercises and the text of the mid-semester exam as well as the final exam.

This paper is concerned with the existence theory of isometric immersions of surfaces with negative Gaussian curvature into the three-dimensional Euclidean space. We reformulate the Gauss–Codazzi equations, i.e., the partial differential equations for isometric immersions, into hyperbolic conservation laws for the flows of Chaplygin gas with nonzero source terms. Then, employing theories of invariant regions and compensated compactness, we establish for any given $p \in [2,\infty[$ the existence of $W^{2,p}_{\rm loc}$-isometric immersions of various general families of metrics over infinite strips $\mathbb{R}\times [0,T]$ with arbitrarily large $T$. Such metrics include those of various classical minimal surfaces: helicoid, catenoid, and Enneper surfaces, as well as metrics in isothermal coordinates or of the “reciprocal-type”. In our fluid dynamical formulation of the isometric immersion problem, we specialise in the case that the two Riemann invariants for the associated hyperbolic conservation law remain bounded and of distinctive signs, and obtain $L^p$-solutions to the initial-boundary value problem via the method of relative entropy with respect to a nonstationary ODE background. The isometric immersions constructed in this paper have second fundamental forms belonging to $L^p_{\rm loc}\setminus L^\infty_{\rm loc}$.

We consider a Cauchy-Dirichlet problem for a heat equation with variable coefficients in non-divergence form. The initial values are assumed to be homogeneous, while the Dirichlet boundary values are non-constant. In this setting, we derive an asymptotic formula for the short-time behavior of the solution, which extends the celebrated Varadhan formula. We stress the fact that the boundary values are allowed to vanish or change sign. The formula is obtained by combining the original ideas of Varadhan with those of Evans and Ishii, pertaining to the theory of viscosity solutions, and some further remarks. In passing, we prove the elliptic counterpart of the formula. This concerns the slow-diffusion behavior of the solutions of the resolvent equation. Furthermore, for the case of the classical Laplace operator, we prove asymptotic formulas for the heat content and the mean value of the solution of the resolvent equation on spheres touching the boundary of the domain. These extend to the case of non-constant boundary values, certain formulas previously obtained by the second author and S. Sakaguchi. The formulas involve the principal curvatures at the touching point and the Dirichlet boundary values. We then use our formulas to describe time-invariant surfaces for the heat equation, in presence of non-constant Dirichlet boundary values. We give two symmetry results in case of non-negative Dirichlet boundary values and a non-existence result, when those values change sign.

We investigate subsonic time-periodic solutions to the radially symmetric non-isentropic Euler system with nonzero angular velocity in an annulus. Under the assumption that the boundary conditions are dissipative on the inner circle and non-dissipative on the outer circle, we establish the existence and stability of time-periodic solutions which are close to steady radially symmetric subsonic spiral flows. It is worth noting that there is no smallness restriction on the background subsonic solutions. The main difficulties arise from the derivative loss caused by the coupling with the radial entropy derivative and the additional zeroth-order terms induced by the nonzero angular velocity in the diagonalized system. One of the key ingredients of the analysis is to rewrite the radial derivative of the entropy as a derivative along the first or fourth characteristic direction. Another one is to control the accumulation of perturbations along characteristic curves by suitably restricting the width of the annulus.

We study nonlinear parabolic equations with delayed diffusion terms governed by finite signed measure kernels. The atom of the kernel at the origin is absorbed into the present-time operator, while the remaining part is treated as a residual delay kernel. Under structural assumptions on the effective present-time operators and a pathwise coercivity condition for the total memory operator, we prove the existence and uniqueness of weak solutions and their stability under weak-star convergence of the kernels. The stability result covers collapsing delayed atoms, whose mass is transferred to the present-time diffusion coefficient in the limit. We verify the assumptions for p-Laplacian type examples, including separated kernels and a regularized class of kernels reaching the origin.

The compressible fluid-solid interaction (FSI) with a thermomechanical phase transition is formulated at large strains within the Eulerian frame. For the deviatoric part, the Jeffreys (also called anti-Zener) rheology with an additional viscosity is adopted. The core philosophy governing the mechanical solid-liquid transition is that the viscous (or viscoplastic) response is temperature-dependent and may fully degenerate to a viscoelastic fluid during thawing, so that there is no elastic response on the shear distortion. This behavior enables the free flow of the fluid, its subsequent freezing into a new configuration, and potential re-melting back into a fluid, allowing such cycles to repeat indefinitely. The classical Stefan problem, associated with the latent heat of the first-order (thawing-freezing) phase transition, is augmented by incorporating kinetic overheating and undercooling. The analysis by a time discretization with an appropriate truncation is applied to a higher-gradient modification of the original formulation, utilizing the concept of multipolar nonsimple continua.

We study a boundary value problem for the $p$-Laplacian in the perforated domain $B(0,\rho)\setminus\Gamma\subset \mathbb{R}^d$, where $\rho>1$ and $\Gamma$ is the union of many small compact cavities placed near the unit sphere. The cavities are separated at scale $\varepsilon$, asymptotically equidistributed on the sphere, and have cardinality of order $\varepsilon^{1-d}$. The cavities have diameters of order $\alpha(\varepsilon)\varepsilon$, where $\alpha(\varepsilon)\to0$, and their relative $p$-capacity is comparable to the relative p-capacity of a ball of the same diameter. The solution is required to equal $1$ on all cavities and $0$ on $\partial B(0,\rho)$. We focus on the critical case $p=d>1$. We identify the critical scale through the parameter $\tau=\lim_{\varepsilon\downarrow0}[\varepsilon\log(1/\alpha(\varepsilon))]^{-1}\in[0,\infty]$. Thus, $\alpha(\varepsilon)=\exp[-(1+o(1))/(\tau\varepsilon)]$ when $0<\tau<\infty$. Away from the unit sphere, the solutions converge to $A_*U_\rho$, where $U_\rho(x)=\min\{1,1-\log |x|/\log\rho\}$ is the radial $d$-harmonic potential of the unit ball in $B(0,\rho)$. The constant $A_*$ equals $0$ when $\tau=0$, equals $1$ when $\tau=\infty$, and is explicit for $0<\tau<\infty$. We construct an explicit ansatz that approximates the solution for sufficiently small $\varepsilon$ in both $L^{\infty}$ and in terms of $d$-capacity.

Given $n,m\in\mathbb{N}$ such that $n\ge2m\ge4$, letting $g$ be a conformally Euclidean metric on $\mathbb{R}^n$, we consider the question of positivity of the lower-order $Q$-curvatures $Q_g^{(2k)}$ for $k\in\left\{1,\dotsc,m-1\right\}$ when $Q_g^{(2m)}$ is assumed to be nonnegative and not identically zero. We assume moreover that the scalar curvature of the metric $g$ is nonnegative near infinity if $n=2m$ or that $Q_g^{(2m)}$ satisfies a slow decay barrier condition near infinity if $n>2m$. Positive results for this question have been obtained by Gursky and Malchiodi for $m=2$ and $k=1$ in the context of closed manifolds with nonnegative scalar curvature and by Li and Xu and Li, Wei, and Xu for $m\ge2$ and $k\in\left\{1,\min(m-1,2)\right\}$ in the context of conformally Euclidean metrics on $\mathbb{R}^n$. These results hold for all $n\ge2m$. Considering the case where $m\ge4$ and $k=3$, we obtain a positive result for this question when $n\in\left\{2m,2m+1,\dotsc,4m-6\right\}$, namely for these dimensions, we obtain that if $Q_g^{(2m)}\ge0$ and $Q_g^{(2m)}\not\equiv0$ in $\mathbb{R}^n$, then $Q_g^{(6)}>0$. On the other hand, in surprising contrast with the results of Gursky and Malchiodi, Li and Xu, and Li, Wei, and Xu, we find that the answer to this question is negative when $k=3$ and $n\ge N_m$ for some $N_m\in\mathbb{R}$. In this case, we are able to construct examples of conformally Euclidean metrics such that $Q_g^{(2m)}$ is positive everywhere, but $Q_g^{(6)}$ is negative at some point. By stereographic projection, our examples extend to metrics conformal to the standard metric on $\mathbb{S}^n$.

We study the small-regularization limit for $L^p$-regularized optimal transport with $1<p<\infty$ and for entropically regularized optimal transport (EOT). The exact first-order (respectively, second-order) asymptotics are determined explicitly under mild assumptions on the source and target measures. Our work generalizes the existing results for quadratic and entropic regularization, and connects them by a natural interpolation via $p\in(1,2)$. We derive all these asymptotics in a unified manner by a novel approach that separates the local computation of the optimal profile from the global enforcement of the marginal constraints: convex duality leads to Gaussian profiles for entropy and Barenblatt profiles for $L^p$-regularization, while a quantization construction turns these local profiles into couplings.

Let $(0,L)\times E_a\subset \R^{d+1}$ denote the cylinder of length $L$, and base $E_a\subset \R^{d},$ where $E_a$ is an open ellipsoid with semi-axes $a=(a_1,...,a_d)$. Let $w_{(0,L)\times E_a}$ denote its torsion function. Heat equation tools are used to show that (i) if $E_a$ is has small eccentricity and $L$ is sufficiently large, then the maxima of $|\nabla w_{(0,L)\times E_a}|$ are located at the centres $(0,0)$ and $(L,0)$ respectively, (ii) if $E_a$ has large eccentricity and $L$ is sufficiently large, then the maxima are located on the lateral side of the cylinder.

Accurately solving partial differential equations (PDEs) on arbitrary geometries and a variety of meshes is an important task in science and engineering applications. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the kernel integral formulation of SSMs. This is achieved by constructing Takenaka-Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and can approximate the solution manifold of PDEs on a wide range of geometries and meshes. In several challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and irregular domains, AMO consistently outperforms state-of-the-art solvers in terms of relative $L^2$ error. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.

We consider repulsive Euler-Poisson equations in both one-dimensional space and in the multidimensional case with radial symmetry, assuming a power-law velocity-dependent damping coefficient. We show that under this assumption, in the one-dimensional case the set of initial data corresponding to a globally time-smooth solution expands, whereas in other dimensions (except for dimension 4) the damping does not influence on improving of smooth properties of the solution. Namely, any arbitrary small perturbation of the steady state blow up, despite of its amplitude of oscillations decays.

In this paper, on a bounded domain $\Omega\subset {\bf R}^n$, we consider a nonlocal problem of the type $$\cases{-\Delta u=Q(u,\lambda)f(u) & in $\Omega$ \cr & \cr u=0 & on $\partial\Omega$\cr}$$ where $Q:H^1_0(\Omega)\times {\bf R}\to {\bf R}$, proving, under suitable assumptions, the existence of at least three weak solutions for each $\lambda$ running in a suitable interval.

We carry on a study of the point-wise regularity theory for the nonlocal Trudinger equation, with measurable and bounded singular kernel. We establish refined quantitative upper bounds, weak and strong parabolic Harnack inequalities, both under optimal tail conditions. Our analysis relies on the adaptation of a refined De Giorgi-Moser machinery based on the parabolic approach á la Di Benedetto, specifically designed to overcome the technical intricacies of the nonlocal, doubly nonlinear framework.

In this paper, we study critical points of second Neumann eigenfunctions on convex domains in two-dimensional space forms from three complementary perspectives: spectral and geometric criteria for the absence of interior critical points, quantitative localization of possible critical points, and explicit bounds for the hot spots constant. Precisely, we first establish monotonicity-radius localization principles in $\mathbb S^2$ and $\mathbb H^2$. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if $\mu_2(\Omega)D^2\le j_{1,1}^2,$ then every second Neumann eigenfunction on $\Omega$ has no interior critical points. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain diameters in $\mathbb{S}^{2}$ and $\mathbb{H}^{2}$. Finally, we develop an analytic approach to study the hot spots constant $\mathfrak{C}(\Omega)$ on convex domains. For planar convex domains, we improve the Euclidean upper bound to $\mathfrak{C}(\Omega)<1.48$. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, $\mathfrak{C}(\Omega) < 4$ for $\Omega\subset\mathbb{S}^{2}$ contained in a hemisphere; $\mathfrak{C}(\Omega) < 11.2$ for $\Omega\subset\mathbb{H}^{2}$. Our proofs combine the properties of Bessel and Legendre functions, eigenvalue estimates, and Green's identity. Our results quantitatively measure “how wrong” the hot spots conjecture can be.

In this paper, we consider the almost sure nonlinear smoothing, the almost sure spatial decay and the almost sure uniform convergence of the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. Firstly, for initial data $g\in H^{s}(\mathbb{R})(s\geq\frac{1}{4})$ and $\Phi_{2}\in L_{2}^{0,s}$, we prove the local well-posedness for the stochastic cubic KdV-Benjamin-Ono equation. Secondly, we establish the almost sure nonlinear smoothing of the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. Finally, by using the almost sure nonlinear smoothing, we obtain the almost sure spatial decay and the almost sure uniform convergence of the integral term in the pathwise solutions to the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. More precisely, we have the following results: for the stochastic mKdV equation, let $s>\frac{1}{3}$, $f\in H^{s}(\mathbb{R})$ and $\Phi_{1}\in L_{2}^{0,s}$. Then, the local pathwise solution $u$ satisfies \begin{eqnarray*} &&\mathbb{P}\Big(\Big\{\omega: \lim_{t\rightarrow0}\Big\|u-U(t)f-\int_{0}^{t}U(t-s)\Phi_{1}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\ &&\mathbb{P}\Big(\Big\{\omega: \forall t\in[0,T_{\omega}], \lim_{|x|\rightarrow\infty}\Big(u-U(t)f-\int_{0}^{t}U(t-s)\Phi_{1}dW(s)\Big)=0\Big\}\Big)=1. \end{eqnarray*} For the stochastic cubic KdV-Benjamin-Ono equation, let $s>\frac{1}{3}$, $g\in H^{s}(\mathbb{R})$ and $\Phi_{2}\in L_{2}^{0,s}$. Then, the local pathwise solution $v$ satisfies \begin{eqnarray*} &&\mathbb{P}\Big(\Big\{\omega: \lim_{t\rightarrow0}\Big\|v-V(t)g-\int_{0}^{t}V(t-s)\Phi_{2}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\ &&\mathbb{P}\Big(\Big\{\omega: \forall t\in[0,T_{\omega}],\lim_{|x|\rightarrow\infty}\Big(v-V(t)g-\int_{0}^{t}V(t-s)\Phi_{2}dW(s)\Big)=0\Big\}\Big)=1. \end{eqnarray*}

This paper investigates the Dirichlet problem for the anisotropic minimal surface equation in a bounded domain. Under the natural assumption of non-negative boundary anisotropic mean curvature, we establish the unique solvability of the Dirichlet problem for continuous boundary data. To achieve this, an essential a priori gradient estimate is established, which also allows us to prove a weak version of Bernstein's theorem for entire solutions under a sharp, one-sided linear growth assumption. Moreover, using the direct method in the calculus of variations, we prove the existence and local Lipschitz regularity of generalized minimizers in $BV(\Omega)$ with $L^1(\partial\Omega)$ boundary data. We also find that this variational formulation naturally yields a Neumann-type boundary condition, geometrically explaining why the Neumann problem requires no boundary curvature constraints.

We consider time-independent solutions with constant vorticity of the free boundary Euler equations for a 3D liquid drop with capillarity. A rigidity result for the solutions of this problem has been recently proved with variational methods: if a certain quantity involving the vorticity parameter, the capillarity coefficient and the area of the equatorial section of the drop is below a certain value, then the solution has necessarily cylindrical symmetry, the shape of the drop is an oblate spheroid, flattened at the poles and bulged at the equator, and each fluid particle moves along a horizontal, circular trajectory with constant angular velocity. In this paper we develop a perturbation analysis of the problem for fluid domains whose equatorial section is close in C2 norm to a disc, and we show that a rigidity result holds also above the threshold obtained with variational methods.

We show a Liouville theorem for domains in graphs with Dirichlet boundary conditions. More specifically, we characterize the non-existence of non-zero bounded harmonic functions. Since Dirichlet boundary conditions give rise to Laplacians with a positive killing term, we can first characterize the validity of the Liouville theorem by the fact that the Green operator applied to the killing terms is equal to 1, or in other words, that the constant function $1$ is a potential. Secondly, we derive a characterization in terms of stochastic completeness at infinity and and total loss of heat. Thirdly, we give a characterization in terms of a Green formula for superharmonic potentials. Finally, we investigate the Liouville property in terms of recurrence and transience of the graph without killing term. As an application, we consider subsets of the Euclidean space such as cones and percolation clusters, and weakly spherically symmetric graphs.

We investigate the relation between minimizers and weak solutions for a class of singular functionals arising from Born–Infeld type theories $\mathcal{L}(s)$. In the setting of an electrostatic field $s=\frac{1}{2}|\nabla\phi|^2$, $\mathcal{L}(s)$ satisfies $\lim_{s\to(1/2)^-}\mathcal{L}(s)=+\infty$, which naturally enforces the finite gradient bound $|\nabla\phi|\le 1$, also called the truncation threshold. For a prescribed extended charge density $\rho$, we consider the relation between the weak solution of the system \begin{equation} \begin{cases} -div\left(b\left(\frac12|\nabla\phi|^2\right)\nabla\phi\right)=\rho,& \text{in }\mathbb{R}^N,\\ b(s)=\mathcal{L}'(s),\quad\lim_{s\to\frac12^-} b(s)=+\infty,\\ \lim_{|x|\to\infty}\phi(x)=0 \end{cases} \end{equation} and the minimizer $\phi_0$ of the singular functional. We propose a monotonic approximation method to handle the intrinsic singularities of $\mathcal{L}(s)$. We prove that the gradient of the minimizer never touches the singular boundary $|\nabla\phi|^2=1$; this structural result yields a key integrability property, the existence and uniqueness of the minimizer, and the corresponding variational inequality. Under the additional assumption that $\rho$ is radially distributed, we show that the minimizer is the unique weak solution. Furthermore, we establish the $C^1$ and $C^2$ regularity of the minimizer under suitable integrability conditions on $\rho$, and provide a uniform estimate for the strict spacelikeness condition $|\nabla\phi_0|\le 1-\epsilon$, where the parameter $\epsilon>0$ is explicitly characterized in terms of the spatial dimension, the spatial region, and $\rho$. Our results extend the classical Born–Infeld theory to a general class of singular Born–Infeld type theories, thereby providing a unified framework for the variational analysis and regularity of such singular functionals and systems.

Let \(n\ge3\) and \(1<p<n\). We prove a quantitative stability estimate for the critical Sobolev trace inequality on the upper half-space. More precisely, the Sobolev trace deficit controls the \(\max\{2,p\}\)-th power of the gradient distance to the manifold of trace bubbles. A central part of the proof is the spectral nondegeneracy of the trace bubbles: the first two eigenspaces of the linearized weighted Steklov problem are exactly the amplitude, dilation, and tangential translation modes.

In this paper, we prove a sharp Fefferman–Stein-type estimate for the fractional Schrödinger equation, which can be regarded as a generalized Strichartz estimate for data in the Fourier Lebesgue space $\widehat{L^p}$. Then, as an application of the Fefferman–Stein inequality and its off-diagonal generalization, we prove large data local well-posedness and small data global well-posedness results for the one dimensional fractional nonlinear Schrödinger equation with pure power nonlinearities in the homonegeneous and inhomogeneous Fourier–Sobolev spaces $\widehat{\dot{H}^s_p},\widehat{H^s_p}$. Solutions are established in $L_x^r(\mathbb{R} ;L^q_t(I))$ spaces in order to overcome the difficulty of a loss of derivatives in the standard Strichartz estimates.

This paper establishes limitations on observability inequality and unique continuation for Schrödinger equations on fractal sets. We prove that, in contrast to the heat equation, such properties can fail in fractal settings. To achieve this, we first extend the classical Turán–Nazarov inequality, which provides lower bounds of trigonometric polynomials of the form $\sum_{k=1}^nc_ke^{2\pi im_kt}$ on sets of positive measure, to the fractal setting. Unlike in the classical case, the constant in the inequality loses uniformity in the degree $n$, and we obtain sharp bounds depending on both $n$ and the frequency difference $m_n-m_1$. These refinements then enable us to construct explicit counterexamples, showing that observability and unique continuation may fail for Schrödinger equations when the observation set is fractal.

We study the recovery of a smooth nonlinearity in semilinear wave equations on globally hyperbolic Lorentzian manifolds. Our approach combines higher-order linearization with distorted plane waves and Gaussian beams, using three-wave, four-wave, and m-wave interactions to recover different Taylor coefficients.

We prove that every smooth entire solution $ u\colon\mathbb{R}^2\to\mathbb{R}^2 $ of the Ginzburg–Landau equation $ -\Delta u=u(1-|u|^2) $ with $ |u(x)|\to1 $ as $ |x|\to\infty $ has finite potential energy, i.e., \begin{equation*} \int_{\mathbb{R}^2}(1-|u|^2)^2 \mathrm{d}x<+\infty, \end{equation*} thereby resolving Brezis' Open Problem 2.5 in [4]. The main difficulty stems from the possible presence of a curl-free mode that carries nonzero circulation and decays only like $ |x|^{-1} $; such a mode lies outside $ L^2 $ and does not admit a single-valued potential. By minimizing over $ L^2 $ gradient corrections, we construct a comparison field that solves the homogeneous equation and inherits the same circulation. The Kelvin inversion, combined with the De Giorgi–Nash–Moser theory for quasilinear elliptic equations, then produces the optimal decay $ O(|x|^{-1}) $. For a Ginzburg–Landau solution, the Bernstein estimate and the coercivity of the Jacobi form produce an $ L^2 $ forcing term in the exterior phase equation. The resulting $ L^4 $ bound on the phase field implies $1-|u|^2\in L^2(\mathbb{R}^2)$, and therefore the potential energy is finite.

This article is the first paper in the series. In this series of articles, we will examine the influence of the friction factor $\alpha$ at the solid–fluid boundary on the stability of Couette flow. Specifically, we consider the stability of Couette flow in a bounded periodic channel $\mathbb{T} \times [-1,1]$ under Robin-type boundary conditions ($u^2|_{y=\pm 1} = 0$, $[\alpha \partial_n u^1 + u^1]|_{y=\pm1} = f $), where $\alpha$ is the friction factor and $n$ is the unit outer normal vector. In this article, we prove that for a given friction factor $\alpha$, as long as the fluid viscosity coefficient $\nu\ll \alpha$ is sufficiently small, the system is asymptotically stable if the initial perturbation satisfies $\|\omega_{\rm in}\| \leq \epsilon \nu^{1/3}$. Moreover, inviscid damping and enhanced dissipation hold.

This manuscript investigates the Cauchy problem for a class of nonlinear integro-differential equations governing anomalous super-diffusive transport in $\mathbb{R}^N$. The linear dynamics are driven by a dual-scale memory kernel whose Laplace transform is sectorial and exhibits distinct power-law asymptotics at high and low frequencies. This super-diffusive structure precludes the infinite regularizing capacity characteristic of classical parabolic theory; consequently, the associated resolvent operator possesses a heavy algebraic tail in Fourier space, acting as a pseudo-differential operator in the Hörmander class $S^{-2}_{1,0}$ and restricting spatial smoothing. By establishing rigorous $L^q-L^p$ multiplier estimates, the critical Lebesgue threshold $q_c$ for local well-posedness is determined. To demonstrate the sharpness of this threshold, instantaneous norm inflation – and consequent ill-posedness – is proven in the supercritical regime $1 < q < q_c$. Furthermore, tracking the structural crossover to the long-time relaxation parameter resolves the global asymptotic dynamics. The nonlocal Fujita-type critical exponent $\rho_F$ is identified, and global-in-time existence along with algebraic decay is established for small initial data in intersection spaces, provided the nonlinearity remains supercritical and overcomes the structural algebraic barrier connecting the dual scales. This general framework applies directly to canonical physical models, including Cole-Cole fractional retardation and multi-scale Prabhakar memory.