Papers

We study finite-time blow-up for the one-dimensional focusing mass-critical half-wave equation \begin{equation*} i\partial_tu=|D|u-|u|^2u. \end{equation*} For even initial data with negative energy and mass slightly above the ground-state mass, we prove the log-log upper bound \begin{equation*} \|u(t)\|_{\dot H^{1/2}}\lesssim \left(\frac{\log|\log(T-t)|}{T-t}\right)^{1/2} \quad \text{as}\quad t\uparrow T. \end{equation*} This gives, for the half-wave equation, the same log-log law upper bound as in the mass-critical nonlinear Schrödinger equation. The proof follows a similar strategy developed by Merle and Raphaël, but requires a new construction of the blow-up profile. Main difficulty arises from the nonlocal operator $|D|$ and the absence of pseudo-conformal symmetry. We construct an almost self-similar profile with exponentially small error by combining tail computations carried out to arbitrary order, depending on a dynamical parameter, with Borel integral summation in $\Lambda$-analytic spaces. Then, in the modulation analysis, we use a local-virial spectral property proved in the companion paper \cite{Park2026arXiv}.

In this note we extend the Ambrosio-DiPerna-Lions theory on the well-posedness of the transport equation to the case in which the vector field satisfies an $L^p$ Osgood condition. In particular we show that bounded distributional solutions to the transport equation are unique and thus renormalized. As opposed to the classical proof, we do not rely on the vanishing of a commutator, but we show that a weighted energy built out of the Littlewood-Paley decomposition of the solution is conserved.

We introduce the convolution between abstractly given anisotropic weight matrices and investigate properties of this new construction. Further, we apply the knowledge to the particular but interesting case when the (isotropic) matrices are associated with weight functions in the sense of Braun-Meise-Taylor and show how the convolution of associated weight matrices modifies the underlying weight functions. Finally, we give a concrete application of the convolution when studying the concept of local solvability for a certain hyperbolic PDE. Indeed, the convolution allows to treat in a natural way a mixed setting; i.e. having a controlled loss of regularity expressed in terms of two, in general different, weight sequences.

It is well known that Newton's method converges faster when the initial guess is closer to a root of a system of nonlinear equations. In this paper, a two-stage Newton initial guess strategy is proposed by learning features from a parameter-space sampling and a database of precomputed solutions. The method uses discrete Newton trajectories to construct two complementary reduced spaces: a solution feature space, built from converged states, and a corrective search direction feature space, built from intermediate Newton increments. For an unseen parameter, a regression model is used to predict a surrogate solution approximation. Then, in a second step, a residual-minimizing correction is computed using a dedicated GMRES-based approach. The resulting state is then used as an initial guess for the high-fidelity Newton method, which completes convergence. The corrective step is computationally inexpensive since it only requires residual evaluations and the solution of a small least-squares problem. The methodology is weakly intrusive once the high-fidelity residual fields and a script-based programming interface are available. This strategy reduces the number of Newton iterations and decreases the overall CPU time. Numerical experiments on representative PDE problems show quantifiable speedups compared with standalone surrogate initialization. Significant speedups are observed. This generic approach can be applied to a broad class of large-scale nonlinear problems.

We prove in detail how to construct the parametrix of a parameter-dependent family of pseudodifferential operators, appearing in the analysis of time-fractional partial differential equations. In particular, we perform a precise study of the dependence from the parameter of the corresponding asymptotic expansion terms, as well as of the smoothing remainders. Moreover, we provide some results about the Laplace transform of vector-valued distributions, also appearing in the analysis of time-fractional partial differential equations.

We investigate positive solutions of semilinear equations driven by uniformly elliptic strictly $2s$-stable Lévy operators, where $s\in (0,1)$. We first prove that every positive distributional solution of $-Lu=u^p$ in a punctured domain $D\setminus\{0\}$ satisfies $-Lu=u^p+k\delta_0$ in $D$ for some $k\ge0$, and that necessarily $k=0$ whenever $p\ge d/(d-2s)$. We then study the corresponding Dirichlet problem in which the Dirac mass is replaced by a bounded positive measure, and establish the existence of a critical parameter $k_\mu$: below this threshold minimal positive solutions exist, whereas above it the problem admits no solution. In the symmetric case, we further prove multiplicity below the threshold, as well as existence and uniqueness at the threshold itself.

We investigate a doubly degenerate nutrient-taxis system of the form \begin{equation*} \begin{cases} u_t = \nabla \cdot (u v \nabla u) - \chi \nabla \cdot (u^\alpha v \nabla v) + \ell u v, \qquad &x \in \Omega, \ t > 0, v_t = \Delta v - u v, \qquad &x \in \Omega, \ t > 0, \end{cases} \end{equation*} subject to the homogeneous Neumann boundary conditions in a smoothly bounded convex domain $\Omega \subset \mathbb{R}^n$ with $n\in \left \{ 3,4,5 \right \}$, where $\alpha \geq 1$, $\chi>0$ and $\ell \geq 0$. For any suitably regular initial data, we establish the global existence of a weak solution that remains uniformly bounded in time, provided that $\alpha$ lies in the range $\left[1, \frac{5}{2} - \frac{n}{4}\right)$, and we also determine the large-time behavior of these solutions. Our proof relies on several novel functional inequalities, a bootstrap argument, and a Moser iteration method.

The compressible Euler–Riesz equations arise in the modelling of a wide range of physical phenomena, including stellar dynamics, plasma physics, and mathematical biology. We study rotating steady states of the attractive compressible Euler–Riesz equations, which we call rotating Riesz stars, and establish existence and nonlinear stability or instability results in both the mass-subcritical and mass-supercritical regimes. In the mass-subcritical regime, we prove the existence of rotating Riesz stars under suitable subhomogeneity assumptions on the angular momentum profile. We then establish their nonlinear stability through a concentration compactness argument adapted to the axisymmetric setting. Rotation creates new compactness difficulties, most notably the possibility that minimising sequences are tight along rings whose radii diverge to infinity. In the mass-supercritical regime, we prove existence in the polytropic setting under suitable superhomogeneity assumptions on the angular momentum profile, thereby extending the theory beyond the small angular velocity regime. The proof requires a careful analysis of mass-preserving scalings, which are more delicate than in the non-rotating case. Finally, by analysing the concavity of the free-energy along these scalings, we establish the instability of the resulting mass-supercritical rotating Riesz stars. Our results show that rotation can have either a stabilising or a destabilising effect, depending on the singularity of the Riesz interaction.

We study fractional parabolic initial-value problems with lower-order drift and potential terms in anisotropic spectral Barron spaces, defined by weighted space–time Fourier $L^1$ norms adapted to parabolic scaling. We prove existence, uniqueness, and maximal regularity with a gain of one derivative in time and $\gamma$ derivatives in space, where $\gamma>0$ is the order of the fractional Laplacian. The evolution is defined only for $t\geq0$, whereas the finite-time norm requires a global extension with sufficient temporal Fourier decay. We construct a finite reflected semigroup extension using a Vandermonde system to match derivatives at $t=0$, obtaining temporal Fourier estimates uniform in the semigroup parameter. Combined with Fourier multiplier estimates for the damped principal operator, it yields maximal regularity. Dimension-independent multiplication estimates support a finite regularity bootstrap, while interpolation and sufficient damping absorb the lower-order terms in the base estimate. The a priori estimate and the method of continuity yield maximal regularity without smallness assumptions on the lower-order coefficients. A frequency-localized counterexample shows that a uniform-in-time spatial Barron bound on the forcing does not imply the corresponding two-derivative solution bound, even for the one-dimensional heat equation. Using this regularity, Fourier sampling yields $n^{-1/2}$ approximation rates for the solution in mixed space–time Sobolev norms using shallow networks with suitable activations. Sampling in a product Hilbert space yields a population-level PINN consistency estimate for shallow cosine networks on a bounded cylinder. There exists a single width-$n$ network for which the sum of the squared mixed-Sobolev solution error, the squared $L^2$-norm of the residual for the whole-space fractional equation, and the squared initial-data error is $O(n^{-1})$.

In this paper, we study the primitive equations in a collar neighborhood of a smooth closed hypersurface in ${\mathbb R}^3$. Using the framework of maximal $L_q$-regularity and the hydrostatic Helmholtz projection, we establish existence, uniqueness, and regularity results for strong solutions. Building on these local well-posedness results, we derive suitable a priori estimates and prove the global existence of strong solutions without imposing any smallness assumptions for the initial data. Moreover, we show that the solutions are $C^\infty$ jointly in time and space.

We establish a quantitative Landis estimate for the one-dimensional fractional Schrödinger equation $(-\Delta)^{1/4}u+V(x)u=0$ in $\mathbb R$ with a real-valued bounded potential. If $\|V\|_{L^\infty}\le 1$, $\|u\|_{L^\infty}\le C_0$, and $\|u\|_{L^2(-1,1)}\ge 1$, then \[ \inf_{|x_0|=R}\|u\|_{L^\infty(x_0-1,x_0+1)} \ge \exp(-CR\log R) \] for all sufficiently large $R$. After the Caffarelli–Silvestre extension and the substitution $y=z^2/2$, the equation becomes a Grushin equation with a weak Robin condition on the degeneracy line. The corresponding angular operator has the arithmetic spectrum $\kappa_n=2n+\tfrac12$ after half-density conjugation. The central spectral estimate is \[ \sup_{\xi\in\mathbb R} \bigl\|C\bigl((\tau+i\xi)^2-L_0\bigr)^{-1}C^*\bigr\| \le C\tau^{-1/2} \] for parameters separated from the angular lattice. It yields a linear-weight Carleman estimate for measurable Robin feedback with absorption threshold $\tau\ge C(1+\|V\|_\infty^2)$. Quantitative inward propagation, fixed-scale Grushin-ball propagation, and an interior-cylinder interpolation estimate then transfer bulk non-vanishing to the boundary. The Landis rescaling converts the local potential dependence $C\|q\|_\infty^2$ into the global rate $CR\log R$.

We prove a sharp volume-stability theorem for closed hyperbolic three-manifolds. Let \((M,h)\) be closed hyperbolic with \(\operatorname{Ric}_h=-2h\), and let \(g_i\) be smooth metrics on \(M\) satisfying $R(g_i)\geq -6, \operatorname{Vol}_{g_i}(M)\longrightarrow \operatorname{Vol}_h(M)$. After passing to a subsequence, there exist \(Z_i\subset M\), smooth domains \(K_i\subset M\), and diffeomorphisms $\psi_i:K_i\longrightarrow M\setminus Z_i $ such that $\operatorname{Vol}_{g_i}(Z_i)\longrightarrow0, \operatorname{Vol}_h(M\setminus K_i)\longrightarrow0, $ and $ \|\psi_i^*g_i-h\|_{C^0(K_i,h)}\longrightarrow0. $ Thus near-equality in the sharp hyperbolic volume bound forces tensorial \(C^0\)-convergence to the hyperbolic metric outside regions of vanishing volume. As an application, we establish stability of the Fischer–Moncrief reduced Hamiltonian at the Lorentz-cone ground state: after CMC normalization, near-minimizing compact vacuum data in the hyperbolic topological class converge, modulo sets of vanishing volume, to the hyperbolic Lorentz-cone geometry in tensorial \(C^0\). This provides a rigorous volume-dominance formulation of the Fischer–Moncrief asymptotic picture.

In this paper, we pose a new class of Minkowski problems corresponding to the mixed Euclidean-Gaussian volume. Using the flow method, we prove the existence of smooth normalized solutions for the corresponding Monge-Ampère-type equation in the symmetric case. Then, by approximation, we obtain the existence of origin-symmetric solutions to the mixed Euclidean-Gaussian Minkowski problem.

We derive the incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation with Bose-Einstein or Fermi-Dirac statistics. The model has a self-consistent collision structure, with the local density acting as the collision frequency and the bulk velocity and temperature determined by nonlinear quantum-weighted moments of the distribution. We work near a global quantum equilibrium under the diffusive scaling and keep the quantum parameter fixed. Uniform estimates with respect to the Knudsen number yield strong microscopic relaxation and identify the limiting infinitesimal quantum equilibrium. Using the local conservation laws, we prove the incompressibility condition, the Boussinesq relation, and strong compactness of the divergence-free velocity component and a quantum-adapted thermal mode, while the acoustic modes vanish locally by a dispersive estimate. The limiting viscous stress tensor and heat flux are identified by solving auxiliary equations for the linearized quantum Fokker-Planck operator and by expanding the local quantum equilibrium manifold. The resulting incompressible Navier-Stokes-Fourier system retains the effect of quantum statistics through its normalization constants and transport coefficients.

For $d\geq 4$, we consider incompressible Euler flows in $\mathbb{R}^{d}$ with bi-rotational symmetry and without swirl. Our first result gives the local wellposedness of the Yudovich-type solution. The second result provides global wellposedness up to $d\leq 6$. In particular, it shows that the rate of growth of the vorticity maximum coincides with the rate from axisymmetric flows without swirl, which was obtained in the paper by the second author and Jeong (Arch. Ration. Mech. Anal. 249(3):32, 2025) and Shao–Wei–Zhang (Acta Math. Sin. (Engl. Ser.), 42(3):663-679, 2026).

We study the one-dimensional reaction-diffusion equation \[ u_t=[A(u)]_{xx}+f(x,u). \] The diffusion operator belongs to a broad class of nonlinear degenerate diffusion operators that includes the porous medium operator as a special case. We develop a systematic regularity theory for the solutions and their free boundaries. First, we establish the $C^1$ regularity of the pressure variable $v$, together with a lower bound for its second spatial derivative. Next, we prove Darcy's law and that, after the waiting time, a right (resp.\ left) free boundary moves with strictly positive (resp.\ negative) velocity (Theorem 3.1), thereby strengthening the previously known results which only gave nonnegativity (resp.\ nonpositivity). Finally, under additional structural assumptions on the diffusion and reaction terms, we obtain higher regularity for both the solution and its free boundaries (Theorems 4.5 and 4.6). These results extend several classical regularity properties of the porous medium equation to a much broader class of degenerate diffusion equations with reactions.

We consider the limit of small Mach number and small vertical-to-horizontal aspect ratio for the isothermal compressible Navier-Stokes system. In addition, we consider the scale in which the stratification is weak. Owing to the anisotropic nature of the problem, the dynamics exhibit a three-wave separation phenomenon, consistent of a slow wave, a fast horizontal acoustic wave, and an even faster vertical acoustic wave. These three waves, unfortunately, are not mutually orthogonal, and the corresponding projections are parametrized by the small parameter, which significantly complicates the nonlinear analysis. Without any restriction on the size of the initial waves, we establish the uniform existence and uniqueness of solutions to the compressible Navier-Stokes system for any fixed small parameter, by carefully analyzing the evolutions of both the energy and the acoustic waves. Moreover, we prove that, as the small parameter tends to zero, the solutions converge to that of the incompressible primitive equations governing atmospheric and oceanic flows.

We study stochastic, parabolic-parabolic Keller–Segel equations on the $d$-dimensional torus in scaling critical Besov spaces, for $d \geq 3$. Using stochastic maximal regularity estimates, we prove local well-posedness of the equation, i.e., that there exists a unique solution up to a maximal time of existence. We show that the time of existence can be made arbitrarily large with arbitrarily high probability, provided that the initial data is sufficiently small in these critical spaces. Contribution to the Oberwolfach Report for the Seminar 'Stochastic Partial Differential Equations in Critical Spaces' organized by Antonio Agresti and Mark Veraar.

This note proves existence and uniqueness of strong solutions to a broad class of fourth-order quasi-linear parabolic equations in the class of Wiener spaces. It improves upon previous results in the literature, devoted to some special cases falling within the general framework developed here, in which uniqueness held true in a smaller class than the space where existence was proven. Gain of analyticity, as well as generalisations to higher-order equations and to propagation of higher regularities, are also discussed.

We prove that a complete, connected, embedded, minimal hypersurface in $\mathbb{R}^{n+1}$ with finite total curvature and Morse index one is the higher-dimensional catenoid.

This paper concerns a variational system of nonlinear elliptic equations that generalizes the classical Brezis-Nirenberg problem. In addition to considering vector-valued unknown functions in place of scalar-valued unknown functions, the problem under consideration generalizes the Brezis-Nirenberg problem in two directions. First, the nonlinearity is the sum of two homogeneous functions of suitable homogeneity degrees. Second, the Sobolev-critical term in the nonlinearity is nonlocal. We establish conditions under which a nontrivial solution to the system under consideration is guaranteed to exist and conditions under which the system under consideration admits only the trivial solution.

Consider a material point in finite dimensions moving under the influence of a potential force according to Newton's laws. Suppose the potential energy function is a generalized well, strictly convex transverse to a smooth submanifold $M$ on which it is minimal. If the potential is steep, one expects the particle will oscillate rapidly about this submanifold and, if the initial displacement is not too great, should approximately move along $M$ as if it were ideally constrained (e.g. geodesic). This expectation is true if the initial conditions are very well prepared but may fail otherwise - additional potential forces determined by how the Hessian of the potential varies along $M$ may be present. The origin of this force is that the transversal motion acts as a simple harmonic oscillator with a slowly varying frequency, which approximately conserves action, not energy. In this work, we regard continuum mechanical systems such as the elastic thread or compressible fluid as material points moving in an infinite dimensional space according to Newton's laws for appropriate potential energy functionals. We show how to arrive at ideally constrained systems such as the inextensible thread and incompressible fluid as a limit of a strong potential force, computing also corrections to the naive predictions when the data is not very well prepared. For example, for the thread we find a resistance to bending emerge from a strong resistance to compression/expansion. For the fluid, the effective incompressible dynamics may be driven by a remnant acoustical wavefield. Both of these emergent features are nonlinear and non-local. Finally, we give examples of some limits for which the naive models robustly hold because the additional force is trivial. These include the homogeneous incompressible Euler, anelastic Euler, as well as the lake and great lake equations.

Let $\Omega\subset\mathbb H^2$ be a bounded smooth horoconvex domain and let $\psi_1>0$ be its first Dirichlet eigenfunction. We prove that \[ \operatorname{Hess}_{\mathbb H^2}(-\log\psi_1)>0 \] throughout $\Omega$, with no restriction on the diameter or the first eigenvalue. The proof is by contradiction. A degenerate Hessian would yield a shifted translation Killing derivative with a singular interior zero. Then the boundary-zero theorem of Grossi and Provenzano shows that the shifted Killing derivative has exactly two zeros on the boundary. A nodal-domain argument on the surface rules this out. As an application we prove that every superlevel set of $\psi_1$ is horoconvex: every level curve has geodesic curvature at least $1$. The Hessian bound makes the shifted construction available for Killing fields with nonvanishing rotation part, and yields the pointwise inequality $|(\operatorname{Hess} u)^{-1}J\nabla u|\le1$ for $u=-\log\psi_1$, where $J$ is rotation by $\pi/2$; a boundary-zero count for translation fields with arbitrary axis completes the argument.

We obtain novel integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous initial-boundary-value as well as interface problems for the Barenblatt-Zheltov-Kochina pseudo-parabolic equation of Sobolev-Galpern type formulated on the real line, half-line and finite interval. This fundamental partial differential equation (PDE) of mathematical physics emerges in a wide variety of natural phenomena and applied sciences including continuum mechanics, thermodynamics, chemical engineering, solid-state electronics, semi-conductor devices, battery research, and nanotechnology. A suitable implementation of a modern methodology, known as Unified Transform Method, is in force throughout this study, with particular challenges arising due to the higher-order mixed-derivative term of the PDE and the generality of the problems under consideration altogether. The boundary and interface conditions appear to be non-standard but are naturally dictated by the structure of the PDE itself. Our explicit analytical formulae directly lend themselves to future explorations of the solutions' qualitative properties such as asymptotic behavior, spatio-temporal dynamics, regularity and well-posedness. This work is expected to be of utility also in the investigation of nonlinear counterparts as well as towards the study of phase-transition phenomena and free-boundary problems, where the interface evolves dynamically according to energy balance laws.

In this paper, we show the $L^2$ contraction property of the planar oscillatory or monotone dispersive shock profiles of the dissipative Kadomtsev-Petviashvili (KP) equation modelling water waves and the multi-dimensional Korteweg-de Vries (KdV) Burgers equation under arbitrarily large perturbations in two space dimensions, up to Lipschitz time-dependent shifts. This stability result extends the results of two recent papers by Chen, Eun, Kang, and Shen on $L^2$ contraction for the KdV-Burgers equation.

We consider the $k$-generalized Korteweg-de Vries equation \begin{equation*} \partial_{t}v+\partial_{x}^{3}v +\partial_{x}(v^{k+1})=0, \qquad (t,x)\in\mathbb R\times\mathbb R, \qquad k\in\mathbb Z_{+}, \end{equation*} emphasizing the modified case $k=2$ and the cubic case $k=3$. We prove that solutions from low-regularity, possibly singular, data $u_{0}$ become real analytic in $(t,x)$ for all $t\neq0$, whenever $u_0$ satisfies a Nelson-type condition \begin{equation*} \sum_{k=0}^{\infty}\frac{\alpha^{k}}{k!}\,\big\|(x\partial_x)^{k}u_0\big\|_{X}<\infty, \end{equation*} for some $\alpha>0$. For the cubic equation, this includes data such as $u_0=x_{+}^{\lambda}$, singular at the origin; for mKdV even discontinuous data yield analytic solutions e.g $u_{0}(x)=\sgn(x)e^{-x^{2}}$.For mKdV we work in the sharp well-posedness space $X=\widehat H^{r}_{s}(\mathbb R)$, $r\in(1,2]$, $s\geq\frac12-\frac1{2r}$, with $r=2$ recovering analyticity on $H^s(\mathbb R)$, $s\geq\frac14$, the best mKdV space in the sense of Kato; for the cubic equation we work in $X=H^{s}(\mathbb R)$, $s>-\frac16$, approaching the critical exponent $s=-\frac16$ from above. Analyticity thus holds on the largest known data class for which mKdV is well-posed, and on data approaching the corresponding threshold for the cubic equation, extending a known smoothing effect for KdV ($k=1$) to the modified and cubic nonlinearities and to a broader class of singular profiles, avoiding pseudo-differential calculus via Lorentz-space refinements replacing Bourgain-space localization. The mechanism is dispersive: analyticity is generated by the flow, symmetrically in time, and singular profiles become instantaneously analytic for $t\neq0$.

In this work, we propose a general discretization strategy for solving the London equation for type-II superconductors in the whole space $\mathbb{R}^3$. To compute the magnetic field $H_0$, we reformulate the problem for the magnetic potential as a transmission problem and discretize it through a nonstandard FEM-BEM coupling. This formulation accounts for both the bounded interior domain and the unbounded exterior domain without introducing an artificial truncation. We then compute the vector field $B_0$, which arises from the Helmholtz-Hodge decomposition of the magnetic potential in the superconducting sample. This field enters the isoflux problem, which identifies the curves along which vortex nucleation first becomes energetically favorable in the Ginzburg–Landau model of superconductivity. We recast the equations for $B_0$ using the mixed formulation of Kikuchi, in which the divergence-free constraint is imposed weakly, and discretize the resulting problem using a classical $H(\operatorname{curl})$-conforming finite element discretization. We validate our discretization strategy through convergence tests and conclude with an application to the isoflux problem. For a ball under a constant applied magnetic field, the unique maximizer is the diameter aligned with the field. For ellipsoids under a constant applied magnetic field aligned with their major axis, our computations provide numerical evidence of a different behavior in sufficiently elongated, cigar-shaped geometries: off-axis competitors reminiscent of U-shaped vortex configurations attain a larger isoflux ratio than the major axis. Since the major axis is therefore not a maximizer, any off-axis maximizer generates, by rotational symmetry, a continuous family of equivalent configurations, implying non-uniqueness and the presence of a degenerate rotational direction in the isoflux problem.

We study singularity formation for the heat flow of harmonic maps from $\R^d$ into $\mathbb{S}^d$ in supercritical dimensions $d \in \{3,4,5,6\}$. It is well known that in each of these dimensions there exist infinitely many self-similar solutions that provide examples of loss of regularity in finite time. In this paper, we extend the results of \cite{BieDon18}, \cite{BieDonSch17} for $d=3$ to higher space dimensions $d \in \{4,5,6\}$ and prove the existence of a monotonically increasing self-similar profile $f_0$, which is asymptotically stable under small corotational perturbations. To construct the solution and resolve the spectral problem, we use rigorous computer assistance. As a byproduct of our stability analysis, we also obtain finite-codimension stability of arbitrary self-similar profiles within the corotational class.

We study the asymptotic expansion of p-Green functions and their derivatives near the pole. In the Euclidean setting, we strengthen a theorem from [arXiv:2203.01206] by establishing improved integrability properties and estimates for the first and second derivatives. In the more general Riemannian setting, we derive a refined asymptotic expansion of the p-Green function near its pole, thereby improving the corresponding result proved in [arXiv:2512.14591].

For every $\alpha\in(0,1)$ we construct an autonomous, divergence-free vector field $u \in C^\alpha_c(\mathbb{R}^2,\mathbb{R}^2)$ that does not have the weak Sard property and, nonetheless, does not induce anomalous dissipation of $L^2(\mathbb{R}^2)$ norm for solutions to the associated advection-diffusion equation. This disproves a conjecture proposed by Bagnara, Boutros, De Lellis and Mayboroda in \cite{BaBoDeMa26}.