Papers

We show that any Serrin domain in $\mathbb{R}^2$ with two boundary components is a member of the canonical unduloidal family

We study the long-time validity of the inviscid limit problem by measuring when the two-dimensional ideal fluid ceases to accurately approximate a slightly viscous fluid with the same initial datum. We construct a stationary solution of the Euler equation from which the Navier–Stokes solution separates at a time scale proportional to the logarithm of the inverse viscosity, achieving the fastest separation from Euler steady states.

We prove that, for every real Lipschitz function $A$ and every integer $n\ge1$, $\|T_{A,n}\|_{L^2(\mathbb{R})\to L^2(\mathbb{R})}\le C\,\|A'\|_\infty^n$, where $T_{A,n}$ is the $n$-th Calderón commutator and $C$ is an absolute constant. The essential tool is the Coifman–McIntosh–Meyer theorem, applied to a holomorphic family of complex curves, together with the Cauchy integral formula on a contour surrounding the interval of secant slopes. We also include a simpler version of the argument that gives $O(\log n)$ and explains why the contour has the shape it has.

In this article, we study the fractal refinement of the Carleson's problem: (1) We introduce an equivalence relation on linear dispersive PDEs on $\mathbb{R}^{n}$ based on the high-frequency oscillation of their Fourier multipliers, proving that equations in the same class share the same Hausdorff dimension of divergence sets. As applications, we estimate these dimensions for abstract dispersive families. Consequently, we recover known $1$-dimensional results for linearized Boussinesq and Beam propagations, while establishing new results for both in higher dimensions and for Klein-Gordon propagations in all dimensions. (2) Next, we study the effect of radial symmetry by obtaining dimension estimates for fractional Schrödinger equations with concave phase and radial initial data. Applying a transference principle, dimension estimates are extended to families of abstract dispersive equations. As special cases, we recover the known result for the Beam propagation and also obtain new results for the linearized Boussinesq and the Klein-Gordon propagations in the radial setting. (3) We study analogues of (2) for the $3$-dimensional real hyperbolic space by obtaining dimension estimates for a wide class of dispersive propagations: the Schrödinger, fractional Schrödinger (with convex and concave phases), Beam, linearized Boussinesq, Wave and Klein-Gordon equations. This is the first result of its kind, outside the setting of flat Euclidean geometry. Our arguments are based on an inter-spatial transference principle of geometric-analytic flavour. In the process, we also obtain a quantitative result on the singular sets of Sobolev functions, in the spirit of Žubrinić, which may be of independent interest.

We prove a new extension principle for the spherically symmetric Einstein–Maxwell–charged scalar field system. Rather than imposing conditions for mass ratio $2m/r$ as in Christodoulou's small-mass-ratio criterion or as in An–Chen–Singh's proof [1] with $2m/r<3/8$, here we work through a proposal by An-Tan [2] with finite Doppler exponent (blueshift). By conducting two successive blow-up analyses, followed by a stationary comparison and a rigidity argument, we establish the uniform boundness of $\alpha'$ with $\alpha'=\partial_v\alpha/\partial_vr$ and $\alpha=\partial_v(r\phi)/\partial_vr$. This provides the desired extension principle in An-Tan [2] and completes the picture of weak cosmic censorship developed therein. As we were about to release our preprint, Cicortas communicated to us that together with his collaborators, they concomitantly used a different approach to obtain the corresponding extension principle with finite blueshift (Doppler exponent) for the uncharged Einstein-scalar field system. Hence, we decided to put the current paper as an appendix to An-Tan [2], thereby completing the proof of the weak cosmic censorship for the Einstein-Maxwell-charged scalar field system. At the same time, we are working with Cicortas and his collaborators to write a new preprint containing both approaches for the Einstein-scalar field system.

We present a new quantitative, scale-critical $C^1$ extension principle for the spherically symmetric Einstein–scalar field system. We derive estimates for solutions in the causal past of a possibly singular endpoint $\mathcal{O}$ of the center. Assume that there is no trapped surface in this region and assume that the Doppler exponent (blue shift) along each incoming null cone is uniformly bounded. We then prove that the solution can extend regularly across $\mathcal{O}$ provided the mass ratio $\mu=2m/r$ satisfies $\sup\mu<\frac38$. This improves the small-mass-ratio condition in Christodoulou's extension principle by an explicit threshold of order one. Our proof has three new ingredients. First, we derive several new uniform scale-critical pointwise estimates for the scalar field and its derivatives. Second, we obtain refined estimates in an exterior self-similar domain within the triangular causal past region. Third, we establish a family of higher-order Morawetz-type identities and identify the ranges of $\mu$, with which there are coercive spacetime estimates. The unweighted identity yields the threshold $\mu\leq\frac14$, while employing extra self-similar weight raises the threshold to $\mu<\frac38$ and provides a weighted control, which implies that the solution's BV norm along an outgoing null cone is small, so that Christodoulou's theory of bounded variation solutions gives the extension.

In this short note, using OpenAI's resolution of the Solomon–Yau conjecture, we extend our joint work with Pedro Gaspar to all dimensions $n\geq2$. We construct eternal Allen–Cahn flows on $S^{n+1}$ connecting approximations of any prescribed least-area Clifford hypersurface and equatorial sphere. We also construct infinitely many eternal integral Brakke flows from least-area Clifford hypersurfaces to equatorial spheres.

We study the Cauchy problem for the energy-critical Schrödinger map into the 2-sphere: \[ \partial_{t}u = u\times\Delta u, \quad (t,x)\in\mathbb{R}^{1+2}, \quad u(t,x)\in\mathbb{S}^2 \] for equivariant initial data of homotopy degree $k=1$. Given any integer $L\in\mathbb{N}_+$, we exhibit a set of initial data arbitrarily close to the ground state harmonic map in the energy norm such that the corresponding solution blows up in finite time with a quantized rate \[ \lambda(t)\sim \frac{\left(T-t\right)^L}{\left|\log(T-t)\right|^{\frac{2L}{2L-1}}} \quad\mbox{as}\ \ t\uparrow T. \] Our proof is inspired by the techniques developed for the energy-critical harmonic map heat flow in a similar context by Raphaël and Schweyer~[Anal. PDE 7 (2014), 1713–1805]. However, its adaptation to a Schrödinger-type equation requires the introduction of a new energy-Virial-Morawetz Lyapunov functional tailored to quantized slow blowup dynamics.

We determine the Leray–Schauder degree of the singular Dirichlet $N$-mean field equation on a smooth bounded domain in $\mathbb{R}^N$, $N\geq2$, for every positive nonresonant parameter and singular exponents $-1<\alpha_i\leq0$. The formula depends only on the Euler characteristic of the domain and the weights $(1+\alpha_i)^{N-1}$. We also identify the homotopy type of every sufficiently negative variational sublevel with the corresponding weighted barycenter space. We prove quantization of interior concentration masses and exclude boundary concentration and residual diffuse mass. The degree computation uses common descent fields for nearby parameters, bounds for their trajectories, and a finite-dimensional comparison with the Euler characteristic of negative sublevels. Annular averaging and dilation provide local contractions, which are extended to establish the sublevel homotopy equivalence. The arguments apply to the nonlinear, possibly degenerate $N$-Laplacian with singular weights, without requiring a global Palais–Smale condition or isolation and nondegeneracy of solutions.

We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed. The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre. For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.

We propose and analyze a projection-based model reduction method for the rapid and reliable solution of time-harmonic acoustic wave propagation problems over a frequency band, with particular emphasis on seismic applications. The high-fidelity model is based on a hybridizable discontinuous Galerkin (HDG) discretization, formulated as a generalized saddle-point system coupling the volume and trace unknowns. The basis construction combines residual-based greedy sampling of the frequency parameter, enrichment with frequency derivatives of the HDG solution, and proper orthogonal decomposition (POD) compression. We establish the well posedness of both the high-fidelity and reduced problems and characterize their stability with inf-sup constants. We investigate the interaction between static condensation, inherent to the HDG discretization, and projection onto the reduced spaces, characterizing its implications for the reduced formulation and approximation errors. Since these operations do not commute in general, we derive a priori error estimates for both the trace and volume variables to quantify the gap between the two operations. We further identify a lifting condition on the volume reduced space that eliminates these gaps and yields a trace error estimate independent of the volume approximation error. Numerical experiments on the two-dimensional Marmousi model and a three-dimensional heterogeneous medium, with more than $10^{7}$ unknowns, illustrate the theoretical estimates and show that the reduced model achieves speedups of several orders of magnitude compared to the high-fidelity solver.

We study the representation of bilinear forms associated with Riesz fractional gradients by symmetric integral kernels. Motivated by [SS18, Open Problem 1.10], we give a complete description of the constant coefficient case. We prove that any symmetric representing kernel is uniquely determined and we derive its explicit formula in terms of the constant matrix $A$. This yields a sharp characterization of the matrices for which the kernel is either nonnegative or uniformly positive. In particular, [SS18, Open Problem 1.10] has a negative answer (even for constant matrices). We also identify a class of nonconstant antisymmetric divergence free perturbations that are invisible to the fractional bilinear form and hence leave the representing kernel unchanged. The general case of nonconstant matrices exhibits a substantially different behavior and will be the main subject of our forthcoming paper [DSV26].

We approximate the solution operator $(a,f)\mapsto u$ of $-\operatorname{div}(a\nabla u)=f$ by operator networks whose only primitives are heat-semigroup functions of a fixed reference operator and pointwise products. The main result is an emulation theorem on strongly local Dirichlet spaces with a spectral gap and a Bakry–Émery bound $\mathrm{BE}(-K,\infty)$: an explicit scheme reaches $\dot H^1$-error $\varepsilon$ uniformly over coefficients with bounded carré du champ and bounded generator image and over $L^2$ data, with parameters depending only on the coefficient bounds, the accuracy and $K$; on the torus this gives a $C^2$-type coefficient class with constants independent of the dimension. The torus version refines this to Wiener classes of fractional regularity $A^{1+\theta}$, with an explicit rate attained by the scheme's product-defect block. The scheme unrolls the Laplacian-preconditioned Richardson iteration, controls each inexact step in the operator norm of $\dot H^1$ without Galerkin truncation, and synthesises the coupling $\nabla a\cdot\nabla u$ from the product defect of one heat layer. All parameters are explicit; the results are approximation theorems for continuum operators.

We show that Payne's conjecture on the comparison between Dirichlet Laplacian eigenvalues and buckling eigenvalues does not hold on flat and hyperbolic cylindrical surfaces and on (round) hemispheres.

We consider a reaction–diffusion equation with Kolmogorov–Petrovskii–Piskunov (KPP) nonlinearity and Neumann boundary conditions in a narrow tube $G^{\varepsilon,\delta}\subset\mathbb{R}\times\mathbb{R}^{n}$ of width of order $\varepsilon$. The tube is chosen from a family indexed by a second parameter $\delta>0$, whose boundary $\partial G^{\delta}$ is smooth for each $\delta>0$ but degenerates as $\delta\downarrow0$. In the iterated limit $\varepsilon\downarrow0$ followed by $\delta\downarrow0$ the problem becomes a Cauchy problem on the line whose spatial operator is a generalized second order differential operator $D_{v}D_{u}$ in the sense of Feller, rather than a classical second order elliptic operator. Solutions of the limiting problem need not be differentiable in $x$, so they must be understood in a generalized sense. Using the Feynman–Kac representation, we prove that the limiting problem has a unique generalized solution. When the cross sectional volume is bounded above and below, we also prove that this solution is Hölder continuous in $t$ with exponent $1/2$, locally uniformly in $x$, whenever it is Lipschitz continuous in $x$. Under the same volume bounds, this Lipschitz condition holds when the initial function and the nonlinearity are Lipschitz continuous in $x$. Finally we prove that the solutions in the narrow tubes converge to the generalized solution uniformly on compact time intervals in the same iterated limit if the tube, the initial function, and the nonlinearity are periodic in $x$.

The horizontal mean curvature flow is a geometric evolution associated with the horizontal perimeter in sub-Riemannian spaces. In this paper, we introduce a minimizing movement scheme for this flow in the first Heisenberg group $\mathbb{H}$, and prove the existence of a flat flow which is locally $1/2$-Hölder continuous in time, with respect to the $L^1$-distance. The construction applies to arbitrary bounded sets of finite horizontal perimeter and does not require smoothness or symmetry of the initial datum.

In this paper, we study the background stability problem for three-dimensional compressible adiabatic flows through porous media. For two given equilibrium states $(\bar{P}^k,\vec{0},\bar{S}^k)$ ($k=0,1$), under the assumption of small initial perturbations around each equilibrium state, pointwise estimates of solutions for individual perturbation problems are obtained. Based on these pointwise structures, a comparison between two individual solutions could be applied and it is proved that the difference between two solutions depends continuously on the equilibrium distance $|\bar{P}^0-\bar{P}^1|+|\bar{S}^0-\bar{S}^1|$ as well as the initial perturbation difference, with additional decay rates, thereby establishing a nonlinear comparison principle for this system. Compared with previous works on background stability problems, it is the first time to obtain stabilities in the pointwise sense. Our main tools are Green's functions, a priori energy estimates, and refined decay estimates for higher-order derivatives.

We study radially nonincreasing, positive blow-up solutions of a class of superlinear heat equations in the non-scale-invariant setting. This class contains nonlinearities with arbitrarily fast, super-exponential growth. Under suitable structural assumptions on the nonlinearity, we derive the sharp global blow-up profile in the original space-time variables. More precisely, for positive nonlinearities $f$ satisfying $$ \lim_{X\to \infty}f'(X)G(X)= q\ge1\quad \quad \hbox{ where }\quad\quad G(X)=\int_{X}^{\infty}\frac{ ds}{f(s)}<\infty\quad\forall\ X>0, $$ and assuming either the time monotonicity $u_t\ge0$ holds or $q>\frac{p_S}{p_S-1}$, we have $$ u(x,t)=(1+o(1))\,G^{-1}\bigg(T-t+\frac{|x|^2}{4q|\log |x|^2|}\bigg), \quad \hbox{as $(x,t)\to (0,T)$.} $$ This estimate in particular provides the sharp final space profile and the corresponding refined space-time profile. Moreover, our result recovers the universal structure of the global blow-up profile, given by the resolvent $G^{-1}$ of the associated ODE evaluated at a time-space building block, which only depends on the limit $q$.

Let $n\geq2$ and $1<p<n$. We study sharp Sobolev inequalities on the round sphere $\mathbb{S}^n$ for functions $u\in W^{1,p}(\mathbb{S}^n)$ satisfying $\int_{\mathbb{S}^n}x|u|^{np/(n-p)}\,d\sigma=0$. We prove that \[ \|u\|_{L^{np/(n-p)}(\mathbb{S}^n)}^p \leq 2^{-p/n}S_{n,p}^p\|\nabla u\|_{L^p(\mathbb{S}^n)}^p+B\|u\|_{L^p(\mathbb{S}^n)}^p \] holds for all such functions with some $B=B(n,p)<\infty$ if and only if $1<p\leq2$ or $2<p<n<3p-2$, where $S_{n,p}$ is the optimal Euclidean Sobolev constant. This determines the exact range in which Aubin's inequality holds with the sharp gradient coefficient. For every $2\leq p<n$, we also establish the corresponding inequality with squared norms and gradient coefficient $2^{-2/n}S_{n,p}^2$. The gradient coefficient in each inequality is optimal. The proof combines stereographic averaging with quantitative stability of the Euclidean Sobolev inequality to obtain local second moment estimates that allow absorption of the localization errors.

We study finite-maturity American puts in the Heston model without the Feller condition. At positive variance, the exercise boundary is smooth and strictly decreasing in remaining maturity and variance. At zero variance, the endpoint is locally Lipschitz in remaining maturity, the boundary moves with bounded speed in maturity uniformly down to zero variance, and it extends continuously to the axis. Each boundary slice at positive maturity has a regular $C^1$ closure tangent to the zero-variance axis, its stock threshold has variance derivative tending to minus infinity, and both partial derivatives of the inverse boundary graph vanish along the slice. The continuation premium at the axis contact stock is of order variance, and as a consequence smooth fit fails in the variance direction at the axis. At expiry the whole boundary, including its axis edge, converges to the strike. We give quantitative boundary gap bounds, an exact limiting weighted boundary Hessian, and vertical identities retaining the maturity term. Exact boundary gap equivalents remain open.

Inspired by a pioneering work by Caffarelli, Gidas, and Spruck, we study positive solutions to the Yamabe equation with nonremovable isolated singularities. It is known that there is a formal one-to-one correspondence between the collection of the solutions to the Yamabe equation with nonremovable isolated singularities and the collection of harmonic functions through formal infinite series determined by the harmonic functions. In this paper, we prove that these formal infinite series yield actual solutions to the Yamabe equation with a nonremovable isolated singularity. These solutions to the Yamabe equation have designated asymptotic behaviors determined by the harmonic functions. In a broader context, we provide a general method of constructing all solutions near their isolated singular points and all solutions near infinity.

We develop a boundary value and potential theory for elliptic operators \[ L=-\Div(A\nabla)+\B\cdot\nabla \] with form-bounded drift in one-sided chord-arc domains. Assuming that the positive part of $\Div\B$ has sufficiently small form bound, we establish local regularity estimates and solvability of the Dirichlet problem, and construct the associated harmonic measure. Under the additional assumption $\Div\B\le0$, we construct a Green's function and prove pointwise and gradient estimates. For the characteristic distributional identity with a Dirac mass at the pole to hold, we will additionally assume either $\Div\B=0$ or $\B\in L^{n,1}_{\textup{loc}}(\Omega)$. Moreover, in the divergence-free case, we obtain a local lower bound for the Green's function, CFMS estimates, doubling of harmonic measure, a comparison principle for nonnegative solutions with zero trace, and a change of pole formula. Finally, using AI, we construct, in the upper half-space, an operator of the form $-\Delta+\B\cdot\nabla$ with $\B$ form-bounded and divergence-free, whose harmonic measure exhibits dimension drop.

We study a fast reaction limit for a two-component reaction-diffusion system in which one component diffuses and the other does not. In the regime considered here, the nondiffusing component converges to zero as the reaction rate tends to infinity, while the reaction term has a nonzero limit. As a result, the limiting equation for the diffusing component retains a time-integral term. Our results cover both homogeneous Dirichlet and Neumann boundary conditions. We prove convergence of the diffusing component and determine the limiting behavior of the reaction term and the rescaled nondiffusing component. The proof combines monotonicity with respect to the reaction rate with barrier estimates adapted to the time-integral term. Under an additional separation condition on the initial data, we also prove uniqueness of the limiting initial-boundary value problem.

We construct a bounded Lipschitz domain in $\mathbb{R}^3$ that is starlike about the origin and whose first positive Neumann eigenvalue is simple. A corresponding eigenfunction, after a choice of sign, has its unique global maximum at the origin. The construction uses domains with increasingly fine angular structure and convergence to a weighted eigenvalue problem whose first positive eigenfunction is radial.

This paper studies resolvent expansions at all four spectral thresholds for the discrete Schrödinger operator $H=-\Delta+V $ on $\Z^3$, where $\Delta$ denotes the discrete Laplacian on $\mathbb{Z}^3$ and $V(n)$ is a real-valued decaying potential of $\mathbb{Z}^3$. At the thresholds 0, 4, 8, and 12, we first give a complete classification of the threshold types and derive the corresponding resolvent expansions in weighted $\ell^2$-spaces, including the non-generic cases of resonances and eigenvalues. As a result, we establish the Jensen-Kato type time-decay estimates of the evolution $e^{itH} P_{ac}(H)$ to discrete Schrödinger equations with thresholds singularities on $\Z^3$, with decay $O(t^{-3/2})$ when all thresholds are regular and $O(t^{-1/2})$ otherwise. These results provide a three-dimensional lattice analogue of the Jensen–Kato classical analysis at zero energy threshold on $\mathbb{R}^3$.

Let $M$ be a rank-one symmetric space of compact or noncompact type and real dimension $m=kn$, where $k\in\{1,2,4,8\}$ is the real dimension of the underlying division algebra. Meng and Wang proved sharp isoperimetric inequalities for reciprocal sums of the first $p$ or $\ell$ nonzero Neumann eigenvalues, with indices strictly below $m$ in the nonconstant-curvature cases. We keep the whole $m$-dimensional transplanted first eigenspace of an equimeasurable geodesic ball and treat its mass and stiffness forms as a coupled generalized eigenvalue problem. For every bounded connected smooth domain in a noncompact rank-one symmetricspace we prove the full sharp estimate \[ \sum_{j=1}^{m}\frac{1}{\mu_j(\Omega)} \ge \frac{m}{\mu_1(B)}, \] where $B$ is a geodesic ball with $|B|=|\Omega|$; equality characterizes geodesic balls. The Hopf-direction contribution is encoded by a positive, unital, trace-preserving projection on symmetric matrices, and the finite-dimensional closure is supplied by a positive-projection Jensen inequality. In compact type, under the localization hypothesis of Meng–Wang, the same full estimate is obtained whenever $\cot^2 R\ge k-1$, where $R$ is the radius of the comparison ball, as well as in the isotropic endpoint cases and for Hopf-balanced angular second moment. Consequently the complete $m$-term inequality holds for all complex and quaternionic projective spaces under the original localization assumption, and for the Cayley plane when $R\le \arctan(1/\sqrt7)$. In the remaining admissible Cayley interval we exhibit a finite-dimensional obstruction showing that the affine tangent-matrix bounds alone cannot yield the full trace estimate. This obstruction concerns the proof mechanism and is not a counterexample to the unresolved spectral inequality.

We study free-material optimization for nonlinear bond-based nonlocal diffusion and its localization as the interaction horizon $\delta$ vanishes. Bond conductivities are subject to pointwise bounds and a global material budget constraint. After normalization, the asymptotic behavior of these problems is governed by the fraction $\theta_\delta$ of potentially active bonds that can be assigned the maximal conductivity. We identify three distinct limiting regimes. If $\theta_\delta\to0$, the normalized optimal values converge to those of the classical vanishing-material optimal design problem, equivalently formulated as a Lipschitz-constrained state problem or a measure-valued flux problem. For $\theta_\delta\to\theta\in(0,1)$, we obtain a directional local free-material problem governed by the upper tail of the directional $p$-strain over position-direction space. Thus the state variable localizes, while the limiting constitutive law retains an internal directional variable inherited from the bonds. If $\theta_\delta\to\theta\geq1$, the limit is the isotropic local $p$-Dirichlet problem. A convex minimax argument eliminates the normalized conductivity variable exactly, after which the analysis combines convex duality, nonlocal compactness, and $\Gamma$-convergence for convex integrands. We prove a unified $\Gamma$-convergence result for the reduced state energies whenever $\theta_\delta\to\theta\in[0,+\infty)$, yielding convergence of normalized optimal values and compactness and identification of optimal states throughout the full parameter range. At the vanishing-material endpoint, we additionally recover optimal measure-valued local fluxes from optimal nonlocal two-point fluxes.

We study finite-time singularity formation for the scalar defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=u^p$. First, for fixed $d\geq4$ and odd $p\geq3$, we establish a conditional blow-up result under the assumption that a smooth self-similar imploding profile exists for the leading-order relativistic Euler system, with associated normalized coefficients satisfying quantitative analytic radius conditions. Second, we verify both the profile and the analytic radius conditions in dimensions four and five for all sufficiently large odd $p$. Using finite speed of propagation, we obtain a dimension-independent threshold $p_0$ such that, for every $d\geq4$ and every odd $p\geq p_0$, finite-time blow-up occurs for some smooth, compactly supported, real-valued initial data. The proof combines a normalized formulation that separates the leading periodic oscillation from the unknown modulation, growth of the spatial analytic radius induced by an outgoing property, and frequency-dependent matrix multipliers that control the cumulative growth caused by the periodic linearized potential.

This paper establishes global well-posedness for the defocusing energy-critical nonlinear Schrödinger equation on the three-dimensional torus with a class of additive Lévy noises. For arbitrary deterministic initial data in the energy space, we construct probabilistically strong solutions that are pathwise unique in the critical solution class. Continuous dependence on initial data, convergence of finite-activity approximations, and a vanishing-noise limit hold on finite time intervals. When the effective noise is nonzero, quantitative bounds for mass exit times imply almost-sure unboundedness of the mass and exclude free scattering. The expected energy then admits a cubic lower bound in time. The analysis couples pathwise critical perturbation theory with energy estimates adapted to Poisson compensation and stopping-time arguments based on the exact mass balance.

We study the 3D rotating primitive equations with gravity in a horizontally periodic channel, with viscous horizontal velocity and no density diffusion. For every Coriolis parameter, stably stratified equilibria are asymptotically stable. Small compatible \(H^4\) perturbations yield global classical solutions with uniform \(H^4\) bounds. The oscillatory density and velocity decay like \((1+t)^{-1}\); the density converges at the same rate to a stratified profile, its horizontal mean faster. For small data, the limit profile is the decreasing vertical rearrangement of the initial density, the unique potential-energy minimizer among rearrangements, independent of rotation, viscosity, and initial velocity. Kinetic plus available potential energy is dissipated exactly by viscosity and decays like \((1+t)^{-2}\), and each fluid particle settles where the limit density equals its own. These rates are optimal for every rotation rate. The linearized problem has real, slowly decaying normal modes, viscously overdamped high-vertical-wavenumber internal gravity waves, whose small solutions show the \(H^r\) density rates \((1+t)^{-(1-r/4)}\), \(0\le r<4\), pressure rate \((1+t)^{-1}\), and energy rate \((1+t)^{-2}\) cannot be improved uniformly for small data. Key difficulties are the loss of a horizontal derivative in the vertical velocity, the failure of \(H^1(\mathbb T^2)\subset L^\infty\), and rotation-induced coupling of divergent and vortical motion, controlled at top order by a positive power of the initial low-order energy. If the stratification is unstable somewhere, every horizontal mode is linearly unstable with real growth rates and the equilibrium is nonlinearly unstable; rotation strictly lowers each growth rate, with an explicit Taylor-number bound depending on the horizontal wavelength, but leaves their supremum unchanged.