Papers

Let $N\in\mathbb{N}\cap[2,\infty),$ $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^N$, and $X(\Omega)$ be a ball Banach function space on $\Omega.$ In this article, under some mild assumptions, we establish a compactness theorem for Sobolev spaces associated with $X(\Omega)$. Different from the fractional Sobolev space, our proof is based on an elaborate decomposition of bounded Lipschitz domains and its corresponding weighted fractional Poincaré inequality on each piece. As applications, we obtain the fractional Poincaré inequality in $X(\Omega)$ that, for any $s\in (s_0,1)$ and $f\in X(\Omega)$, \begin{align*} \|f-f_\Omega\|_{X(\Omega)} \lesssim(1-s)^{\frac 1q} \left\|\left[\int_\Omega \frac{|f(\cdot)-f(y)|^q}{|\cdot-y|^{N+sq}}\,dy \right]^{\frac{1}{q}} \right\|_{X(\Omega)}, \end{align*} where $s_0$ is a given positive constant and the implicit positive constant is independent of $s$ and $f$. Using this, we further establish the well-posedness of a weighted Triebel–Lizorkin type nonlocal variational problem. These results are of wide generality and, even when they are applied to Morrey spaces, weighted Lebesgue spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, and Orlicz-slice spaces, the obtained results are also new.

Deep learning models, particularly Transformers and Neural Operators, exhibit a well-documented "spectral bias," effectively acting as low-pass filters that smooth out high-frequency information. While benign in fluid dynamics, this bias is catastrophic for Chaotic Dynamical Systems, where the underlying strange attractor is characterized by fractal geometry and infinite spectral density. We introduce the Fractal Neural Operator (FNO), a novel architecture that utilizes a non-resonant prime number basis to approximate continuous dynamical systems. Unlike geometric encodings ($2^k$), which suffer from spectral gaps and resonance, our Harmonic Weierstrass Encoder injects infinite spectral resolution into the latent space. We demonstrate that FNO extends the valid prediction horizon of the Lorenz-63 system to 347 Lyapunov times, exceeding state-of-the-art Reservoir Computing baselines by a factor of 2.3x. These results suggest that "chaos" is not inherently unpredictable to neural networks, but rather requires non-differentiable, fractal embedding manifolds.

We study a deterministic filtering problem formulated directly on the Wasserstein space of probability measures with finite second moment. Motivated by the Mortensen minimum-energy observer, we consider the reconstruction of an evolving probability density from partial observations by minimizing an action functional combining a kinetic transport cost and a time-dependent observation mismatch. The resulting value function is defined on the infinite-dimensional manifold $(P_2(R^d), W_2)$ and satisfies a Hamilton-Jacobi-Bellman equation involving the Wasserstein gradient. Under suitable regularity and growth assumptions on the observation functional, we establish dynamic programming principles, continuity of the value function, existence of minimizing trajectories, and viscosity solution properties of the associated Hamilton-Jacobi equation. We provide two complementary notions of viscosity solutions: a geometric formulation based on subdifferentials in Wasserstein space, and a Hilbertian formulation inspired by Lions' lifting approach. This allows us to prove a comparison principle and uniqueness of solutions. Extensions to transport equations with drift are also discussed. Finally, we introduce a semi-Lagrangian scheme in order to approximate the value function, and show $\Gamma$-convergence of the scheme.

We study a spatial version of the Smoluchowski coagulation equation in which particles carry both an integer mass and a spatial location, and coagulate according to a mass-space product kernel. After coagulation, the resulting particle inherits the sum of the parent masses and a single spatial location. We provide a recursive formula for the solution and show that, while the first local mass moments are conserved up to gelation time at each location, the second local mass moments evolve according to a closed system of differential equations. When the spatial part of the kernel is of Mercer type, we further show that this system reduces to a replicator equation under a time rescaling and normalisation. We then use the fact that the replicator equation admits monotone functionals to quantify how spatial heterogeneity influences the rate of gelation and derive bounds on the gelation time. We also discuss the implications of these results for several important classes of Mercer kernels, including radial, diffusion, and translation-invariant kernels.

We recently proved that if their initial velocity and magnetic field both are the sum of a classical 3D-part and some 2D-part (i.-e. depending only on the horizontal space variables), the solutions of the 3D-rotating magnetohydrodynamic (MHD) system converge (when the Rossby number goes to zero) towards those of a 2D-MHD system with six components and an additional 3D magnetic field transported by the 2D limit velocity. We also provided explicit convergence rates for large ill-prepared initial data. In this article, thanks to new estimates, we improve the convergence rates and allow larger initial data. Finally these improvements are adapted to the rotating fluids system.

We analyze a generalized incompressible model proposed by Ohkitani [19]. This model is based on the observation that the two-dimensional Burgers' equation can be related to the incompressible Navier-Stokes equations by rotating the velocity gradient by 90 degrees. We present several results with initial data in $H^{3}$. First of all, we examine the inviscid model and show the existence and uniqueness of a local-in-time solution that blows up in finite time if the initial vorticity contains a negative part. In the presence of viscosity, we show the existence of a unique global-in-time solution without requiring a sign condition on the initial vorticity, establish the long-time behavior of the difference between two solutions, and derive temporal decay rates for the velocity field when the initial vorticity is non-positive.

We consider Neumann boundary value problems for the $p$-Laplace equation on Riemannian manifolds with nonnegative Ricci curvature. Using spherical symmetrization under appropriate constraints, we derive Talenti-type comparison results in Lorentz spaces. We further show that, in contrast to the Robin case, the Neumann setting admits weaker constraints, which yields stronger comparison principles.

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in $\mathbb{R}^{n}$, as an application we get the Brunn–Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in $\mathbb{R}^{n}$.

This paper focuses on the free boundary problem of the three-dimensional compressible Navier-Stokes-Poisson equations with degenerate viscosities for self-gravitating viscous gaseous stars. For spherically symmetric barotropic motion, we establish the local well-posedness of classical solutions. The solutions obtained here are smooth all the way up to the moving boundary and capture the physical vacuum boundary behavior of the Lane-Emden star configuration for all adiabatic exponents $\gamma>\frac{4}{3}$.

This paper investigates the dynamical behaviors of solutions to the Yamabe flow via the modified potential well method. We first establish the local existence and regularity of weak solutions for the flow. Several new results concerning global existence and blowup are obtained by classifying initial data into stable and unstable sets. Specifically, solutions with initial data in the stable set exist globally and extinguish in finite time, whereas those originating from unstable initial data blow up in infinite time. For certain high-energy initial data, we show that the solution decays to zero as time tends to infinity and undergoes finite-time blowup. In addition, we analyze Palais-Smale sequences to reveal the intrinsic relationship between the long-time asymptotic behavior of solutions and steady states. Finally, we derive the Pohozaev identity for the equation and prove the corresponding nonexistence theorem.

The system coupled by the Schrödinger equation and the Intermediate Long Wave (ILW) equation is a particular model describing the interaction of long and short waves. We prove local and global well-posedness of the initial value problem associated with this system in Sobolev spaces of low regularity. Our approach relies on the use of energy estimates, Bourgain spaces, and Tao's gauge transformation.

This paper studies the long-time dynamics of a coupled hyperbolic system for magnetizable piezoelectric beams with infinite viscoelastic memory. Memory dissipation is characterized by $A^\alpha$, the fractional power of a positive self-adjoint operator $A$ with $\alpha\in[0,1)$. An asymmetric fractional magnetoelectric coupling is adopted: the mechanical-to-magnetic coupling uses integer-order operator $A$, while the magnetic feedback to mechanics is governed by fractional operator $A^\beta$ ($\beta\in[0,1)$). This model bridges the gap between the well-studied integer coupling case ($\beta=1$) and the unsolved fully fractional symmetric coupling problem, offering a universal framework for related coupled systems. Under mild assumptions on memory kernels and system parameters, we prove well-posedness via semigroup theory and derive an explicit polynomial decay estimate for smooth initial data: \[ \|X(t)\|_{\mathcal H} \le C t^{-\frac{1}{4-2\beta-2\alpha}}\|X_0\|_{D(\mathcal A)},\quad \forall\,t\ge 1, \] where the decay exponent is explicitly determined by $\alpha$ and $\beta$. For exponentially decaying memory kernels, the decay rate is sharp if stiffness coefficients satisfy $\alpha_1\ne\alpha_2$. When $\alpha_1=\alpha_2$, we only obtain an upper bound $\delta\le \frac{1}{3-\beta-2\alpha}$ for decay index $\delta$, leaving the optimal rate open. Comparisons with integer feedback coupling ($\beta=1$) show fractional feedback ($\beta<1$) slows energy decay. It demonstrates that $\beta$ weakens indirect damping and degrades structural stabilization. The results reveal the intrinsic interaction between fractional memory dissipation and fractional coupling in strongly coupled dissipative systems.

We analyze the stability of the Hagstrom-Warburton nonreflecting boundary conditions (HW-NRBCs) for the first-order time-dependent Maxwell equations. The HW-NRBCs enjoy very small reflection coefficients and do not use high-order derivatives or nonlocal boundary operators, which makes them well-suited for high-order accurate numerical discretizations. The main result of this paper is an $L^2$ a-priori bound on the solution in terms of initial and boundary data and volume sources in a half-space. To obtain this result, we first derive a mapping that takes outgoing components to ingoing components of the solution of the Maxwell equations with HW-NRBCs, which shows that the Kreiss condition does not hold uniformly. Next, to prove stability, several symmetrizers are constructed, which establishes well-posedness in a generalized sense.

We study the existence of normalized solutions for the nonlinear Dirac equation \[ \begin{cases} -i\sum\limits_{k=1}^3\alpha_k\partial_k u + m\beta u - |x|^{-b}|u|^{p-2}u = \mu u, \quad x\in\mathbb{R}^3, \int_{\mathbb{R}^3}|u|^2 dx = a, \end{cases} \] where $b\in(0,1)$, $p\in(2,3-b)$, $a>0$ is a prescribed mass, and $\mu\in\mathbb{R}$ is a Lagrange multiplier. For any $b\in(0,1)$ and $p\in(2,3-b)$, we establish the existence of a normalized solution for all sufficiently small masses $a>0$, with $\mu\in(0,m)$ and $u\in H^{\frac12}(\mathbb{R}^3;\mathbb{C}^4)$. Our results cover the full range of nonlinearities, including mass-subcritical, mass-critical, and mass-supercritical cases. The main challenges are the strongly indefinite nature of the Dirac operator and the loss of translation invariance caused by the singular weight $|x|^{-b}$. We overcome these difficulties by combining a constrained min-max reduction method with a novel weighted compact embedding in $L^p(\mathbb{R}^3,|x|^{-b}\,\mathrm{d}x;\mathbb{C}^4)$. This approach circumvents the singular potential at the origin and yields a unified existence theory valid in the small-mass regime.

We study the polar director structure in thin planar cells filled with bent-core liquid crystals in the ferroelectric smectic-A phase (SmAP$_\text{F}$). We analyze a continuum phenomenological model proposed in the physics literature and present rigorous proofs of the existence and uniqueness of the equilibrium solutions. We further investigate the qualitative properties of nontrivial solutions and examine the effects of a bias electric field, surface anchoring, and cell thickness on the polar director configuration. Our results are consistent with previous experimental and numerical simulations reported in the physics literature. In addition, our analysis reveals new parameter-dependent behaviors supported by our numerical simulations and extends results reported from previous literature.

We study a reactive Maxwell–Stefan-type membrane transport system under a finite-occupancy constraint with explicit vacancies. The admissible state space is $\mathcal{D} = \{u \in (0,1)^n : \varrho(u) := \sum_{i=1}^n u_i < 1\}$, where the vacancy fraction $1-\varrho(u)$ represents the local free volume. This bounded-occupancy geometry induces a Boltzmann–Fermi entropy and a global parametrization by entropy variables. The associated mobility is assumed to split into a composition channel, corresponding to redistribution at fixed total occupancy, and a mass channel, corresponding to variation of the filling fraction. The main structural difficulty is that the unaugmented mobility may lose coercivity in the mass channel. We show that a single rank-one augmentation of the form $\gamma_0 \mathbf{1} \otimes \mathbf{1}$ restores full coercivity while leaving the composition block unchanged. On this basis, we prove four results: a quantitative channel-wise coercivity estimate for the augmented mobility; global existence of entropy weak solutions via an implicit Rothe scheme in entropy variables; a weak–strong stability estimate in relative entropy with uniqueness in the strong class; and convergence of a fully implicit finite-volume approximation that preserves the bounded-occupancy structure and satisfies a discrete entropy inequality.

We prove a sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues of bounded Lipschitz domains in $\mathbb R^N$. More precisely, if $N\ge2$ and $\Omega\subset\mathbb R^N$ is a bounded Lipschitz domain, then $$ \prod_{j=1}^N \sigma_j(\Omega)\le \frac{\omega_N}{|\Omega|}, $$ where $0=\sigma_0(\Omega)<\sigma_1(\Omega)\le\sigma_2(\Omega)\le\cdots$ are the Steklov eigenvalues of $\Omega$, and $\omega_N$ denotes the volume of the unit ball in $\mathbb R^N$. This extends the convex-domain theorem of Henrot, Philippin, and Safoui to arbitrary bounded Lipschitz domains, and in particular settles the remaining higher-dimensional case of a problem posed by Henrot.

We develop an abstract regularity framework for a class of two-dimensional nonlinear elliptic systems, including almost harmonic maps. The approach combines a Campanato-type iteration scheme with a Caccioppoli-type estimate and identifies general assumptions under which local Hölder continuity follows. More precisely, we prove that any class of admissible pairs $(u,f)$ that is stable under rescaling and satisfies an oscillation-decay property consists of locally Hölder continuous maps. The resulting Hölder exponent is explicit and matches the classical Morrey–Campanato threshold determined by the Lebesgue integrability of the source term $f$. The framework is purely analytic and avoids the use of $\mathcal{H}^1$–$\mathrm{BMO}$ duality, Wente's inequality, moving frames, and conformal uniformization. We illustrate the flexibility of the framework through several classes of elliptic systems. As a first example, we recover local Hölder continuity for almost harmonic maps \[ -\Delta u=|\nabla u|^2u+f \] into $\mathbb{S}^n$ with $L^q$-integrable tension fields by means of a direct argument independent of the classical harmonic map regularity theory. We next consider systems of the form \[ -\Delta u=\Omega\cdot\nabla u+f, \] showing that the analytic condition $\operatorname{div}\Omega\in L^q$ for some $q>1$ is sufficient to ensure regularity (for classical harmonic maps, $\operatorname{div}\Omega=0$). We further apply the framework to...

We prove that the support of the canonical space-time measure for a Brakke flow is a parabolic $(k+2)$-rectifiable set. As a consequence, we obtain that at almost all points along the flow, with respect to this canonical space-time measure, there exists a unique, static, planar tangent flow, and that various notions of density for the flow agree at these points. Moreover, following on from our previous work `The space-time-Grassmann measure of the Brakke flow', we continue to develop the approach to the Brakke flow as a space-time-Grassmann measure. We prove that the standard notion of convergence for Brakke flows, coming from the compactness theorem of Ilmanen (7.1 of `Elliptic regularization and partial regularity for motion by mean curvature'), is equivalent to the convergence of these space-time-Grassmann Radon measures. This gives an alternate notion of varifold convergence to the one exhibited in 7.1(ii) of `Elliptic regularization and partial regularity for motion by mean curvature'.

We prove the global well-posedness of small perturbations of two-dimensional sessile droplet equilibria for the incompressible Navier–Stokes equations with surface tension, Navier-slip boundary conditions, and a dynamic contact-point law. The main difficulty is the construction of solutions in the presence of the horizontal translational degeneracy of the equilibrium manifold. To remove this degeneracy, we work in a moving polar coordinate system determined by an orthogonality condition. We then establish local well-posedness through a Galerkin construction of pressureless weak solutions, recovery of the pressure, higher-order estimates, and a contraction argument. Combining this local theory with the global energy–dissipation estimates obtained in our previous work yields a unique global solution and the corresponding exponential decay estimate.

For every $n \ge 3$, we construct uncountably many families of type II ancient solutions to the Yamabe flow on the unit round $n$-sphere $\Ss^n$. These families are pairwise distinct up to conformal equivalence, and no member is conformally equivalent to a rotationally symmetric solution. At every negative time, the Ricci curvature tensor of each solution is indefinite at some point. Moreover, the associated backward limit space is a wedge sum of finitely many isometric copies of $\Ss^n$. These examples show that the collection of ancient Yamabe flows on $\Ss^n$ has a much richer structure than suggested by two natural comparison problems: the compact ancient Ricci flows on $\Ss^2$, all of which are known to be rotationally symmetric, and the elliptic Yamabe equation on $\R^n$, whose positive entire solutions are only the standard bubbles. The construction uses a non-radial inner–outer gluing scheme. After stereographic projection, we reformulate the flow as a conformally invariant parabolic problem on $\R^n$. By exploiting Kelvin invariance and switching between the Euclidean and spherical formulations as needed, we control the non-radial modes directly without reducing the problem to one space dimension. Weighted Hölder estimates provide the pointwise control needed to establish the Type II behavior, the Ricci-sign property, conformal inequivalence, and the description of the backward limits in a straightforward manner.

This work investigates linear elliptic equations with multiple nonlocal nonlinearities in a bounded domain, which takes the form \begin{equation*} -\nabla\cdot(\od\nabla u) +\lambda fu= \eta \sum_{j=1}^kh_j\boldsymbol{\mathsf{N}}_j[u]+h_0. \end{equation*} Here $\lambda$ and $\eta$ are positive parameters, and $\boldsymbol{\mathsf{N}}_j[u]$ represents a nonlocal term dependent on the unknown solution~$u$. All coefficients are defined in the context of a wide range of applications. As such equations generally lack a variational structure, a new approach is developed that combines fixed-point arguments with asymptotic techniques. This method establishes the existence and multiplicity of solutions under specific conditions. Of particular interest is the role of the nonlocal nonlinearities, which lead to diverse structures of solutions. To the best of our knowledge, this property is a new finding not typically observed in related nonlocal elliptic problems.

Landau solutions, when oriented along the vertical axis, represent a one parameter family of exact, self-similar, axisymmetric, swirl-free solutions to the 3D stationary Navier-Stokes equations forced by an upward facing point-source of momentum at the origin. They have an isolated singularity at the origin. Other singular steady state solutions can be derived from a similar framework. Some of these variants have been proposed as models for physical scenarios. For example, a solution first found by Squire has been proposed as a model of a fluid entrained to a radially discharging surface layer of oil. Another, Serrin's swirling vortex, exhibits qualitative features shared with some tornadoes, like a two-cell structure consisting of a central downdraft and peripheral updraft as well as swirl. The first objective of this paper is to provide a detailed analysis of these and other examples, especially when boundaries are present. In this direction we find several new, physically motivated classes of solutions and identify new connections between the physics literature and the mathematics literature. Many of these examples are formulated on the half-space, but much of the mathematical literature on singular steady-state solutions is for the whole-space. The second objective of this paper is to establish asymptotic stability for many of these solutions by formulating the problem in domains with boundaries. Our most general result requires a new approach to asymptotic stability that is based on eventual regularity. As a special case, we prove that a class of steady-state solutions motivated by Serrin's swirling vortex are stable under axisymmetric perturbations. This requires a novel observation because these solutions are too singular – they are called “Type III” in the literature – to be directly amenable to existing approaches.

We study finite-time blow-up for the nonlinear wave equation \begin{equation*} v_{tt}-\Delta v=|\nabla_x v|^2 \end{equation*} in dimensions $n\geq2$, under radial symmetry. For every prescribed radius $r_0>0$, we construct solutions which blow up in finite time $T>0$ on the sphere $\{|x|=r_0\}$ with logarithmic Type-I rate. The leading singular dynamics are governed by “generalised self-similar” profiles of the associated one-dimensional equation, while the radial geometry generates a curvature correction of size $\mathcal{O}((\frac{T}{r_0})^2)$. A key simplification in our approach is a logarithmic radial correction which removes the first-order radial drift and reduces the geometry to a decaying inverse-square forcing. We further prove asymptotic stability of the resulting family under radial perturbations. A new feature compared with the one-dimensional theory is that the stable blow-up family is not fully explicit. To overcome this, we develop spectral and semigroup estimates on an extended light cone, together with Lipschitz dependence on the modulation parameters for the spectral projections, the stable flow, and the non-explicit correction.

We study the problem of cooking the yolk and albumen of a hen's egg to their respective optimal temperatures of $T_Y^* = 65^\circ$C and $T_W^* = 85^\circ$C, subject to the requirement that neither temperature ever exceed its target at any time during cooking, since temporary overshoot still overcooks the egg even if the final reading is correct. We model the egg as a two-domain sphere with distinct thermal diffusivities, and take the Laplace transform of the heat equation in each domain, reducing the problem to a $3 \times 3$ linear system in the transform variable $s$ with hyperbolic-trigonometric solutions. The resulting transform is inverted numerically via Talbot's method and validated against a finite-difference solver. A single boiling phase cannot satisfy the no-overshoot requirement: the thin outer albumen heats far faster than the insulated yolk and necessarily overshoots $T_W^*$ before the yolk approaches $T_Y^*$. We show that a three-phase protocol resolves this: a sous-vide pre-soak at exactly $65^\circ$C (which cannot overshoot since the bath temperature equals the target), a short boil to bring the albumen toward $T_W^*$, and an ice-water bath that arrests the albumen's residual overshoot while residual heat continues raising the yolk to its target. Optimizing the phase durations gives $17.26$ minutes of sous-vide, $66$ seconds of boiling, and an ice bath, achieving both targets at $T^* \approx 20.67$ minutes with neither constraint violated at any time. This compares favorably with the periodic protocol of Di Lorenzo et al. (2025), which requires 32 minutes and misses both targets substantially.

We study a homogenization problem for first-order Hamilton-Jacobi equations in the Wasserstein space with a convex Hamiltonian. We show that the solution $U^\varepsilon$, which is the value function of a mean field control problem, converges uniformly as $\varepsilon \to 0$ to the solution of a limiting Hamilton-Jacobi equation whose Hamiltonian is obtained through a suitable cell problem. Furthermore, we establish quantitative rates of convergence. Under general assumptions with multiscale dependence, we prove that the rate of convergence is $O(\sqrt{\varepsilon})$. When the Hamiltonian depends only on the fast variable and the momentum, we establish the sharp convergence rate $O(\varepsilon)$. To the best of our knowledge, this is the first quantitative convergence result extending the optimal rate for first-order Hamilton-Jacobi equations in finite dimensions to the Wasserstein space. Finally, we show that our analysis extends to dynamic optimal transport problems, where the terminal condition imposes a constraint on the final distribution.

We show that spatial zero extension across the boundary may fail in parabolic Sobolev spaces $\mathring{\mathcal{H}}^1_p((0,T) \times \Omega)$, which can also be characterized as $$ L_p(0,T;\mathring{W}^1_p(\Omega))\cap W^1_p(0,T; W^{-1}_{p}(\Omega)). $$ More precisely, for any $p\in [1, \infty)$, we construct a function $u\in \mathring{\mathcal{H}}^1_p((0,T)\times \mathbb{R}^d_+)$ whose zero extension does not belong to $\mathcal{H}^1_p((0,T)\times \mathbb{R}^d)$. The obstruction occurs even for a flat boundary and is caused by a self-similar boundary layer concentrated at the initial-boundary corner, which produces a boundary supported normal flux defect after zero extension. We also discuss the suitability of various Sobolev-type spaces as solution spaces for parabolic equations in divergence form.

We consider quasilinear conservative systems \[ (\phi(|\dot q|)\dot q)'=\nabla V(q), \qquad q\in\R^{N}, \] with $\Phi$-growth kinetic term and potential $V\in C^{1}(\R^{N};\R)$. Assuming that for some $c\in\R$ the sublevel set $\{V\le c\}$ splits into two disjoint closed subsets $\mathcal V_c^{-}$ and $\mathcal V_c^{+}$, we prove the existence of trajectories $q_c$ with prescribed energy $-c$ connecting $\mathcal V_c^{-}$ and $\mathcal V_c^{+}$, obtained through an energy-constrained variational method. Although the construction yields weak solutions in an Orlicz-Sobolev setting, minimal $c$-connections are shown to be classical $C^2$ trajectories satisfying the strong energy identity $E_{q_c}\equiv -c$. The resulting entire trajectories include heteroclinic, homoclinic, and brake-type orbits. Applications to double-well, Duffing-type, and multiple pendulum systems are discussed.

We study the wave equation in the exterior of a strictly convex bounded domain $K \subset {\mathbb R}^d, d \geq 3,$ odd, with dissipative boundary condition $\partial_{\nu} u - \gamma(x) \partial_t u = 0$ on the boundary $\Gamma$ and $0 < \gamma(x) <1, \:\forall x \in \Gamma.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ In [10] we established that for $\gamma \equiv const$ and $K = \{x \in {\mathbb R}^3: \:|x| \leq 1\}$ the operator $G$ has no eigenvalues and we conjectured that the same result holds for every strictly convex obstacle. In this paper we prove this conjecture.

We study the Wasserstein gradient flow generated by a family of extended generalized variance functionals, defined as the expected squared $n$-dimensional volume of a simplex, which includes the classical variance-type interaction and generalized variance as special cases. The key structural observation is that this functional depends only on the covariance matrix. Consequently, the Wasserstein gradient flow reduces to a finite-dimensional system for the eigenvalues of the covariance matrix, and the full measure-valued solution can be recovered through an explicit linear pushforward representation. Using this representation, we establish global well-posedness for arbitrary initial data in $\mathcal P_2(\mathbb R^d)$ without assuming compact support. We also study the long-time behavior of the flow. For every initial datum in $\mathcal P_2(\mathbb R^d)$, the solution converges to a limiting equilibrium measure whose covariance has rank strictly less than $n$. Moreover, we obtain sharp convergence rates in all spectral regimes: exponential in the non-degenerate case and algebraic with the optimal exponent in the degenerate case.