Papers

We prove that a smooth toric Fano fibration admits a complete gradient shrinking Kähler-Ricci soliton.

We prove global well-posedness and scattering for the three-dimensional defocusing cubic nonlinear Schrödinger equation with arbitrary initial data in H^s(R^3), $s>\frac{1}{2}$. The proof combines the $I$-method with improved long-time bilinear $L^2_{t,x}$ estimates for frequency-localized components of the solution. The key high–low frequency estimate follows from a directional interaction identity and an induction on frequency.

A vanishing sequence $V_n$ of divergence-free matrix fields is one that is bounded in $L^1$ and is carried by open sets $A_n$ of vanishing volume. Recently established, Bouchitté's vanishing mass conjecture says that the directions such a sequence can carry are rigidly constrained: their limiting distribution must be a superposition of microstructures whose barycenters are singular matrices. We prove the converse in the non-symmetric setting: every such superposition is attained by a vanishing sequence. In fact, we are able to construct divergence-free fields $V_n$ $supported$ on $A_n$, whose relative boundary is a smooth compact manifold.

For each sufficiently large $N$, we construct smooth solutions of the magnetohydrostatic equations on embedded solid tori whose regular pressure levels are nested tori, whose magnetic field vanishes exactly on a round magnetic axis, and whose group of Euclidean symmetries, even when isometries reversing the sign of the field are admitted, is exactly the cyclic group $C_N$ of rotations through multiples of $2\pi/N$. For each such $N$, these equilibria form a smooth one-parameter family which is locally nontrivial in the moduli space of embedded configurations and have none of the plane-reflection, axial, or helical symmetries conjectured by Grad. A Lean 4 certification of the above results is also provided.

We study the Cauchy problem for the one-dimensional full compressible Navier–Stokes equations with viscosity $\mu(\theta)=\tilde\mu\theta^\alpha$ and heat conductivity $\kappa(\theta)=\tilde\kappa\theta^\beta$. For each fixed $\beta\ge0$, we prove the global stability of the combination of a viscous contact wave with rarefaction waves, provided that the viscosity exponent $\alpha$ and the total wave strength are sufficiently small. The initial perturbation may be large in $H^1$, and the specific volume and temperature are assumed to have positive initial lower bounds. The solution remains uniformly bounded in $H^1$ relative to the wave, and converges uniformly to that wave as time tends to infinity. We show that both the specific volume and the temperature admit time-uniform lower and upper bound. Estimates for the logarithmic volume derivative and the temperature gradient then close the $H^1$ estimates without requiring a second derivative of the initial specific volume.

We establish arbitrary-order a priori regularity estimates for the reverse heat flow in the Ornstein–Uhlenbeck setting, that is, for the probability-flow ODE of score-based diffusion models. For compactly supported, possibly singular targets satisfying a dimension-uniform doubling condition on supporting-cap masses, we bound the iterated material derivatives of the flow and their first spatial derivatives pointwise, with constants independent of the dimension $d$ and explicit in time and support radius. The proof rests on two ingredients: a spatially uniform bound on normal-direction posterior fluctuations, which is the only point where the geometry of the target enters, and a centered posterior-moment normal form closed under material differentiation. This algebraic structure, together with a weighted regularity calculus, yields quantitative bounds at every order, including the mixed space–time directional derivatives required by Runge–Kutta stages. As an application, for exact-score sampling stopped at forward time $\delta>0$, order-$p$ Taylor and explicit Runge–Kutta schemes reach accuracy $\varepsilon$ in $\widetilde O(d^{1/p}\varepsilon^{-1/p})$ steps in total variation, and the Taylor scheme in $\widetilde O(d^{1/(2p)}\varepsilon^{-1/p})$ steps in $W_2$.

We study strict ground-level gaps for autonomous nonlinear Dirac equations on noncompact metric graphs. The relevant level at infinity is the autonomous full-line level \(d_\lambda^\infty=d_{\lambda,\mathbb R}\), since a concentrating sequence escaping along a half-line sees the real line after recentering far from all vertices. We prove that, uniformly for \(\lambda\) in a compact subinterval \(I\Subset(-mc^2,mc^2)\), the presence of a sufficiently long pendant path in the essential reduction of the graph implies \[ d_\lambda(\mathcal G)<d_\lambda^\infty. \] The proof uses a half-line restriction of a symmetric full-line ground state, uniform exponential decay, resolvent localization for massive Dirac-Kirchhoff spectral projections, and a compactness argument for generalized Nehari fibres. We also identify elementary graph classes for which no such gap can hold: pure transmission graphs whose essential reduction is isometric to the full line satisfy \(d_\lambda(\mathcal G)=d_\lambda^\infty\).

We study the Cauchy problem for the Boltzmann-BGK model in the three-dimensional whole space with nontrivial collision frequency $\nu(f)=\rho^\alpha T^{\beta}$, where $\alpha\in(1/3,1]$, $\beta\in[0,1]$, and $3\alpha+2\beta\geq 2$. Existing well-posedness results for such models have largely been confined to near-equilibrium regimes and stationary problems. For nonnegative initial data with finite mass and energy that are sufficiently small in suitable polynomial-weighted $L^{\infty}$ norms, we establish global existence and uniqueness of mild solutions near vacuum. No uniform positive lower bound on the macroscopic density or temperature is imposed. The analysis relies on two complementary mechanisms. First, a phase-space weight invariant along free-transport characteristics yields dispersive estimates for the collision frequency and the gain term. The resulting time-integrable decay controls the nonlinear growth and closes the global weighted estimates. Second, we establish a Lipschitz estimate for the relaxation operator in a weighted $L^1$ space controlling mass and energy, with a Lipschitz constant depending only on weighted upper bounds for the distribution functions. The relaxation operator $\nu\mathcal{M}$ has a more favorable structure for Lipschitz estimates than the local Maxwellian $\mathcal{M}$ alone: the collision frequency compensates for the singular dependence on the macroscopic fields, allowing us to establish Lipschitz continuity without uniform positive lower bounds for the density or temperature. Together, these estimates yield a unique global mild solution with uniform weighted bounds.

We prove interior Hessian estimates in arbitrary dimensions for admissible solutions of Hessian quotient equations with constant right-hand side, when the numerator and denominator differ in degree by one or two. Counterexamples show that this degree range is optimal. The proof combines a new nonlinear comparison function, a doubling maximum principle, and a Pogorelov estimate.

We study finite-time singularity formation for a one-dimensional Euler–Poisson system in Lagrangian coordinates with Cattaneo heat conduction and quadratic heat-flux corrections in the pressure and internal energy. Under the structural choice $\kappa=\kappa_0v$, two different breakdown mechanisms are obtained. First, an explicit affine solution reaches $v=0$ in finite time, so the Eulerian density $\rho=1/v$ diverges while the remaining state variables stay finite at each fixed spatial point. Second, for $a'(1)<0$, the Poisson field is localized near equilibrium by $F=\phi_x/v$, producing a strictly hyperbolic balance law subject to the propagated constraint $F_x=1-v$. We compute the characteristic speeds and an explicit genuine-nonlinearity coefficient and show that genuine nonlinearity fails for all propagating families only when two explicit algebraic degeneracy conditions hold simultaneously. For neutral compactly supported data satisfying the Poisson constraint exactly, a narrow compressive wave generates only an $O(\varepsilon^2\eta)$ zero-speed Poisson mode. A constraint-compatible John–Hörmander–Bärlin bootstrap and a Riccati comparison then yield finite-time gradient catastrophe while the solution remains uniformly close to equilibrium and $F_x$ stays bounded. The two results exhibit density collapse and small-amplitude gradient blow-up as distinct breakdown mechanisms within the same Euler–Poisson–Cattaneo model.

In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping $f\colon\mathbb{R}^n\to\mathbb{R}^m$ lies in the local Sobolev space $W^{1,p}_\text{loc}(\mathbb{R}^n\colon\mathbb{R}^m)$ if and only if $f$ is a $(p,1-n/p)$-compactly Hölder mapping, where $p>n$. In fact, we prove the following more general result: whenever $X$ is a complete $Q$-Ahlfors regular metric space supporting a $p$-Poincaré inequality with $p>Q$, and $V$ is any Banach space, the compactly Hölder class $CH^{p,1-Q/p}(X\colon V)$ consists precisely of the continuous representatives of the local Hajłasz and Newtonian Sobolev spaces $M^{1,p}_\text{loc} (X \colon V)$ and $N^{1,p}_\text{loc}(X\colon V)$. We also establish quantitative comparisons between the corresponding seminorms.

We prove sharp two-eigenvalue isoperimetric inequalities for Neumann and Aharonov–Bohm Neumann spectra on surfaces. If $\Omega\subset\mathbb S^2$ is smooth, simply connected and proper, then the harmonic mean of $\mu_2(\Omega)$ and $\mu_3(\Omega)$ is bounded above by the first positive Neumann eigenvalue of the equal-area geodesic disk, with equality only for disks. For simply connected surfaces with Gaussian curvature bounded above, we obtain the magnetic analogue for the first two Aharonov–Bohm eigenvalues. The proof combines a two-dimensional reciprocal Rayleigh–Ritz principle with Green-level comparison. A key additional ingredient is a spectral ordering theorem for magnetic spherical caps: for $0<\nu<1/2$, the first two eigenvalues lie in the angular sectors of effective orders $\nu$ and $1-\nu$. We prove this by the factorization $L_0=T^*T$, $L_1=TT^*$ and an exact Neumann–Dirichlet spectral shift. We also obtain sharp full-sphere and closed-surface bounds, and an annular inequality in terms of conformal modulus and flux.

We study the two-dimensional incompressible Euler-Maxwell system under the normal geometry in which the velocity and electric field are planar and the magnetic field is normal to the plane. We identify a structural distinction between the projected, charge-free formulation and the unprojected system. For the projected system, we establish a local well-posedness theory for initial velocities in $H^1(\mathbb R^2)$ with bounded initial vorticity and electromagnetic data in $H^{\frac 32}(\mathbb R^2)$, under a strict sub-luminal condition on the velocity. The main ingredient is an endpoint time-like trace estimate for half-waves, which controls the magnetic gradient in $L_t^2$ along every fluid trajectory. This closes the Yudovich vorticity estimate without requiring an Eulerian Lipschitz bound on the electromagnetic field. For the unprojected system, the longitudinal electric mode produces an additional vorticity source term containing the charge density $\operatorname{div }E$. In this case, we construct smooth, compactly supported radial electromagnetic data converging to zero in $H^{\frac32}(\mathbb R^2)$, with zero initial velocity, whose unique global smooth solution exhibits a vorticity norm inflation in $L^\infty_x$. Thus, at the same electromagnetic Sobolev regularity, the Gauss constraint separates endpoint regularity propagation from charge-driven vorticity norm inflation.

On a $d$-dimensional, $d\geq 2$, globally hyperbolic spacetime with a timelike boundary $(\mathcal{M},g)$, we investigate the Klein-Gordon equation with Robin boundary conditions. First of all, we prove existence and uniqueness of the advanced and retarded fundamental solutions, discussing in addition their structural properties. Secondly, under the hypothesis of infinitesimal convexity of $\partial\mathcal{M}$ and of local finiteness of boundary reflections for broken null bicharacteristics, we characterize their wavefront set in the interior $\mathring{\mathcal{M}}$, using the propagation of singularities theorem of Melrose and Sjöstrand. In the second part of the paper, we introduce a notion of Robin-Hadamard two-point correlation function as a positive distribution on $\mathcal{M}\times\mathcal{M}$ with prescribed antisymmetric part and singular structure, the latter being characterized in terms of its wavefront set. We complement this definition with a local Hadamard form. Up to a smooth remainder, this consists of the standard directed Hadamard parametrix together with additional contributions associated with the reflected null rays. Their coefficients are determined by transport equations supplemented by Robin matching conditions at the boundary. The construction applies to an arbitrary, but finite number of reflections. We prove that this local formulation is equivalent to the wavefront set condition, thereby establishing a counterpart of Radzikowski's theorem in the presence of a timelike boundary. Finally, starting from the ground state on suitable static backgrounds and using a deformation argument, we prove existence of Robin-Hadamard two-point correlation functions on the class of spacetimes considered.

The branched transport problem is a nonconvex and nonsmooth variational optimization problem on normal $1$-currents in $\mathbb{R}^n$ with prescribed boundary. The optimality is with respect to some non-decreasing, lower semicontinuous, and subadditive function $\tau:\mathbb{R}_+\to\mathbb{R}_+$ with $\tau(0)=0$ describing the cost $\tau(m)$ to move an amount of mass $m$ per unit distance. The subadditivity leads to complicated, hierarchically ramified patterns in the support of (suboptimal) solutions. These network-like sets appear to have regularity properties similar to those of the singular surfaces arising in the so-called Brakke flow, a weak generalization of the mean curvature flow. We construct a geometric flow of transportation networks, which correspond to normal real $1$-rectifiable currents, by modifying Brakke's variational approximation scheme. We prove the existence of a limit that is Hölder continuous with respect to the flat norm and whose branched transport cost decreases along the geometric evolution. We further analyze a closely related geometric flow approximated by $1$-varifolds, whose weight measures model the branched transport cost, and establish its $1$-rectifiability together with a motion law analogous to Brakke's inequality.

The Whitham equation has given its name to a wider family of nonlocal, nonlinear equations with very weak dispersion, which all feature highest waves. Recent work by Ehrnström, Maehlen and Varholm establishes leading-order asymptotics at the crest of solutions to such Whitham-type equations, and conjectures similar expansions for all derivatives of the solution. By extending their techniques we confirm the conjecture for a range of equations with highest waves of Hölder regularity $C^{s}$ for $s\in[0.35,1)$. We do this by strong induction, redistributing difference operators to deal with singularities in higher-order derivatives arising from integral convolution kernels. The restriction $s\geq0.35$ arises solely from the separate argument establishing the zeroth-order asymptotics, which requires a uniform sign estimate that we verify using rigorous interval arithmetic. The higher-order induction itself applies for every $s\in(0,1)$, so any extension of the zeroth-order result immediately yields the corresponding higher-order expansions.

We consider initial and initial boundary value problems for the one-dimensional compressible Navier–Stokes equations of an ideal polytropic gas in unbounded domains. The viscosity and heat conductivity are proportional to $\theta^\alpha$ and $\theta^\beta$, respectively. For every fixed $\beta\geq0$ and sufficiently small $\alpha\geq0$, depending on the initial data, we prove the global existence and asymptotic stability of a strong solution for arbitrarily large $H^1$ initial data. Dividing the momentum equation by the viscosity allows us to estimate the specific-volume derivative without an initial $H^2$ assumption.

We study global strong solutions and their large-time behavior for the one-dimensional outer pressure problem of a viscous, heat-conducting ideal polytropic gas. The viscosity and heat conductivity satisfy $\mu=\tilde\mu\theta^\alpha$ and $\kappa=\tilde\kappa\theta^\beta$. For each prescribed nonnegative pressure and each fixed $\beta\geq0$, we allow large $H^1$ initial data with positive specific volume and temperature, provided that $\alpha\geq0$ is sufficiently small. When the limiting outer pressure is positive, the solution has uniform bounds and converges to a stationary state. When the limiting outer pressure is zero and the stated weighted pressure conditions hold, the normalized solution converges to an expanding state at an algebraic rate in physical time.

We consider a generalisation of the probabilistic reformulation of the supercooled Stefan problem from arXiv:1902.05174 allowing for non-integrable initial data. From an application of Ito's formula, we obtain a relationship between the Laplace transform of the initial data and a transform of the solution, together with its possibly discontinuous free boundary. Using this relationship we give a sharp existence result for global solutions, we extend previous results on the uniqueness of solutions, and we determine the asymptotic behaviour of solutions for a certain class of initial data.

Landau solution is a special family of solutions to the stationary Navier Stokes equations, which is important for the study of stationary problems such as asymptotic behavior and regularity. In this paper we prove that the Landau solution is rigid in rotated self-similar class when rotation parameter $\alpha$ sufficiently small or large. The arguments in the two parts rely on different approaches. For small rotation, we use the compactness argument and establish a classification lemma of the self-similar kernel of the linearization around any Landau solution, this method can also be extended to DSS and RDSS cases. For large rotation, the strong angular dissipation produces additional coercivity on the non-axisymmetric component, which forces the solution to reduce into Landau solution.

Recently, the existence of regular solutions $(u,p)$ to the stationary incompressible Navier–Stokes equations on $\mathbb R^n$ with bounded compactly supported forces $f$ for $5\le n\le15$ was established by Li–Yang (Comm. Math. Phys., 2022). However, it is still unknown whether there exists a regular solution for dimensions $n>15$. Here we answer this question and obtain the existence of regular solutions to the stationary Navier–Stokes equations in arbitrarily high dimensions. The key innovation is to combine an averaged signed pressure-potential identity with $L^2$-BMO estimates for the Bernoulli source flux, yielding a Bernoulli supremum bound independent of the size of the skew coefficient representing the drift. Together with the energy estimate for the comparison solution, this yields a sublinear bound for the Newtonian potential of the positive Bernoulli function, removing the upper dimension restriction in the earlier existence argument. We also construct periodic solutions and whole-space solutions for forces in $L^\infty\cap L^{2n/(n+2)}$. For compactly supported forces, we record two far-field profiles, their velocity, gradient and pressure remainders, and the corresponding cancellation criteria.

In this paper, we study the separation and stability properties of radial solutions to a Hénon-type equation with exponential nonlinearity on the hyperbolic space. By transforming the radial hyperbolic equation into a weighted Euclidean equation and applying well-established separation results in Euclidean space, we classify the solution structures or establish sharp alternatives for radial solutions throughout the full range $n\ge2$ and $\alpha>-2$. We also give an affirmative answer to the open question recently posed by Huang and Zhao.

We introduce determinantal ellipticity, or det-ellipticity, a quantitative structural condition for fully nonlinear elliptic operators on compact Kähler manifolds that provides the link between the $L^\infty$-estimates and algebraic/combinatorial properties of a large class of Hessian elliptic operators. On the analytic side, we introduce $\mathcal D$-subsolutions extending the determinant sublevel condition of Sui–Sun. Using the auxiliary comparison method of Guo–Phong–Tong and pluripotential theory for complex Monge–Ampère equations, we obtain relative $L^\infty$-estimates in big cohomology classes under a determinant-entropy bound for viscosity supersolutions and singular reference potentials. Det-ellipticity provides a systematic construction of such subsolutions. On the algebraic side, we study Gårding elliptic polynomial operators of degree $d$. We characterize the determinant increment bound with exponent $d/n$ by positivity of the determinantal capacity of the top homogeneous part. This is equivalent to balanced-point conditions for the Newton polytopes of rank-one scalarizations and to slope semistability of an associated subspace polymatroid. Together with the analytic hypotheses, these criteria yield relative $L^\infty$ estimates for the corresponding fully nonlinear equations.

This paper is concerned with the following planar Schrodinger-Poisson system: $\begin{cases} -\Delta u +V(x)u+\lambda\phi u=u\ln u^2, & \text{in }{R}^2,\\ \Delta\phi=u^2, & \text{in }{R}^2. \end{cases}$ where $\lambda\in{R}$ is a parameter and $V\in C^1({R}^2,R}^{+})$ is a coercive potential. Due to the presence of the logarithmic nonlinearity, the manifold method developed by Ruiz [R] is not applicable here. We apply a general minimax principle to an auxiliary functional and prove the existence of a nontrivial solution. Then, by minimizing the energy functional over the set of all nontrivial solutions, we succeed in showing that the system admits a ground state solution. If, in addition, $V$ is radially symmetric, we obtain infinitely many non radial sign-changing solutions.

We study a generalized vanishing discount problem for Hamilton–Jacobi equations without assuming monotonicity either pointwise or in the averaged sense with respect to all Mather measures. Specifically, we consider \[ \lambda a(x)u(x)+H\big(x,Du(x)\big)=c_0, \] and assume that, for every sufficiently small $\lambda>0$, this equation admits a strict subsolution bounded from below uniformly with respect to $\lambda$. We prove that, under a smallness assumption on the Mather quotient, the maximal viscosity solution converges uniformly as $\lambda\to 0^+$ under a sign condition on the discount coefficient $a(x)$ determined by a prescribed part of the Aubry set. The limit is characterized explicitly in terms of the Peierls barrier and the selected Mather measures. As an application, the elementary solution of the critical Hamilton–Jacobi equation \[ H\big(x,Du(x)\big)=c_0 \] associated with any isolated static class can be realized as the vanishing discount limit. This provides the first mechanism for selecting multiple critical solutions in vanishing discount problems beyond the classical monotonicity framework, showing that different families of Mather measures yield different limiting critical solutions. We finally propose a variant of the discounted problems for which the strict subsolution hypothesis is automatically verified and hence applies to more general Hamiltonians $H$ and functions $a$.

In this paper, we study the Rochet-Choné model of monopolist pricing in the seller's market limit. We introduce a monotonicity formula which yields an asymptotic description of solutions, and prove that the proportion of consumers priced out of the market vanishes as overall consumer demand increases, providing a counterpoint to Armstrong's desirability of exclusion.

We study simultaneous vanishing discount and viscosity limits for mechanical Hamilton–Jacobi equations on the $n$-dimensional torus. In the critical regime, we explicitly identify the selected limit in terms of the critical Mañé potential and the quadratic behavior of the potential near its wells. We also obtain quantitative and sharp convergence estimates in the subcritical regime. In the supercritical regime, we prove convergence when the local harmonic ground-state energy of the potential has a unique minimizer among the wells. Finally, we establish a concentration phenomenon of the associated Gibbs measures.

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior, and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. The $L_p$ Minkowski problem for $C$-close sets is to determine, for a real number $p$ and a nonzero finite Borel measure $\mu$ defined on $\Omega_{C^\circ}=S^{n-1}\cap \mathrm{int} C^{\circ}$, whether there exists a $C$-close set $\mathds{A}$ such that $\mu$ is the $L_p$ surface area measure of $\mathds{A}$. In this paper, we will solve the problem for $p\in (0,1)$ and for $\mu$ being a nonzero finite Borel measure on $\Omega_{C^\circ}$. Moreover, we establish the continuity of solutions to the $L_p$ Minkowski problem for $p\in [0, 1]$ in several settings.

Let $\mathbb H^n$ be the $n$-dimensional hyperbolic space, and $\Delta_{\mathbb H^n}$ be the Laplace–Beltrami operator on $\mathbb H^n$. Let $u=u(t,\cdot)$ be the solution to the wave equation $\partial^2_{t}u-\big(\Delta_{\mathbb H^n}+\frac{1}{4}(n-1)^2\operatorname{Id}\big)u=0$ with initial conditions $u(0,x)=f(x)$ and $\partial_tu(0,x)=0$. In this article we show that for a radial function $f$, $2n/(n-1)\leq p<\infty$, $\gamma<1/p$ and $T\geq 0$, there exists a constant $C_p$ independent of $f$ and $T$ such that $$ \left(\int_T^{T+1}\|u(t,\cdot)\|^p_p\mathrm d t\right)^{\frac1p}\leq C_{p,\gamma}\,\mathrm e^{(n-1)\left(\frac12-\frac1p\right)T}\left\|(-\Delta_{\mathbb H^n})^{\frac{n-1}2\left(\frac12-\frac1p\right)-\frac\gamma2}f\right\|_p. $$ We also show that the exponential part $\mathrm e^{(n-1)(1/2-1/p)T}$ of the bound is optimal.

We consider the three-dimensional incompressible Navier–Stokes–Cahn–Hilliard (NSCH) system in domains perforated by no-slip obstacles of diameter of order \(\varepsilon^3\) separated by distances of order \(\varepsilon\). Both viscosity and mobility may depend on the phase variable. At this critical Stokes-capacity scale, the limiting velocity \(\boldsymbol u\) and phase field \(\phi\) satisfy an NSCH system with the additional Brinkman resistance \(\nu(\phi)\mathbf B\boldsymbol u\), where \(\mathbf B\) is determined by the exterior Stokes capacity of the reference obstacle. The principal analytical difficulty is the interaction between the phase-dependent viscosity and the order-one energy concentration of the critical Stokes correctors. The variational chemical-potential identity yields, after scalar extension, strong convergence of the phase in \(L^2(0,T;H^1(\Omega))\), and hence strong convergence of the phase-dependent viscosity in the same topology. Combined with a cellwise Hardy multiplier estimate, this compactness can be transferred through the concentrated corrector layer. The resulting weighted-capacity statement applies to uniformly positive and bounded coefficient sequences converging strongly in \(L^2(0,T;H^1(\Omega))\) and simultaneously identifies the effective Brinkman force and the corresponding viscous dissipation lower bound. We also investigate a vanishing capillary coefficient \(\lambda_\varepsilon\to0\). After the natural velocity scaling, the limit is an unsteady Stokes–Brinkman equation coupled to an unadvected Cahn–Hilliard equation.