We prove short-time well-posedness for the Muskat problem with surface tension in the full two-phase setting, allowing different viscosities, arbitrary density contrast, and rigid boundaries. In particular, no Rayleigh--Taylor sign condition on the density contrast is imposed. The interface is assumed to be a graph, uniformly separated from the fixed boundaries, and the initial data may be large in the log-critical class $\dot C^{1,\log^\varkappa}\cap H^1$, with $\varkappa>1$. Thus the result reaches the natural Lipschitz threshold up to a logarithmic correction. The main difficulty is that, in the presence of viscosity jump and boundaries, the interface equation is not given by a closed explicit contour dynamics law. Instead, the normal velocity is recovered through an elliptic transmission problem in moving domains, and the resulting evolution is a genuinely nonlocal quasilinear equation. We derive sharp Schauder-type estimates, adapted to the log-critical scale, for the transmission operators generated by the bulk Darcy flow. These estimates identify the third-order parabolic mechanism produced by surface tension and control the nonlinear coupling between the interface geometry and the elliptic transmission structure. The proof builds on the Schauder framework developed in Part~I of this series \cite{CHN1}, but requires a new analysis of the Muskat transmission problem in moving domains. Combining this elliptic theory with the contour formulation and time-weighted Hölder estimates, we obtain existence, uniqueness, smoothing, and stability for large interfaces in arbitrary dimension.

We study the finite-scale one-component degeneration problem for suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations under a scale-invariant bound and smallness of the vertical component. Qualitative compactness gives convergence, in the harmonic-pressure quotient, toward the strict two-and-a-half-dimensional boundary, but it does not provide a quantitative rate. This paper proves, in an explicitly abstract trace-obstruction skeleton associated with the old observable closure, that the standard old observable package is insufficient to force a logarithmic or power selected-trace rate. The negative result is an envelope/skeleton theorem, not a Navier--Stokes counterexample. After excluding elementary high-frequency escape by parabolic trace drop and fixed-window analytic obstruction by finite-dimensional Lojasiewicz control, the remaining obstruction is an all-order finite-mode flat branch. We identify the Navier--Stokes-specific mechanism needed to control this branch: strict Schur trace-projectability may fail, but the resulting defect can be visible through the relaxed vertical-pressure channel. In active finite-window models, strict Schur phantoms are relaxed-visible. The final theorem is a conditional dichotomy: either relaxed anti-phantom closure holds and yields conditional logarithmic strict-shadow selection, or there exists an NS-realizable, cleaned, relaxed-invisible, unaligned left-singular cascade.

In this paper, we prove the asymptotic stability of the critical constant steady state for a simplified parabolic--elliptic Keller--Segel system in $\mathbb{R}^N$ ($N \ge 3$), which admits a one-parameter family of constant steady states. Although the stability threshold for constant steady states is known, the critical case has remained open. We also show that the convergence rate in the critical case differs from the rates obtained for previously studied subcritical constant steady states.

In this paper, we develop a continuity method for the Dirichlet problem of Hessian equations on Riemannian manifolds. Such equations, introduced by Caffarelli, Nirenberg and Spruck, are defined in terms of the eigenvalues of the Hessian and a given pair $(f,\Gamma)$, where $f$ is a symmetric function defined in a symmetric cone $\Gamma\subset\mathbb{R}^n$, and $\Gamma$ specifies the set of admissible eigenvalues for the solution. Our method combines techniques from Morse theory with a characterization of the pair $(f,\Gamma)$. More precisely, in the type 2 case, we first construct admissible functions using Morse theory, and then solve the Dirichlet problem without any additional assumptions on the boundary or the subsolution. Building on this characterization of the pair, we can approximate the type 1 equation by a family of type 2 equations.

In this paper a general framework for convex integration is developed, in order to construct weak solutions to the Cauchy problem, by building on ideas from [C. De Lellis and L. Székelyhidi, Arch. Ration. Mech. Anal., 195 (2010)] and [S. Markfelder, Nonlinearity, 37 (2024)]. This framework may be applied to a large family of partial differential equations in order to construct weak solutions in $L^\infty ((0,T) \times \Omega)$ (for a bounded domain $\Omega)$ which are weakly continuous in time with respect to the weak topology of $L^r (\Omega)$ for some $r \in (1,\infty)$. This allows us to construct solutions which obey an energy inequality. In the second part of the paper we apply the framework to several inviscid models appearing in the field of geophysical fluid mechanics in order to show existence of weak solutions for all initial data, and to prove that there exist initial data for which there are infinitely many solutions which satisfy an energy inequality. We first consider the incompressible and the barotropic compressible Euler equations to recover the corresponding results from the literature. In addition, the framework allows us to prove a new result for the incompressible Euler equations, namely the global existence for the Cauchy problem in $L^\infty$. Moreover, we use the framework in the context of the hydrostatic Euler equations (also known as the incompressible inviscid primitive equations), which leads to the first convex integration approach which is able to construct admissible solutions with the natural energy for this system. A crucial ingredient in the proof of this result is the computation of a large subset of the convex hull. Finally, we apply the framework to the compressible inviscid primitive equations and to the inviscid quasi-geostrophic equations to obtain the first results on existence of wild data for these two geophysical models.

We present almost sure polynomial bounds for Hölder norms of solutions of the 1d periodic fractional Benjamin-Bona-Mahony (BBM) equation. Namely, we apply quantitative quasi-invariance of certain Gaussian measures with energy cutoff using the strategy from Tzvetkov (2015) and the globalization argument from Bourgain (1994) in order to extend, almost surely, the $L^2$-based deterministic control to the $L^{\infty}$-based setting.

We study Dirichlet Green's functions associated with second-order parabolic systems with rapidly oscillating periodic coefficients that are symmetric and independent of time. For bounded $C^{1,1}$ domains, we obtain a sharp zeroth-order convergence estimate from the oscillating Green's function to its homogenized counterpart, with the optimal rate $O(\varepsilon)$ and Gaussian off-diagonal decay. For bounded $C^{2,1}$ domains, we also prove a first-order expansion for the spatial gradient in terms of Dirichlet correctors, with an $O(\varepsilon)$ error up to a logarithmic factor. In this time-independent symmetric setting, these results improve the convergence rates established by Geng in [Calc. Var. Partial Differ. Equ., 62(6), 2023] for parabolic systems with time-dependent periodic coefficient matrices.

In this chapter we review some results describing the high-frequency eigenmodes of the Laplacian on compact manifolds, or Euclidean domains, for which the geodesic flow is chaotic. We focus on the macroscopic distribution of these eigenmodes, which is described by the concept of semiclassical measure. The main result on the question is the Quantum Ergodicity theorem, originally due to Schnirelman. We provide the detailed proof of this theorem, including the adjustments necessary to treat the case of manifolds with boundary. We also discuss the Quantum Unique Ergodicity conjecture, and some progress towards this conjecture for strongly chaotic (Anosov) systems. In particular, we describe the constraints on admissible semiclassical measures, in terms of their Kolmogorov-Sinai entropy, as well as more recent delocalization results.

We characterize complex-elliptic operators $\mathbb A(D)$ through a hierarchy of overdeterminacy ($\ell$-vanishing) quantifying the structural twisting of their symbols. This framework yields the optimal dimensional estimate for $BV^{\mathbb A}$-functions: a measure ${\mathbb A} u$ cannot concentrate on sets of dimension below $n-1$. Consequently, the jump part of ${\mathbb A} u$ is characterized as an $(n-1)$-dimensional surface measure with density given by the symbol and the two-sided traces. Building on this dimensional bound, we prove that measures satisfying $\frac{{\mathbb A} u}{|{\mathbb A} u|} \in \mathrm{span}\{P_0\}$ precisely decompose into finite sums of one-dimensional $BV$ profiles. Ultimately, these results reveal that complex-ellipticity strictly enforces a plane-wave structure on tangent measures.

We study Hartree-type nonlinear heat equations associated with fractional generalised anharmonic oscillators \(A_{k,l}\) in weighted modulation spaces. We first derive Strichartz-type estimates for the associated heat semigroup and then apply them to establish global well-posedness for small initial data. For \(s>\frac{d}{q'}\), the result is obtained via trilinear estimates exploiting the algebra property of the weighted modulation space, \(M^{p,q}_s\). We further establish refined trilinear estimates that bypass the algebra structure, thereby extending the global well-posedness theory to the wider range \(s\ge 0\).

In the present paper we are dealing with the following quasilinear elliptic problem: \begin{equation*} \begin{cases} -\mathrm{div}(\rho(x_N) |\nabla u|^{p-2}\nabla u) =a|u|^{s-2}u &\mbox{in }&\ \mathbb{R}^N_+, -|\nabla u|^{p-2}\frac{\partial u}{\partial x_N}=b|u|^{q-2}u&\mbox{on }&\ \mathbb{R}^{N-1}, \end{cases}\ \end{equation*} where $a,b\in \mathbb{R}$, $p,q,s\in(1,\infty)$ and $\rho$ is a continuous positive function on $[0,+\infty)$. We first prove new and sharp embedding results that we establish for the associted weighted energy spaces. In application, we establish existence and regularity of weak solutions to the above problem. We also prove for this problem the nonexistence of nontrivial weak solutions by a new Pohozaev-type identity we obtain. The new results about existence and nonexistence highlight the role of the weight $\rho$ on the solvability of the problem contrasting strongly with those when $\rho$ is constant.

The emergence of cooperation in structured populations is fundamental to the success of human societies. Physical and online networks can drive behavioural change by altering who people interact with, thereby modifying social pressures. In this paper, we study imitation dynamics in a population of self-interested agents playing a continuous strategy Prisoner's Dilemma on a dynamically evolving weighted network. In the fast-network regime, we incorporate the edge weights into the strategy evolution before deriving and analysing the large population mean-field limit. Without noise, we establish well-posedness and show the solution collapses to a single Dirac mass. For initially separated clusters, we identify a payoff threshold and sufficient conditions for the overall level of cooperation to increase. We then introduce stochastic strategy updates, and obtain a non-local Fokker-Planck equation in the mean-field limit. We rigorously prove existence and uniqueness of stationary distributions, and show linear stability under sufficient noise. Numerics illustrate that noise can transform the deterministic consensus into stable cooperative stationary behaviour. These findings show that the fast adaptive interactions and stochastic exploration can jointly support the emergence of stable cooperation at a population level.

In this article, we study some classes of solutions for a stationary Navier-Stokes equation where we consider a rough drift given by a singular integral operator which does not belong to the classical Calder{ó}n-Zygmund family of singular integral operators. Given a small external force, we will construct solutions to this system in the framework of weighted Morrey-Sobolev spaces. The use of Morrey-based Sobolev spaces provides a more general setting than the usual Lebesgue-based Sobolev spaces, and the presence of Muckenhoupt weights will allow us to present some existence and uniqueness results from several points of view.

This paper investigates the well-posedness issue for a reduced two-dimensional model of micropolar fluids. This reduced model presents a coupling between an Euler-type equation for the velocity field of the fluid and a transport-diffusion equation for the microrotation field (which is a scalar field, in this setting). We establish the local existence and uniqueness of strong solutions in the scale of Besov space $B^s_{p,1}$ having regularity index $s\geq1+2/p$. Furthermore, in the subcritical case when $s>1+2/p$, we prove that these solutions exist globally in time. The global persistence of regularity in the critical setting $s=1+2/p$ remains open.

Let $D \subset \mathbb{R}^d$ ($d \geq 2$) be a bounded connected open set of class $C^m$ with $m \geq 3$. We prove that, for a generic domain $\Omega \in \mathcal{O}^m(D)$ in the sense of Baire category for the Micheletti topology, every Dirichlet-Laplacian eigenfunction has nonzero mean, $\int_\Omega \varphi_n \neq 0$ for all $n \geq 1$. This answers a question raised by Steinerberger and Venkatraman. The proof rests on a shape-derivative computation whose degeneracy is identified with an overdetermined elliptic problem of Schiffer type. Rather than resolving this problem, we bypass it through a Baire-category argument based on the meagerness of domains with analytic boundary. As a consequence, we obtain that for a generic domain the Dirichlet heat equation is approximately controllable, and rapidly stabilizable, i.e. stabilizable at any prescribed exponential decay rate, by a single spatially homogeneous scalar control, via the Fattorini-Hautus criterion.

Global nonnegative martingale solutions are shown to exist for a stochastic Keller-Segel system with a nonlocal Fisher-KPP source and super-linear multiplicative noise. The result is obtained for nonnegative initial data with no smallness assumption, provided that the nonlocal source term is dominant. The main difficulty stems from the absence of a coercive structure and the super-linear nature of the noise. An additional cut-off with finite L^2 norm in the classical Galerkin method is added to establish a well-posed approximation problem. Moreover, due to the nonlocal Fisher-KPP structure, it is necessary to prove the positivity of the approximating solution in order to obtain uniform estimates. In the compactness arguments, the usual tightness argument in the framework of Hilbert spaces cannot be directly applied to the uniform estimates obtained in this paper. As a result, we develop a more general version of the compactness argument and tightness criterion, presented in the appendix, which will be applied throughout the paper. This allows for the global existence of nonnegative martingale solutions to be derived from Jakubowski's version of the Skorokhod Theorem, along with a thorough discussion of the convergence properties.

We formulate and prove a conditional finite-scale reduction theorem for the local one-component regularity problem for suitable weak solutions of the three-dimensional Navier--Stokes equations. Starting from a scale-invariant bound Phi(1) <= M and smallness of the critical vertical component C_3(1) = delta, the argument compares the solution with a strict two-and-a-half-dimensional shadow class. The comparison is made in the harmonic-pressure quotient, which is the natural local topology for pressure compactness. The Reynolds commutator produced by coarse graining is treated as a positive covariance stress and is absorbed by an unresolved-variance buffer; consequently this stress contributes additively, while the genuinely vertical residuals carry a positive power of delta and may pass through finite-stage exponential constants. The theorem is deliberately stated as a reduction theorem. Under the explicitly listed structural inputs--prepared pressure-covariance closure, weak horizontal-defect admissibility, sharp admissible-time trace tightness, singular-stratum tangent-cone inputs, strict limiting smoothing and decay, finite-window trace-cost/Newton solvability, and the vertical-duality active-residual estimate--we derive r_reg(0,0) >= c_{M,theta} |log C_3(1)|^{-sigma/3}. The paper does not constitute an unconditional resolution of the logarithmic one-component regularity problem. Its contribution is a theorem-driven reduction: strict-shadow selection failure is reduced to a finite-mode flat trace obstruction, and that obstruction is eliminated, conditionally, by vertical duality forced by the full three-dimensional vertical momentum equation.

We consider the homogenization of three dimensional viscous incompressible non-Newtonian flows satisfying certain general $r$-structure in perforated domains. We focus on the case of `small holes' by assuming the holes under consideration are of size $\varepsilon^{\alpha}$ with $\alpha >3$, where $\varepsilon$ is the perforation parameter used to measure the mutual distance between the holes. We show the limit equations remain unchanged in the homogenization limit under the constraint $ \frac{6(\alpha - 1)}{4\alpha-5}< r<3-\frac{3}{\alpha}$, which seems optimal in the sense of Sobolev capacity of holes as explained in Remark 1.3. Quantitative convergence rates are further derived for both the velocity field and the pressure. To the best of our knowledge, both the qualitative and quantitative homogenization results are firstly given for non-Newtonian flows in the `small holes' case.

In this paper we establish sharp global $C^{1,\alpha}$ regularity for a class of degenerate/singular fully nonlinear elliptic equations featuring Hamiltonian structures. The results presented in this paper serve as the boundary counterpart to the interior regularity estimates recently derived in \cite{AN25b, HJMZ26}, and substantially generalize the existing regularity theory developed in \cite{BBLL24b, BSRR23, BD14}. The optimal regularity is established within the framework of Caffarelli's perturbation argument \cite{C89, CC95}.

This paper investigates the three-dimensional compressible Navier--Stokes system with a polytropic pressure law and its pressureless counterpart arising from the high Mach number limit. We focus on the different critical scaling structures of these two models. In the presence of the pressure term, the pressure gradient is balanced with the inertial and viscous effects, and thereby selects a fixed critical scaling for the pressure system. In contrast, once the pressure term is ignored, the pressureless system admits a more flexible one-parameter family of invariant scalings. For both systems, we establish the global well-posedness of strong solutions under low-regularity assumptions on the initial data, allowing vacuum and large oscillations. This improves the global result of Wen (Adv. Math. 482 (2025), Paper No. 110628), where higher regularity assumptions on the initial data are required. A central feature of our result is that the smallness conditions are {\it exactly invariant} under the intrinsic critical scalings of the corresponding systems. These scaling structures differ from the usual parabolic scaling used in the critical-space framework of Danchin (Invent. Math. 141 (2000), pp. 579--614), where the system is reformulated around a reference state. We also derive uniform a \textit{priori} estimates and obtain exponential decay estimates for the global strong solutions. The results show that the pressure term not only changes the analytic estimates, but also plays a decisive role in selecting the critical scaling structure and in determining the dynamical behavior of compressible flows.

This paper investigates an inverse problem for the simultaneous reconstruction of two spatially varying reaction coefficients, the local proliferation rate and the competition (saturation) coefficient, together with the unknown initial condition, in a nonlinear, density-dependent reaction-diffusion model motivated by cell invasion and tumor growth dynamics. Using Carleman estimates, we establish a global uniqueness result together with a Lipschitz-type stability estimate for the reaction coefficients and a weaker, logarithmic stability estimate for the initial condition. For the numerical reconstructions, we develop a two-stage algorithm employing a time-shift strategy to decouple the coefficient and the initial condition. Numerical experiments are presented to illustrate the feasibility, accuracy, and robustness of the proposed inversion method.

We investigate the Schrödinger bridge problem on a finite state space with a strictly positive Markov reference kernel. Starting from the semi-dual convex formulation, we introduce a continuous-time evolution for the terminal Schrödinger potential and show that its equilibria coincide with the unique solution of the bridge problem. The proposed dynamics induces an evolution for the terminal marginal. This marginal equation is governed by a state-dependent nonlocal Onsager operator, identified with the Hessian of the semi-dual functional. We derive its associated Dirichlet form, establish coercivity estimates on the appropriate quotient space, and interpret the resulting equation as a nonlocal gradient-flow formulation of relative entropy. Under natural positivity assumptions, we prove global well-posedness of the SBOF, convergence to the Schrödinger bridge, convergence of the induced couplings and path measures, and exponential relaxation of the terminal marginal. The latter follows from a uniform Poincaré inequality on compact sublevel sets together with entropy--variance comparison estimates. We also discuss the connection with finite-state generative modeling through the Doob transform and illustrate the theory on finite-grid examples involving rare states.

We investigate a free boundary PDE model for vascularized tumor growth with two time-periodic coefficients representing nutrient supply $\phi(t)$ and nutrient demand $\psi(t)$. Under radial symmetry, the model reduces to a nonautonomous ODE for the tumor radius. We prove a vanishing-persistence dichotomy governed by the averaged coefficients $\bar{\phi}$ and $\bar{\psi}$: if the average nutrient supply does not exceed the average nutrient demand, namely $\bar{\phi}\le \bar{\psi}$, the tumor radius tends to zero; if the average supply exceeds the average demand, namely $\bar{\phi}>\bar{\psi}$, the tumor persists and converges to a unique positive periodic solution. We also study the linear stability of this periodic solution when nonradial perturbations are imposed. Under a stronger condition on the nutrient supply and demand, we show that the radially symmetric periodic solution is linearly stable when the tumor aggressiveness parameter is sufficiently small. Numerical simulations are provided to illustrate the analytical results.

Let $\Delta$ be the Neumann Laplacian on the interval $(0,\pi)$, and let $T>0$. An integration by parts formula is proven for the spectral fractional Laplacian, $(-\Delta)^\alpha$, for $\alpha \in (0,1)$. As an application, we prove well-posedness results for the structurally damped beam equation $$u_{tt}+\Delta^2 u+\rho (-\Delta)^\alpha u_t=0, x\in (0,\pi),t>0$$ with various boundary conditions including $$ u_x(0,t)=u_{xxx}(0,t)=0;\ u_x(\pi,t)=f(t),\ u_{xxx}(\pi,t)=0, $$ and $f\in L^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null-controllability. Analagous results are proven for higher order controls, and for the Dirichlet Laplacian.

In this paper, we apply the I-method to establish global well-posedness for the fractional nonlinear Schrödinger equation with initial data u_0 \in H^s(R^d) for s <\alpha/2, i.e., below the energy threshold. Moreover, for radial initial data, we combine a modiffed Morawetz estimate-recovered via Balakrishnan's formula-with the I-method to obtain improved results. In the same spirit, we employ the "upside-down" I-method to derive polynomial-in-time growth bounds for the higher-order Sobolev norm. The main difffculty stems from the fact that Strichartz estimates for the fractional Schrödinger equation has a loss of derivatives, and the problem is always L^2-supercritical, thereby requiring more delicate analysis.

We show that for a generic real or complex-valued compactly supported potential, the corresponding Schroedinger operator achieves maximal resonance density, in the sense that its integrated resonance counting function achieves the optimal asymptotic upper bound. For odd dimensions this follows from results of Dinh-Vu once we adapt an argument of Christiansen Hislop. The proof for even dimensions constitutes the bulk of the paper, and we prove several new results on resonances which have analogues in the odd dimensional case. This includes a sharp upper bound on the integrated resonance counting function for any compactly support potential, a proof that the characteristic function of a ball has resonance counting function which achieves the optimal upper bound, and an even-dimensional analogue of the result of Dinh-Vu on asymptotics of the resonance counting functions for complements of pluripolar subsets of analytic families of potentials. We use the characterization of resonances as zeros of certain Fredholm determinant functions related to the scattering matrix, allowing us to apply techniques and results from the theories of one and several complex variables. Our proof that the characteristic function of a ball has counting function achieving the optimal upper bound uses the uniform asymptotics of Bessel functions and follows ideas of Zworski, Christiansen-Hislop, and Dinh-Vu.

We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along oriented, proper mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. This provides a new, finite variational framework for mean curvature flow beyond the finite-volume setting. Our result yields several applications, including rigidity for solitons and convergence for two-dimensional immortal solutions. In particular, every proper self-expander with finite constant-coefficient calibration energy must be a plane in all dimensions and codimensions.

In this paper, we study the noncommutative fractional symmetric Sobolev spaces on noncommutative torus. We prove noncommutative distributional fractional Sobolev inequality and as its application, we obtain Sobolev embeddings. In order to obtain these results, we first prove a noncommutative version of the famous O'Neil inequality for the convolution. As a first application of our main results, we obtain a Cwikel-Solomyak-type estimate. As an another application, we show a $L_2$-time decay for the mild solution of the Cauchy problem for the diffusion equation in this noncommutative setting. When $\theta=0,$ our results recover many known results on Sobolev embedding on the torus.

We consider the periodic quintic nonlinear Schrödinger and prove small-mass global well-posedness in $H^s(\mathbb{T})$ for $s>0$. The proof relies on a new derivative-loss-free $L^6_{t,x}$ Strichartz estimate which is established using the high-low method, an asymmetric superlevel set estimate and a new refined broad-narrow argument. Although our $L^6_{t,x}$ Strichartz estimate is not sharp, being valid on slightly shorter time scales than the optimal logarithmic scale, combining it with the $I$-method enables the extension of local solutions to arbitrary times.

We prove the weak cosmic censorship conjecture in $2+1$ spacetime dimensions for the circularly symmetric Einstein-scalar field system in the presence of a negative cosmological constant $\Lambda<0$. More precisely, we show that for any integer $k\geq2$, the maximal development of generic $C^k$ initial data does not contain naked singularities. An essential step of the proof is establishing the presence of a mass gap. In particular, this implies that all naked singularities have infinite blueshift, which represents the main instability mechanism.