Papers

We continue our study of the well-posedness theory for the intermediate nonlinear Schrödinger equation (INLS). Firstly, we prove that INLS is locally well-posed in $H^s (\mathbb{R})$ for any $s>0$. This improves on our previous result of local well-posedness for any $s>\frac 14$, and covers the full scaling-subcritical range for INLS. In particular, we also obtain the local well-posedness for the continuum Calogero-Moser equation without chirality assumption in the full scaling-subcritical range. Our method relies on a gauge transformation, the derivation of a closed system for four auxiliary variables, and nonlinear smoothing estimates, but not on the completely integrable nature of these equations. Secondly, for the integrable models, we prove global well-posedness for $0<s<\frac12$ for initial data with small $L^2$-norm. Moreover, we show that our global well-posedness result applies whenever $L^2$-equicontinuous sets are preserved by the flow, and so the small-data restriction would be removed by an a-priori equicontinuity result. Our argument relies on a novel family of conserved quantities based upon the Lax pair we discovered in our prior work.

Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain \(\Om\Subset\R^2\) such that the zero-Dirichlet solution of the planar translating mean curvature equation has a nonconvex sublevel set. The construction is based on a corrected near-critical grim-reaper profile and explicit barriers on a long convex channel.

We consider the initial-boundary value problem for the stochastic time-nonlocal telegraph equation with $(\mathcal{PC}_\varepsilon)$-type kernel $a$: \begin{align*} \gamma \partial_t \left( a \ast \partial_t (a \ast v)\right) =\Delta v-\partial_t (a \ast v)+ \Psi(v)+ \Phi(v) \frac{\mathrm{d}W(t)}{\mathrm{d}t}, \end{align*} where $W$ is a space-time Gaussian white noise, $\Psi$ satisfies a linear growth condition, and $\Phi$ is Hölder continuous and uniformly nondegenerate. This model characterizes high-frequency signal propagation in small-scale systems under stochastic fluctuations. We develop a new hereditary phase-space lifting framework for time-nonlocal telegraph equations. In addition, we propose a novel generalized coupling framework, which features a new construction of the damping control term for the velocity. Based on these analytic tools, we prove the first results on weak existence and uniqueness in law for mild solutions, valued in $L_{loc}^2(\mathbb R_+; H^{\delta})$, to the stochastic nonlocal telegraph equation. The regularity index $\delta$ can be arbitrarily close to $\min\{\frac{1}{2},\frac{\varepsilon}{2+\varepsilon} \}$ from below, and the admissible lower bound of the Hölder exponent $\kappa$ is quantitatively determined by the integrability exponent $\varepsilon$. For the corresponding IBVP of the stochastic damped wave equation, obtained by replacing $a$ with the Dirac measure $\delta_0$, $\kappa$ can be improved to any value in $(\frac{3}{5}, 1]$. More significantly, the generalized coupling framework also handles low-regularity nonlinearities depending on both displacement and velocity.

We study the energy $\mathcal{E}(s) := E_{s}[\Phi_s]$ of the one-dimensional fractional Allen–Cahn layer solution $\Phi_s$, defined as the unique odd, increasing solution of $(-\Delta)^{s} \Phi_s = \Phi_s - \Phi_s^{3}$ with $\Phi_s(\pm\infty)=\pm 1$ and $\Phi_s(0)=0$, for $s\in(1/2,1]$. Our main results are a sharp qualitative and quantitative description of the energy $\mathcal{E}$ on this interval. We show that the energy is continuous and strictly decreasing, and we obtain explicit asymptotic expansions at both endpoints. At the upper endpoint we prove $\mathcal{E}(s) = \frac{2\sqrt{2}}{3} + \kappa_1 (1-s) + o(1-s)$ with an explicit formula for $\kappa_1$. At the lower endpoint we prove that the energy goes to infinity as $\mathcal{E}(s)=\frac{1}{\pi(s-1/2)}+O(1)$. The strict decrease is proved with computer assistance. On an interior subinterval it is reduced to finitely many inequalities verified by interval-arithmetic computations. The proofs combine the Cabré–Sire construction of the layer, the minimality theorem of Palatucci–Savin–Valdinoci, an identity for the $s$ derivative of the energy, and a computer-assisted coercivity estimate at explicit approximate layers.

We investigate the vanishing viscosity limit for the one-dimensional compressible Navier-Stokes equations in the regime of two interacting shock waves from the same characteristic family of the underlying Euler equations. Unlike the interaction of two shocks from distinct families, which produces two outgoing shocks in their respective original families, the collision of two same-family shocks generates an outgoing shock in the same family together with a rarefaction wave in the other family. This configuration is more singular because the collision occurs on a larger time scale that depends inversely on the wave strength, making it challenging to justify the vanishing viscosity limit not only in the processes before and after the shock collision but, more crucially, at the collision point. Furthermore, the emergence of both shock and rarefaction waves after the collision introduces an additional difficulty. To overcome these obstacles, we first employ the anti-derivative method before the collision, which allows us to fix the locations of both viscous shocks precisely up to the collision point. However, uniform estimates with respect to the viscosity cannot be closed up to the collision time; we therefore introduce a carefully constructed approximate collision time to obtain the required higher-order energy bounds. After the collision, we apply a weighted relative entropy method, combined with time-dependent shifts, to handle the composite wave structure arising from the coexistence of shock and rarefaction waves. Our main result establishes that, for suitably small wave strengths and viscosity coefficients, there exists a family of global smooth solutions to the Navier-Stokes equations that converge to the entropy solution of the Euler equations with an explicit convergence rate. The techniques are expected to be applicable to other related problems in vanishing viscosity theory.

We study the existence and multiplicity of principal eigenvalues and eigenfunctions for a periodically heterogeneous nonlocal dispersal model with advection. The operator we consider is resolvent-positive but not resolvent-compact; therefore, the classical Krein-Rutman theory cannot be applied directly. When the advection coefficient has a constant sign, we prove the existence and uniqueness of the principal eigenvalue and the corresponding normalized eigenfunction. In sharp contrast, when the advection does not have a constant sign, the problem is more involved and leads to surprising results. Depending on the coefficients of the equation, the principal eigenproblem can either have a unique normalized solution or a continuum of solutions, at the boundary of which there exists a principal eigenvector with a singular measure component. In the latter situation, all the constructed eigenvalues are embedded in the continuous spectrum of our operator. We completely characterize the eigenvalues associated with positive eigenvectors, even when the eigenvector is a Radon measure. Finally, we discuss an application to a nonlinear KPP-type equation with nonlocal dispersal, which possesses a continuum of nontrivial stationary solutions, a different behavior from the classical KPP equation with local diffusion.

The Liouville problem for the three-dimensional stationary Navier–Stokes equations remains open, even for axisymmetric \(D\)-solutions. In this paper, we obtain two results based on decay in the cylindrical radial variable \(r=|x'|\). (i). Using a new pointwise Calderón–Zygmund estimate adapted to cylindrical geometry, we improve the decay estimates of Carrillo–Pan–Zhang (2020, JFA) and prove \[ |\nabla u_r|+|\nabla u_z| \lesssim r^{-5/4}[\log(\mathrm e+r)]^{5/4}, \quad |\omega_r|+|\omega_z| \lesssim r^{-9/8}[\log(\mathrm e+r)]^{9/8}, \quad r\gg1. \] (ii). We develop a new approach to Liouville theorems that improves the axisymmetric criteria of Wang (2019, JDE) and Zhao (2019, Nonlinear Anal.). Without any symmetry assumption, we show that a \(D\)-solution is trivial if one of the following holds: \[ (\mathrm a).\,\sup_{{|x'|=r,\, z\in\mathbb R}} |u(x',z)| \leq Cr^{-2/3}[\log(\mathrm e+r)]^{-\gamma}; \quad (\mathrm b).\, \sup_{{|x'|=r,\, z\in\mathbb R}} |\omega(x',z)| \leq Cr^{-5/3}[\log(\mathrm e+r)]^{-\gamma}, \] for $r\geq1$, where $\gamma>1/3$.

We construct a divergence-free velocity field on the three-dimensional torus that is time-periodic and consists of three alternating piecewise-affine shears. For sufficiently large shear amplitude, every non-zero divergence-free initial field in $L^p$ grows exponentially under the ideal induction equation. The proof establishes a uniform cone condition for the time-one map, and combines it with a bunching inequality to rule out nontrivial divergence-free fields lying almost everywhere in the stable bundle. Additionally, we show that for the time-reversed velocity, whose time-one map is the inverse of the original one, there exist nontrivial, bounded, divergence-free initial configurations taking values almost everywhere in its two-dimensional stable bundle. The corresponding solution decays exponentially in every $L^p$.

We study the simultaneous vanishing-viscosity and vanishing-resistivity limit of steady incompressible MHD flows in a disk. The boundary velocity is a small nonaxisymmetric perturbation of a rigid rotation with mean angular speed \(\alpha\), while the prescribed tangential magnetic trace has mean \(\beta\). Assuming \(\alpha\neq0\) and the non-Alfvénic condition \(|\alpha|\neq|\beta|\), we construct solutions converging on compact interior subdisks to a rigidly rotating ideal MHD core with constant vorticity and out-of-plane current density. A new MHD–Wood law determines the two core rotations from the boundary data: the velocity core is selected by a coupled kinetic–magnetic balance, whereas the magnetic core is fixed by the imposed mean circulation. Consequently, zero circulation gives complete interior magnetic expulsion, while nonzero circulation leaves a uniform magnetic rotation after the nonaxisymmetric modes are confined to a thin boundary layer. This provides a fully coupled realization of Prandtl–Batchelor selection and flux expulsion; unlike classical kinematic models, the magnetic field actively changes the flow and need not be weak. The proof combines a non-Alfvénic coercive theory for a periodic MHD boundary layer, global matching of the two fields, and a coupled stability estimate adapted to the magnetic boundary condition. A separate conditional rigidity argument, using exact viscous identities and local convergence but no interior asymptotic expansion, explains the same core structure for a broader single-eddy family.

This is the first part of a two-part work concerning the boundary layer convergence for chemotaxis-Navier-Stokes system in a two-dimensional half-space. In this paper, we investigate the chemotacxis-Navier-Stokes system with the logarithmic singularity under Navier-slip boundary conditions. More precisely, we perform an exact asymptotic expansion for the chemotaxis Navier-Stokes system with viscous coefficient $\varepsilon>0$, and obtain partial boundary layer profiles, establishing the well-posedness of the corresponding boundary layer profiles. Specially, we also establish the local well-posedness of solutions to the supercritical chemotaxis Euler equation (with $\varepsilon=0$) by overcoming the difficulty from the disappearance of diffusion terms.

In this note, we prove that, on weighted graphs, the Lin–Lu–Yau curvature coincides with the $p$-Ollivier curvature up to scaling whenever the idleness parameter $p\geq 1/2$. Moreover, the threshold $1/2$ is sharp. This extends an earlier result of Bourne et al. (Ollivier–Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin–Lu–Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747).

We study the behavior of the ordinary differential equations, flow maps, and continuity equations associated to autonomous random velocity fields that admit a natural multiscale finite range decomposition. The velocity fields we consider are only Hölder regular in space—$C^{\alpha-}(\mathbb{T}^d)$ for some $\alpha \in (0,1)$—thus the associated ODE and continuity equation are not a priori well-posed. However, above the critical threshold of $\alpha = 1/2$, due to multiscale stochastic cancellations, we prove well-posedness is almost surely restored away from the zero level set of the velocity field. This threshold marks a genuine transition, as demonstrated by examples lying below the threshold that exhibit robust ill-posedness. We additionally provide effective regularity estimates below the critical threshold and prove analogous results in the related "refreshing" regime.

We study a fully implicit Rothe time discretization for a cutoff first-order formulation of the Westervelt equation. The key ingredients are the enthalpy variable and the primitive mobility variable, which turn each nonlinear time step into a uniformly monotone elliptic problem and avoid higher-order energy estimates and inverse inequalities. For every time step, the discrete problem reduces to a monotone elliptic equation, which yields well-posedness of the Rothe scheme by the Browder-Minty theorem. We derive a discrete energy inequality, establish compactness for the transformed variable by an Alt-Luckhaus type argument, and pass to the limit to obtain existence of weak solutions for the cutoff first-order system. We prove a weak-strong uniqueness principle for the cutoff problem and formulate a conditional a posteriori criterion under which the cutoff is inactive. For sufficiently regular solutions of the forced cutoff problem, we establish consistency estimates which identify the temporal residuals produced by the Rothe approximation in the natural discrete-in-time weak norms. These estimates provide a rigorous consistency basis for the observed temporal behavior. Finally, numerical experiments illustrate the stability of the scheme, its observed near first-order temporal behavior, and inactivity of the cutoff along the computed discrete trajectories in the tested regimes.

In this paper, we study vanishing capillary limit for isentropic compressible Navier-Stokes-Korteweg system in a bounded interval. The main challenges focus on the capillary term and the boundary effect. A new uniform dissipative estimate in terms of the third-order derivative of density and some correctors are derived to handle such difficulties. It leads to the optimal convergence rate of the solutions in $L^\infty$ norm globally in time with arbitrarily large initial data. This work provides a rigorous derivation of the isentropic compressible Navier-Stokes system in a bounded interval from the isentropic compressible Navier-Stokes-Korteweg system via vanishing capillary limit.

We study the identification of effective coefficients for wave equations in heterogeneous media. Such equations arise in the modeling of spatio-temporal metamaterials, where the underlying material properties exhibit variations in both space and time. While homogenization provides effective models in the asymptotic regime of vanishing microscopic scales, determining macroscopic parameters from observations of wave propagation remains challenging. We introduce an optimization-based method that identifies a constant effective coefficient by minimizing a cost functional. The approach is designed for situations in which the underlying space-time-dependent coefficient is unknown, while the solution is available in space and time by measurements. We extend optimization-based coefficient identification techniques from elliptic multiscale problems to wave equations and prove that, provided a homogenized limit exists, the identified coefficient converges to the homogenized coefficient as the microscopic scale tends to zero. Furthermore, we establish convergence of the corresponding effective solution towards the heterogeneous solution, including strong convergence in $L^2$ and weak convergence of first-order space and time derivatives. Numerical experiments further demonstrate the performance of the method for non-periodic space-time heterogeneous media, including cases for which a homogenized limit is not known to exist.

In this note, we show that a weak mean curvature flow with critical forcing term obtained by Liu–Tonegawa (2024) satisfies the BV-type area change formula, that is, their flow is not only a Brakke flow but also a generalized BV flow, which is proposed by Stuvard–Tonegawa (2024). To establish this result, we identify minimal conditions under which a Brakke flow satisfies the area-change formula. As an application of our main theorem, we derive a lower bound for the extinction time of generalized BV flows with a critical forcing term. We also outline the proof of a compactness theorem for generalized BV flows.

We establish the existence of a positive solution for a one-dimensional Minkowski-curvature equation with mixed boundary conditions and a sign-changing nonlinearity. A nonnegative linear shift and the Green kernel of the associated linear problem produce an invariant cone. Principal-eigenvalue comparisons and the fixed point index give cone expansion near zero and compression at infinity. A globally defined auxiliary equation and a first-contact argument show that the resulting solution has slope below one and hence solves the original equation. We also derive a directly verifiable asymptotic criterion.

We consider a class of mixed anisotropic and nonlocal singular quasilinear elliptic problem associated with Muckenhoupt weights. The presence of the singular nonlinearity, which blows up near the origin along with its interaction with anisotropy, weighted degeneracy, and nonlocal diffusion creates significant analytical challenges. We employ monotone approximation, weighted Sobolev embeddings, compactness, and variational methods to establish the existence and uniqueness of weak solutions under suitable assumptions on the datum. Further, we characterize the best constant in an associated weighted mixed anisotropic and nonlocal Sobolev-type inequality, prove that it is attained, and show that the normalized weak solution is the unique extremal. These results are new even in the mixed weighted Laplace case \(p=2\).

We study the well-posedness of the boundary layer for a pseudo-plastic fluid by energy methods under Oleinik's monotonicity assumption. We need to address two main difficulties: derivative loss and the difficulty caused by viscosity. For the first difficulty, we borrow the cancellation mechanism proposed by Masmoudi and Wong [CPAM, 2015]. For the second difficulty, we combine the monotonicity assumption and Faà di Bruno formula to obtain precise control over the high-order derivatives of viscosity, which is the main contribution of this paper.

In this paper, we consider the periodic problem to steady viscous Burgers equation with an periodic and external force. Our main aim is to construct periodic solutions to this model which are uniformly bounded with respect to the viscosity via a direct Fourier series approach. Firstly, as an example, starting from a special solution of the non-viscous Burgers equation with a specific force, we solve the approximate solutions to the viscous Burgers equation in an explicit way. Secondly, we construct the solution to the periodic problem of the viscous Burgers equation with the external force by solving the initial value problem to a second ordinary differential equations, which is uniformly bounded with respect to the viscosity. Compared with Jauslin-Kreiss-Moser's result, our approach provides an explicit constructive procedure for the periodic solutions and yields a sharper convergence rate (up to higher order) when the viscosity vanishes.

We consider the Navier-Stokes-Korteweg equations in a bounded domain or a periodic cell. The pressure considered in this paper may not be monotone with respect to the density so that there exist non-constant equilibria allowing two-phases. Using a simple Hilbert space framework, we prove that if an isolated equilibrium is energetically stable and non- degenerate, it is exponentially stable under the isothermal Navier-Stokes-Korteweg flows when the space dimension is less than or equal to three. For non-isolated case, we prove that a global-in-time solution near an energetically stable equilibrium converges to possibly another equilibrium exponentially fast. No smallness assumptions on equilibria are imposed. For the proof we apply a (generalized) stability principle due to J. Prüss, M. Wilke and G. Simonett (2013).

In this paper, we study the time-periodic problem for the Vlasov-Poisson-Boltzmann (VPB) system with a given time-periodic external force in the whole space $\mathbb{R}^3$. The force is allowed to be non-potential. Around the global Maxwellian, we prove the global existence of small solutions in a hybrid function space that combines the low-frequency Besov framework for the forced Boltzmann equation with a corresponding control of the self-consistent electric field. The main novelty lies in the treatment of the nonlinear Vlasov force $-\nabla_x\phi \cdot \nabla_vf + \frac{1}{2}(v \cdot \nabla_x\phi)f$ at low frequencies. Rather than treating it as a generic source term, we exploit the Poisson equation and macroscopic balance laws to recover the structural cancellation required for the VPB semi-group estimates, which combined with high-frequency energy estimates and weighted microscopic propagation, yields a closed global well-posedness theory. We further prove the asymptotic stability of small solutions driven by the same force. When the external force is time-periodic, Serrin's method yields a unique time-periodic solution with the same period, together with its stability. As a direct consequence, our result also gives the existence and stability of stationary solutions when the external force is time-independent.

Let $\mathcal{U} \Subset \mathbb{C}$ be fixed, $\Phi(s)=e^s-s-1$, $F(\nu)=\frac{\nu}{2+|\nu|}$, and let $f^{F(\nu)}$ denote the principal solution of the corresponding Beltrami equation. The identity $K_{F(\nu)}=1+|\nu|$ identifies compactly supported David coefficients with exponential-Orlicz parameters. We prove that, on the open subset of $L^\Phi_{\mathcal{U}}(\mathbb{C})$ where a sufficiently high finite exponential moment is available, the principal solution map is locally real $C^{1,1}$ with values in $W^{1,2}_{\mathrm{loc}}(\mathbb{C})$. The derivative in a direction $\eta \in L^\Phi_{\mathcal{U}}(\mathbb{C})$ is the principally normalized solution of $\bar{\partial} V - F(\nu) \partial_z V = DF_\nu(\eta) \partial_z f^{F(\nu)}$. The proof uses a pullback by the base principal solution. The key estimate is the pointwise cancellation $\frac{\|DF_\nu\|_{\mathrm{op}}}{1-|F(\nu)|^2} \le \frac{1}{2}$, which converts the linearized equation into a $\bar{\partial}$-equation whose source is controlled directly by the $L^\Phi$-norm of the direction. Combined with the principal degenerate $L^2$-resolvent and the optimal Jacobian regularity for exponentially integrable distortion, this yields a uniform quadratic remainder estimate. At the origin one obtains $D \mathrm{Sol}_0[\eta]=(1/2)\mathcal{C}\eta$ in $W^{1,2}_{\mathrm{loc}}$.

We study a coupled magneto-quasistatic model for an eddy current brake. The model leads to a nonlinear infinite-dimensional differential-algebraic system in which the nonlinearity is structural rather than caused by nonlinear material laws. It is generated by the closed electromechanical feedback loop: the angular momentum determines the velocity of the conducting disk, the velocity and the magnetic field generate motional eddy currents, and these currents induce a Lorentz torque acting back on the rotor. We introduce a finite-dissipation solution concept and prove global existence and uniqueness of weak solutions for initial data given by the magnetic field and the angular momentum, and for inputs given by the supplied coil current and the externally applied torque.

The analytic properties of the Dirichlet-to-Neumann operator reflect the geometry of the underlying manifold. As a guiding example of this relationship, consider the Euclidean unit ball. There, the Dirichlet-to-Neumann operator has an explicitly computable spectrum, consisting of the nonnegative integers with multiplicity, and commutes with the boundary Laplacian. In this paper, we study the extent to which these properties distinguish the ball among various classes of manifolds. We show that, in every dimension, Euclidean balls are uniquely determined by their Steklov spectrum among bounded Euclidean domains with smooth and possibly disconnected boundary, answering a well-known open question. We then classify all smooth compact connected oriented surfaces with nonempty boundary whose Dirichlet-to-Neumann operator commutes with the boundary Laplacian. Finally, in dimensions three and above, we prove that, for smooth metrics conformal to the Euclidean ball, the Dirichlet-to-Neumann operator commutes with the Laplacian on the round sphere if and only if the conformal factor is radial.

Recently, novel integrability techniques, and most notably Birkhoff coordinates followed by an explicit formula, have been introduced to solve the Benjamin–Ono (BO) equation, leading to major theoretical and computational advances. We propose to perturb this explicit BO formula via a novel splitting method for numerically solving a class of PDEs, which includes the Intermediate Long Wave (ILW) equation, as well as other quasilinear equations that are not necessarily integrable. Using the Birkhoff coordinates of BO we establish first-order convergence in the $H^s$-norm ($s \ge 0$) while requiring only one additional derivative on the initial data $u_0 \in H^{s+1}_{0}$, significantly improving upon the regularity requirements of typical splitting methods for nonlinear dispersive equations. Furthermore, we prove that in the deep-water limit, the scheme for the ILW equation converges to the BO solution. Computational advantages of these schemes are shown in simulations: unlike classical splitting methods, our approach does not require a restrictive quadratic time step condition to nearly preserve energy over long time scales, thus rendering efficient and accurate long-time simulations feasible. As an application, we numerically explore the soliton resolution conjecture for both the ILW and a KdV–BO equation.

We study the moduli of sourceless framed Beltrami-Vekua equations under recombinations of the unknown, scalings, and orientation-preserving changes of variables. On every bounded simply connected domain, such an equation reduces to $w_{\bar z}=B\bar w$ on the unit disk, with residual symmetries given exactly by zero-free holomorphic gauges and Möbius transformations. When $B$ is zero-free, the equation is completely classified by two data modulo Möbius: the hyperbolic mass density $\vartheta=\tfrac14(1-|z|^2)^2|B|^2$ and the phase-curvature current $K=\Delta\arg B$, whose hyperbolic density at $C^2$ regularity is $\kappa=\Delta_{\mathrm{hyp}}\arg B$. We determine the exact range of these invariants: every positive Hölder density and every phase current arising as the Laplacian of a Hölder phase occur. For fields with zeros, the classification extends to the phase-integrable sector through the triple $(\vartheta,d\eta,d{\star}\eta)$, where $\eta=\operatorname{Im}(dB/B)$. On the tame sector, $d\eta$ is the atomic charge measure, while $d{\star}\eta$ carries the remaining phase curvature. Thus the pseudo-analytic mass and charge are numerical projections of a larger infinite-dimensional moduli space. Explicit equal-mass, equal-charge, inequivalent equations are exhibited.

Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. We prove and widely generalize a conjecture of A.\,I.~Nazarov \cite{Naz21}: for $s\in(1,\frac 32)$ the quadratic form $Q^{\rm SP}_s[u]$ of the spectral fractional Dirichlet Laplacian strictly increases under the map $u\mapsto|u|$ provided $u\in\tilde H^s(\Omega)$ changes sign in $\Omega$.

We establish sharp parameter thresholds governing $H^2$ -regularity of weak solutions in $H_0^1$ for two classes of elliptic problems with critical unbounded perturbations, in dimensions $N\geq 5$. More precisely, we consider two distinct $\lambda$-parametric elliptic problems $ - \Delta v + \lambda \frac{x\cdot \nabla v}{|x|^{2}} =f$ and $-\Delta v + \lambda \frac{v}{|x|^2}=f$ posed in a bounded $C^2$-domain $\Omega\subset \mathbb{R}^N$ containing the origin $x=0$. We observe that the singular perturbations $\frac{x\cdot \nabla v}{|x|^{2}}$ and $\frac{v}{|x|^2}$ are homogeneous operators of order 2 consistent with the scaling of the Laplacian. In view of the Hardy inequality the problems are well-posed in $H_0^1(\Omega)$ for $\lambda<\frac{N-2}{2}$ and $\lambda>-\frac{(N-2)^2}{4}$ respectively. The main results are as follows. For the first problem we show that any solution $v\in H_0^1(\Omega)$ belongs to $H^2(\Omega)$ for any $\lambda< \frac{N-2}{2}$ provided $f\in L^2(\Omega)$. This fully extends the previous $H^2$ regularity properties obtained by Kim and Tsai in \cite{Kim-Tsai} for $\lambda\leq 0$. For the second problem we show that $H^2$ regularity holds for any $\lambda>- \frac{N(N-4)}{4}$ and fails for any $\lambda\in \left(-\frac{(N-2)^2}{4},-\frac{N(N-4)}{4}\right]$. This extends sharply the range of $\lambda \in \left(-\frac{N(N-4)}{4}, \frac{N(N-4)}{4}\right)$ obtained when applying the Kato perturbation theory in \cite{Kato}. In addition, we develop sharp second order Hardy-Rellich type inequalities for the involved elliptic operators which are essential in the above proofs.

We consider a spectrally unstable steady state $(\rho_0,v_0)$ of the incompressible stratified Euler equations on a class of $d$-dimensional domains. Assuming that the linearized equation admits an exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of the background steady flow $v_0$, we construct the local stable and unstable manifolds of $(\rho_0,v_0)$. The proof is based on the Lyapunov–Perron method after reformulating the Euler equation as an ODE on the infinite-dimensional manifold of volume-preserving Lagrangian maps, with the density treated as a frozen Lagrangian parameter as well as the weight in the $L^2$ metric. We also discuss some applications to two-dimensional steady flows.