Papers

We consider the following derivative nonlinear Schrödinger equation with a single power-type perturbation \begin{equation*} i\partial_tu+\partial_x^2u+i|u|^2\partial_xu+b |u|^pu=0, \end{equation*} with $b\geq 0$ and $p\geq 4$. When $b=0$ or $p=4$, the equation possesses a family of two-parameter solitons; see, for instance, \cite{CoOh06,Oh14}. Moreover, the authors established the orbital stability/instability of these solitons. In \cite[Corollary]{CoOh06}, a criterion for orbital of solitons was proved. Using the explicit formula of solitons to compute the necessary quantities, the authors verify that the solitons are orbitally stable across most of their range of existence (i.e., $c^2 < 4\omega$). When $b\neq 0$ or $p>4$, an explicit formula for the soliton profile is unavailable, making it difficult to verify the criterion in \cite{CoOh06}. In this paper, we prove the soliton profile varies smoothly with respect to parameters $b$ and $p$. More precisely, we show that the solitons change slowly when $b$ is sufficiently small or $p$ is sufficiently close to $4$. Consequently, we obtain the orbital stability of solitons in these cases. In the borderline case ($c=2\sqrt{\omega}$), the soliton still depends smoothly on the parameter $b$. However, its orbital stability or instability remains an open problem. Solutions to (dNLS) fail to scatter even for small initial data, a property originally proved for $b=0$ in \cite{BaWuXu20}. We also obtain similar results regarding the existence of modified wave operators, analogous to those in \cite{HaOz94}.

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schrödinger equation \begin{equation*} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=h(u) \quad\mathrm{in}\ \mathbb{R}^{3}, \end{aligned} \end{equation*} where $\lambda$ appears as a Lagrange multiplier, $h$ is a $L^2$-supercritical and Sobolev subcritical nonlinearity. The solutions correspond to critical points of the energy functional subject to the $L^2$-norm constraint $\int_{\mathbb{R}^3}|u|^2dx=a^2>0$. Taking into account the Pohozaev manifold and perturbation method, we obtain the existence of ground state normalized solutions and infinitely many normalized solutions. Moreover, our results cover several relevant existing results in \cite{LZ2023}. And in the end, we get the asymptotic properties of energy as $a$ tends to $+\infty$ and $a$ tends to $0^+$.

We construct infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons–Hawking spaces. We consider circle-invariant Lagrangian $2$-spheres whose quotient curves are concave and are $C^2$-close to a collection of consecutive collinear segments. We prove that the corresponding flow exists smoothly for all time and converges to the associated $A_{n-1}$-chain of special Lagrangian spheres. Although the mean curvature converges uniformly to zero, the second fundamental form becomes unbounded. More precisely, $\log\max |A(\,\cdot\,,t)|$ is comparable to $\sqrt{t}$ as $t\to\infty$. The proof is based on a one-parameter family of barrier curves and a detailed analysis of their asymptotics. In this way, we refine the infinite-time convergence picture arising in the work of Lotay and Oliveira by proving curvature blow-up and estimating its rate in this semi-stable case.

Let $p_1,p_2\in(1,\infty)$ and $M=M_1\times M_2$ be the product of two geodesically complete Riemannian manifolds. In this paper, the authors first develop an anisotropic potential-theoretic framework adapted to the Green operator $G^M$ and the mixed-norm Lebesgue space $L^{p_2}(L^{p_1})(M)$, and then demonstrate that the classical equivalence among parabolicity, Green function integrability, and Liouville property persists in this genuinely anisotropic setting. More precisely, the authors establish the following equivalence: $M$ is $L^{p_2}(L^{p_1})$-parabolic if and only if the Green function $G^M(x;\,\cdot\,)$ fails to belong to $L^{p_2'}(L^{p_1'})(M \setminus B(x,\,r))$, which is in turn equivalent to the $L^{p_2'}(L^{p_1'})$-Liouville property, where $p_i'$ denotes the conjugate exponent of $p_i$. Under a weak radial Harnack-type inequality – in particular, under Li–Yau heat kernel estimates, and hence for products of manifolds with nonnegative Ricci curvature – these conditions are further equivalent to the divergence of the nonlinear mixed-potential $\mathcal{G}_{p_1,p_2}(f)$ for every nonzero nonnegative $f\in {\mathcal C}_c^\infty(M)$. A key feature of this anisotropic theory is its sensitivity to the geometry of each factor \(M_i\), rather than merely to that of the total manifold \(M\). In contrast to the isotropic case, where parabolicity and the classical Liouville property holds on \(\mathbb{R}^n\) precisely when \(n \le 2\), the anisotropic setting exhibits a refined threshold: the \(L^{p_2}(L^{p_1})\)-parabolicity and the \(L^{p_2'}(L^{p_1'})\)-Liouville property holds on \(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2}\) if and only if $ D_{\mathrm{eff}} := \frac{n_1}{p_1} + \frac{n_2}{p_2} \le 2. $ This effective dimension $D_{\mathrm{eff}}$ captures the anisotropic interplay between the exponents \(p_1, p_2\) and the geometries of \(M_1, M_2\).

The dynamical stability of laminates or planar phase boundaries for hyperelastic materials of Hadamard type in two space dimensions is studied. For that purpose, the stability function, known as the Lopatinskii determinant, is computed for states of deformation at both sides of the planar interface that account for the generalized Legendre-Hadamard conditions derived by Grabovsky and Truskinovsky (J. Elast. 123 (2016), 225–243). The sufficient conditions for the dynamical stability of such configurations are described in terms of the physical parameters of the model, such as the shear modulus, and computed under kinetic conditions across the interface of both Maxwell (conservation of energy) or Abeyaratne and Knowles (dissipation of energy) types.

We study exterior Dirichlet problems for \(k\)-Hessian equations with prescribed quadratic asymptotics, allowing the asymptotic matrix to be merely \(k\)-admissible and not necessarily positive definite. The key point is that the correct metric at infinity is not determined by the asymptotic matrix itself, but by the coefficient matrix obtained by linearizing the \(k\)-Hessian operator at this matrix. This gives the exterior barriers and subsolutions needed to solve the Dirichlet problem, both in viscosity and smooth settings, for all sufficiently large asymptotic constants. In the case of smooth, strictly star-shaped domains with strictly \((k-1)\)-convex boundary, we complete the characterization of existence and nonexistence through a linearized capacitary comparison and a tangential-trace contradiction on the inner boundary.

We study a logarithmically hyperviscous Navier-Stokes model on the three-dimensional torus with Lie-transport noise, which includes both transport and stretching. We prove that, for noise of sufficiently large intensity and high frequency, the system admits a unique global smooth solution with probability arbitrarily close to one. Unlike previous works, this physically motivated noise does not preserve energy or enstrophy, but rather circulation. Global well-posedness is established through a probabilistic mechanism that produces effective dissipation via a scaling limit. Crucially, this approach bypasses the lack of conserved quantities and tames the singular nature of stochastic stretching.

Slowness surfaces are algebraic varieties arising from propagation of elastic waves. In dimensions $2$, we completely classify the types of singularities slowness surfaces can have. The two types of possible singularities are a transversal self-intersection and a tangential singularity produced by a concentric circle and ellipse that are tangent to each other. To interpret these results analytically, in the case that the slowness surface has transversal self-intersections, we show that the principal symbol of the elastic wave operator is locally smoothly diagonalizable.

We consider the initial value problem (IVP) for a two-parameter family of derivative nonlinear Schrödinger equations on the torus, known as the Majda-McLaughlin-Tabak (MMT) model arising in weak wave turbulence theory. For positive derivative order, we show that the flow map is not $C^3$ at the origin. Using an enhanced energy method, we prove unconditional local well-posedness in Sobolev spaces. At the energy regularity, conservation of the Hamiltonian and a mass-type quantity yields unconditional global well-posedness.

We settle the global nonlinear stability problem for the family of Kerr black holes in the full subextremal range: spacetimes evolving from initial data close to those of a subextremal Kerr black hole as solutions of the Einstein vacuum equation ${\rm Ric}(g)=0$ settle down to a nearby member of the Kerr family at the rate $\mathcal{O}(t_*^{-2-\epsilon_{\mathcal K}})$ in spatially compact regions. For the initial data, we require $\mathcal{O}(r^{-1-\epsilon_0})$-decay for $\epsilon_0>0$ – more precisely, an arbitrary but finite expansion into terms $r^{-z}(\log r)^k$ where $z>1$, $k\in\mathbb{N}_0$, plus a remainder term with $\mathcal{O}(r^{-3-\epsilon_0})$-decay. We use a generalized wave map gauge modified using gauge source terms that lie in a suitable finite-dimensional space determined by the expansion of the initial data. Like the final black hole parameters (mass and angular momentum) and the gravitational wave tail, the gauge source terms are treated as unknowns in a nonlinear (Nash-Moser) iteration scheme. We work directly with the tensorial equation and in particular do not rely on reductions to scalar equations (except insofar as a reduction to the Teukolsky equation is used in the proof of linear mode stability). This paper relies on two companion papers by the author. The first one introduces a strong form of constraint damping in the full subextremal range, which we use in our formulation of the gauge-fixed Einstein equation as a black box. The second one provides tame estimates (albeit with weak decay) for forward solutions of a general class of wave-type equations, which we show here to include the linearizations of the gauge-fixed Einstein equation arising in our nonlinear iteration scheme; these estimates are the starting point for our detailed asymptotic analysis.

We consider the semilinear parabolic equation \[ \partial_t u = \Delta u + 2\mathfrak{q}\,\delta_{\mathbb{S}}\,\nabla u + |u|^{p-1}u \qquad \text{in } (0,\infty)\times\mathbb{R}^N, \] where $|\mathfrak{q}|\le 1$, $p>1$, and $\mathbb{S}$ is a fixed interface hyperplane. Working in Lebesgue spaces, we first establish local well-posedness of mild solutions. This is achieved by combining Gaussian bounds for the associated fundamental solution with a contraction mapping argument adapted to the lack of spatial homogeneity induced by the interface term. We then prove a sharp Fujita-type dichotomy for nonnegative solutions. Specifically, we show that every nontrivial solution blows up in finite time when $1<p \le 1+\frac{2}{N}$, whereas for $p>1+\frac{2}{N}$ global solutions exist for sufficiently small initial data. The blow-up analysis relies on a suitably adapted test-function method that accounts for the presence of the interface. It is noteworthy that the critical exponent coincides with the classical Fujita exponent for the heat equation, indicating that the Fujita phenomenon remains stable under the presence of discontinuous diffusion effects and interface transmission conditions. To the best of our knowledge, this is the first result of this type for operators involving a singular drift supported on a hypersurface.

We consider solutions of uniformly elliptic equations with measurable coefficients. We assume that the lowest eigenvalue of the coefficient matrix is at least $K^{-1}$ and the largest eigenvalue is at most $K$. In three and higher dimensions we construct $\alpha$-Hölder continuous solutions with $\alpha = \exp(- c_n K)$. This disproves a long-standing conjecture by showing that, except for the two-dimensional case, the Hölder exponent obtained from the Bombieri–Giusti Harnack inequality has the optimal dependence on the ellipticity constant $K$. Using the same construction as a starting point, we disprove a conjecture of De Giorgi about continuity of solutions to non-uniformly elliptic equations.

This work presents the analysis and numerical simulation of a stationary drift-diffusion model for electrical discharge in micro-electro-mechanical systems (MEMS). The model couples Poisson's equation for the electrostatic potential with continuity equations for positive ions and electrons, incorporating a Townsend-type impact ionization source term that depends exponentially on the electric field magnitude. We prove the existence of weak solutions under physically relevant assumptions and establish uniform bounds on the carrier densities. The proof relies on a regularization-approximation scheme with truncated nonlinearities, monotone operator theory (Browder-Minty), iterative energy estimates, and Stampacchia-type truncation arguments. We further develop a robust finite element solver to simulate the carrier density and electrostatic potential profiles for several geometries, including two-dimensional domains and a three-dimensional axisymmetric geometry.

We investigate a variational problem for eigenvalues of the Laplace-Beltrami operator on smooth manifolds with respect to Radon measures belonging to a suitable class; we are motivated by conformal eigenvalues in dimension two. Our main result is a regularity result for stationary measures with respect to outer variations. More precisely, we prove that any sufficiently regular stationary measure is absolutely continuous with respect to the classical volume measure and that its density is induced by an harmonic map. Our result has some interesting applications to Steklov eigenvalues on subdomains.

For minimizers of a degenerate diffusion functional with a singular reaction term, we prove that the free boundary is $(n-1)$-rectifiable. The argument relies on a suitable integrability property, derived from a pointwise gradient estimate, combined with a Hausdorff dimension estimate for a portion of the zero set.

We investigate the existence of segregated rotating waves, arising in the singular limit of competition-diffusion systems of the type \[ \partial_t u_i -\partial_{xx} u_i = f(u_i)-\beta u_i \sum_{j \neq i} a_{ij} u_j,\qquad x\in\mathbb{S}^1,\ t>0, 1\le i,j\le k, \] as $\beta\to+\infty$. Here $k\ge3$, the reaction $f$ is of Fisher-KPP (logistic) type, and the competition coefficients $a_{ij}>0$ are not necessarily symmetric. Assuming that, for every $i$, \[ \dfrac{a_{i+1,i}}{a_{i,i+1}}=\lambda>0, \] we provide a complete characterization of the rotating waves enjoying an equivariant structure, where each density is a suitable rotation of any other one: such waves exist if and only if $\lambda$ belongs to an explicit range, in which case the angular velocity $\omega=\omega(\lambda)$ is uniquely prescribed, as is the rotating profile. In particular, stationary solutions (with $\omega=0$) exist only in the symmetric case $\lambda=1$. This marks a strong difference with the same problem with either Dirichlet or Neumann boundary conditions, where it is known that no periodic in time solution exists, also in the asymmetric case, sheding more light on some conjectures and open problems concerning the long time behavior of competition-diffusion systems.

We prove global-in-time strong pathwise well-posedness for a stochastic fluid-structure interaction problem coupling a two-dimensional incompressible Navier-Stokes fluid to a one-dimensional damped Kirchhoff plate. The coupling is imposed on a fixed interface through continuity of velocities and balance of normal stresses, and stochastic forcing, modeled by a cylindrical Wiener process, acts on both the fluid and structure equations. We split the problem into a linear stochastic part and a nonlinear deterministic remainder. The linear stochastic problem is treated by proving that the associated fluid-structure operator admits a bounded \(\mathcal{H}^\infty\)-calculus, yielding stochastic maximal regularity. This requires a decoupling procedure for the non-diagonal operator domain, and pressure estimates via suitable lifting constructions. The deterministic remainder is solved locally by quasilinear methods, and the resulting blow-up criterion is ruled out by higher-order a priori estimates. This is the first global-in-time strong pathwise well-posedness result for a stochastically forced Navier-Stokes system interacting with a deformable elastic structure.

We study the local controllability near zero of the Burgers equation with a scalar control and a fixed space-dependent source profile, in the case where the linearized system fails to be controllable and a second-order analysis is therefore required. We prove that quadratic obstructions to finite-time controllability can be quantified by Sobolev norms of the control with fractional negative exponents ranging over a full interval. To our knowledge, this is the first example, for a natural physical PDE, of a continuous scale of fractional quadratic obstructions, and the first such continuous scale for finite-time local controllability. Our explicit constructions shed light on the origin of fractional obstructions for partial differential equations, by relating the obstruction exponent to the regularity, and in some cases to the physical-space singularity, of the source profile. We identify the natural structural conditions on the source profile leading to obstructions quantified by the $H^{-1}$ and $H^{-5/4}$ norms of the control, thereby providing a general framework in which the previously studied case of a constant source profile fits naturally. In this constant-profile case, we improve existing results by identifying the arithmetic condition on the Fourier mode which ensures that a small-time obstruction actually persists in finite time. Finally, we derive sharp nonlinear remainder estimates adapted to the precise regularity of the source profile. These estimates make most of our obstruction results optimal with respect to the smallness assumption imposed on the control.

We study small–data solutions of a nonlinear scalar field equation on spatially flat $d$–dimensional FLRW spacetimes ($d\ge4$). In conformal time $\tau$ the field satisfies a damped semilinear wave/Klein–Gordon equation with time–dependent coefficients determined by the scale factor $a(\tau)$ and the conformal Hubble rate $H(\tau)=\dot a/a$. We focus on accelerated conformal expansion of the form $H(\tau)=H_0(1+\tau)^{-\alpha}$ with $H_0>0$ and $0\le\alpha<1$, for which $a(\tau)$ grows stretched–exponentially, and we assume a power potential $V(\varphi)=-\frac{\varepsilon}{m+1}|\varphi|^{m+1}$. For global solutions arising from sufficiently small, spatially localized initial data, we introduce the conformal rescaling $\phi=a^{(d-2)/2}\varphi$, which removes the first–order Hubble damping and exposes the interaction as a time–dependent coupling. In the rescaled equation the nonlinearity is weighted by $g(\tau)=a(\tau)^{\sigma}$ with $\sigma=\frac{d+2-(d-2)m}{2}$, so the conformal power $m_{\mathrm{conf}}=\frac{d+2}{d-2}$ is the sharp threshold for redshift suppression: $g$ decays for $m>m_{\mathrm{conf}}$, is constant for $m=m_{\mathrm{conf}}$ (classical conformal invariance), and grows for $1<m<m_{\mathrm{conf}}$. For accelerated conformal expansion $H(\tau)=H_0(1+\tau)^{-\alpha}$ with $0\le\alpha<1$ and for superconformal interactions $m>m_{\mathrm{conf}}$, we prove that $g\in L^1([0,\infty))$ and deduce small–data global existence together with scattering/asymptotic linearization for $\phi$. As a complementary result in the diffusion–dominated regime $1<m<1+\frac{2}{d-1}$, we adapt a weighted energy method for variable damping to deduce explicit $L^2$ and $L^1$ decay rates. These bounds provide a quantitative PDE formulation of redshift–induced suppression of nonlinear scalar self–interactions at late conformal times.

We study wave-type equations on dynamical spacetimes that settle down to a subextremal Kerr black hole spacetime. We prove strong estimates for solutions of (tensorial) linear wave-type equations when the time-translation-invariant model satisfies a spectral assumption of mode stability type. We allow for this model to admit zero energy bound states; besides the scalar wave operator (which has no bound states), examples include the wave operator on 1-forms and the linearization of the Einstein field equations in generalized harmonic gauge. We demonstrate the utility of our estimates by proving the global existence of solutions to some quasilinear wave equations, including in the presence of zero energy bound states. The results proved here are, moreover, crucial ingredients in the author's proof of the nonlinear stability of subextremal Kerr black holes. Our key novel linear estimate controls linear waves in weighted $L^2$-based spacetime Sobolev spaces that encode b-regularity, by which we mean regularity with respect to spacetime scaling, spatial scaling (in a hyperboloidal foliation of spacetime), and angular derivatives; this estimate is moreover tame in the b-regularity order, as needed for its applicability in a Nash-Moser iteration scheme. Its proof combines four main ingredients: microlocal propagation estimates in the edge-b-setting near null infinity (as introduced by the author with Vasy) and in the author's 3b-setting in the forward cone; estimates for the stationary model operator; energy estimates on edge-b-spaces on finite time intervals; and commutations with b-vector fields. For the nonlinear applications, we moreover develop a dictionary between decay rates in different spacetime regimes on the one hand and weighted low-energy resolvent estimates on the other hand. This paper builds on Part I only a broad conceptual level, and is largely self-contained.

Score-based generative models and Langevin samplers rely on estimating the score function $\nabla_x\log p_t(x)$ of a forward diffusion. Classically this is tractable when the drift is linear: the marginal density is Gaussian and the score is a global conditional expectation. For a general nonlinear, state-dependent drift the marginal density has no closed form, and existing methods–denoising score matching and global Fokker–Planck residual penalties–resort to global averaging that inflates estimation error in low-density regions precisely where accuracy is most critical. We address this by developing a local Fokker–Planck geometric framework that replaces global conditioning with local parabolic averaging. Our approach rests on three contributions. First, a time change to the cumulative-variance coordinate reduces the variable-coefficient Fokker–Planck equation to a standard inhomogeneous heat equation, on which we extend Evans' classical heat-ball monotonicity method to derive exact local mean-value representations for the score $\nabla_x\log p$ together with the density, log-density, and entropy density; local well-posedness is established under an explicit dimension-dependent drift budget. Second, for high-dimensional Monte Carlo evaluation of the resulting heat-ball integrals, we introduce the $\kappa$-measure and derive its exact factorized sampler with unit per-sample weight, $\chi^2_2$ radial concentration. Third, the $r\to0$ limit of the heat-ball residual recovers the pointwise Fokker–Planck residual, showing that the local framework is a one-parameter generalization of global FP-residual methods, and that the DSM population minimizer is feasible for the heat-ball constraint at every scale. We validate the framework on 2D structured data on 256-dimensional MNIST, and on a dedicated sampler study confirming the concentration laws.

We normalize a first-order real planar elliptic system, by pointwise algebra, to a framed Beltrami-Vekua equation $\Phi(w_{\bar z} - \mu w_z) + \Psi(\overline{w_z} - \mu\,\overline{w_{\bar z}}) + \mathfrak{a} w + \mathfrak{b} \bar w = \mathfrak{f}$, with $|\mu| < 1$ and $|\Phi| > |\Psi|$, and compute the closed transformation laws of its data under the recombination of unknowns $w \mapsto \varphi w + \psi \bar w$ and under orientation-preserving $C^1$ changes of variables. The 2-form $\Theta = \frac{\bigl|\,\Phi\,\mathfrak{b} - \Psi\,\mathfrak{a} - (\Phi\, L\Psi - \Psi\, L\Phi)\,\bigr|^2}{\bigl(|\Phi|^2 - |\Psi|^2\bigr)^2\,\bigl(1 - |\mu|^2\bigr)}\; dx\, dy$, with $L = \bar\partial - \mu\,\partial$, is invariant under the recombination and covariant under the changes of variables. The total mass $\mathcal{M} = \int_\Omega \Theta$ is therefore an invariant of the equivalence class. One recombination and one scaling carry any framed equation, in closed form, onto the trivial-frame slice - a Beltrami-Vekua equation over the same $\mu$ - there identifying $\Theta$ with the pseudo-analytic mass density of the unframed equation. We then show all of this persists at measurable regularity: it suffices that $\mu$ be measurable and locally elliptic and that the frame lie in $W^{1,2}_{\mathrm{loc}} \cap L^\infty_{\mathrm{loc}}$, the changes of variables then being quasiconformal homeomorphisms. In that class every equation with $\|\mu\|_\infty < 1$ is quasiconformally equivalent, of equal mass, to one over $\mu = 0$.

In this paper, we introduce a suitable notion of flat solutions for the anisotropic surface diffusion equation with elasticity in three-dimensions, based on a minimizing movement scheme inspired by that introduced by Cahn and Taylor. Using this scheme, we prove the existence of classical solutions to the equation without using a curvature regularization term. Moreover we establish the consistency of the approximation method.

We study small-amplitude solitary waves for two-dimensional capillary–gravity flows with arbitrary vorticity on the equatorial $f$-plane. The steady free-boundary problem is formulated as a reversible Hamiltonian spatial-dynamics system in which rotation enters through the speed-dependent effective gravity $g_*=g-2\Omega c$. A center-manifold reduction reduces the local bifurcation problem to finite-dimensional Hamiltonian systems governed by the low-frequency spectrum of a Sturm–Liouville problem with an eigenvalue-dependent boundary condition. We identify the Hamiltonian $0^2$, real $1{:}1$, and Hamiltonian–Hopf resonance curves and obtain corresponding families of symmetric solitary waves under standard non-degeneracy assumptions. We also show that the weak-effective-gravity threshold $g_*=0$ is separated from the $0^2$ and local Hamiltonian–Hopf resonances for uniformly non-stagnant laminar flows, and can be approached only in a near-stagnation regime.

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^2$, $W_0^{1,2} \left( \mathbb{B} \right)$ is a standard Sobolev space. Suppose a function $h_{\epsilon}(x)$ is radially symmetric, nonnegative, continuous on $\overline{\mathbb{B}}$ and satifies $\underset{x \rightarrow 0}{\lim} h_{\epsilon}(x) |x|^{- 2 \epsilon} =1 $, with $h_{\epsilon} (x) >0$ on $\overline{\mathbb{B}} \setminus \{0\}$. In \citep{26}, Zhang proved that the supremum in the following inequality can be attained by some function $u_{\epsilon}$, i.e. , \begin{align} \int_{ \mathbb{B} } h_{\epsilon} (x) e^{ 4 \pi \left(1 + \epsilon \right) {u_{\epsilon}}^2 } dx = \underset{u \in W_0^{1,2} \left( \mathbb{B} \right) \cap \mathcal{S} \setminus \{0\} , ~ \int_{ \mathbb{B} } |\nabla u|^2 dx \leq 1}{\sup} \int_{\mathbb{B}} h_{\epsilon} (x) e^{4 \pi (1 + \epsilon) u^2 } dx, \label{eq: 0.1} \end{align} where $4 \pi$ is the best constant in the classical Moser-Trudinger inequality, and $\mathcal{S}$ is the set of radially symmetric functions. In this paper, we consider the compactness of the sequence $\{ u_{\epsilon} \}_{\epsilon} $ and prove that the limit of this sequence is a function $u_0 \in C^1 (\overline{ \mathbb{B}} )$. Moreover, the $u_0$ is an extremal function of the supremum \begin{align*} \underset{u \in W_0^{1,2} \left( \mathbb{B} \right) \cap \mathcal{S} \setminus \{0\} , ~ \int_{ \mathbb{B} } |\nabla u|^2 dx \leq 1}{\sup} \int_{\mathbb{B}} e^{4 \pi u^2 } dx. \end{align*}

We study the asymptotic behavior of global minimizers of a Ginzburg–Landau-type functional with general compact vacuum manifold $\mathcal{N}$ on bounded domains in $\mathbb{R}^3$, in the regime where the energy grows at a logarithmic rate. We show that the normalized energy measures converge, up to a subsequence, to a measure supported on a finite union of closed line segments connecting prescribed singularities on the boundary. The limit map is a harmonic map valued locally by minimizing $\mathcal{N}$ away from this singular set. We also establish uniform $W^{1,q}$-estimates with $ q\in(1,2) $ and uniform potential estimates for minimizers, independent of the parameter $\varepsilon$. Finally, we prove that the singular set of the limiting measure solves the homotopical Plateau problem in codimension $2$.

We study the dissipative Boussinesq problem, which extends the "good" Boussinesq equation by incorporating viscosity effects. It is well-known that this model supports monotone decreasing traveling kink solutions. We show that these kinks are orbitally stable in $H^1(\rone)$. Moreover, they are asymptotically stable in $\dot{H}^1(\rone)$ and $L^p(\rone), 2<p\leq \infty$.

We study energy quantization for a class of Dirac systems on compact spin Einstein manifolds of dimension \(n\). For a sequence of solutions to a nonlinear Dirac system with uniformly bounded energy on a fixed spin Riemannian manifold, we first establish an energy identity theorem. We then investigate the more complicated case of underlying domain manifolds being a sequence of non-collapsed degenerating spin Einstein manifolds. At an orbifold singular point, three types of bubble spinors can possibly appear, living respectively on \(\mathbb{R}^n\), on a Ricci-flat ALE bubble space, and on the flat cone \(\mathbb{R}^n/\Gamma\). By developing asymptotic analysis for solutions over degenerating neck regions, we establish that energy identity holds.

In the context of hyperbolic formulations of Einstein's field equations obtained via gauge fixing, constraint damping is a desirable feature that ensures that violations of the gauge condition and thus of the constraint equations are suppressed in evolution. Besides its utility in numerical relativity, it has played a key role in several (linear and nonlinear) stability proofs of spacetimes as solutions of the Einstein equation. In this paper, we show that an enhanced form of constraint damping can be implemented for the linearization of the Einstein equation around any subextremal Kerr black hole metric: the constraint propagation wave operator satisfies mode stability and has an arbitrarily large indicial gap at zero energy. The results proved here are a key ingredient in the author's proof of the nonlinear stability of the subextremal Kerr family.

We study the two-dimensional stochastic Navier-Stokes equations on the torus with horizontal dissipation and additive noise. First, we prove a uniform large deviation principle for the solution paths in the energy space $C([0,T];H).$ The proof combines a large deviation principle for a linear Ornstein-Uhlenbeck process, a new Lipschitz estimate for the nonlinear map, and a contraction principle that avoids any exponential tightness condition. Second, we analyse the long-time dynamics. By exhibiting a deterministic steady state immune to the horizontal dissipation, we show that exponential mixing cannot hold. Finally, we investigate the invariant measures of the system. In the deterministic case, the infinite-dimensional kernel of the horizontal Laplacian contains infinitely many Dirac invariant measures. Under a suitably degenerate noise, however, the kernel supports no invariant measure at all; moreover, any invariant measure of this stochastic system, should it exist, must be orthogonal to this kernel. This sharply distinguishes the anisotropic equations on the torus both from the fully dissipative Navier-Stokes equations and from their counterparts on domains with no-slip boundaries.