Papers

We establish the existence of smooth solutions to the symmetric maximal surface equation with degenerate boundary conditions. Moreover, we prove that these solutions maximize the associated area functionals. This result serves as the Lorentzian analogue of minimal graphs in hyperbolic spaces together with their associated area minimizing problem.

We establish local Calderón–Zygmund estimates near the natural exponent for very weak solutions to matrix-weighted quasilinear equations\[ - \mathrm{div\,} A_{\mathbb{M}}(x,Du) = - \mathrm{div\,} A_{\mathbb{M}}(x,\mathbf{f}), \qquad A_{\mathbb{M}}(x,\xi)=\mathbb{M}(x)A(x,\mathbb{M}(x)\xi), \] where $A$ has $p$-growth and strong monotonicity, and $\mathbb{M}$ is a measurable positive-definite matrix field. We assume that $\mathbb{M}$ has bounded condition number and that \(\omega:=|\mathbb{M}|^p\in A_p\), without imposing uniform upper or lower bounds on $\mathbb{M}$. This extends the near-natural Calderón–Zygmund theory developed by Adimurthi–Phuc \cite{AP15} to the matrix-degenerate setting: There exists $\delta_0>0$ such that every very weak solution $u \in W^{p-\delta_{0}}_{\omega, \mathrm{loc}}$ satisfies \[ \mathbf{f}\in L^\gamma_{\omega,\mathrm{loc}} \Longrightarrow Du\in L^\gamma_{\omega,\mathrm{loc}} \] for $p-\delta_0\le\gamma\le p+\delta_0$. The proof requires handling the lack of energy estimates below the natural exponent and the use of Lipschitz truncation in the weighted setting. We achieve this through comparison estimates, higher integrability, and weighted analysis techniques.

We study high-frequency propagation for a viscoelastic wave equation with spatially dependent hereditary memory in relative-history form. The kernel may have the integrable singularity $\mathfrak m(s,x)=s^{p-1}m(s,x)$, $0<p<1$; the regular case $p=1$ is included. Using the half-amplitude propagation distance and corresponding travel time as units, the wavelength $h\ll1$ yields the memory factor $\varepsilon=h^{1-p}$. We construct exact solutions with full two-scale geometric-optics expansions in powers $h^{k+(1-p)\ell}$. Memory modifies the propagation geometry through the instantaneous modulus $\sigma+\varepsilon\int_0^\infty\mathfrak m(s,\cdot)\,d s$, while the kernel singularity contributes $C_p=\Gamma(p)e^{i\pi p/2}$ to the leading transport equation. For $0<p<1$, this produces fractional scales, frequency-dependent attenuation, and a dispersive phase correction; for $p=1$, the fractional hierarchy disappears, attenuation is frequency independent, and the transport phase correction vanishes. We also derive a local damped wave equation whose incoming high-frequency solutions approximate the hereditary solutions with $O(h)$ error in semiclassical $C^k$ norms. Exterior observations for all incident directions and $0<h\ll1$ uniquely recover $\sigma_{\mathfrak m}$ and the full temporal jet of $m$ at $s=0$, which determine the expansion modulo $O(h^\infty)$. Finally, a contraction-semigroup argument gives well-posedness and arbitrary finite-order Sobolev regularity for spatially dependent weakly singular kernels and prescribed full prehistory, with explicit compatibility conditions and estimates uniform in $\varepsilon$. These estimates justify the geometric-optics construction.

We establish higher regularity of the regular free boundary for non-negative solutions of the Kolmogorov obstacle problem \[ \mathcal K u=\chi_{\{u>0\}}, \qquad \mathcal K=\Delta_x+x\cdot\nabla_y-\partial_t. \] Assume that the contact set satisfies a quantitative thickness condition at a free-boundary point. We prove that, in a neighborhood of that point, the free boundary is a non-characteristic \(C_{\mathcal K}^{1,\beta}\) hypersurface for some \(\beta\in(0,1)\). Equivalently, its inward non-degenerate normal is Hölder continuous with respect to the intrinsic Kolmogorov distance, and the deviation of the graph from its tangent hyperplane is of order \(O(r^{1+\beta})\) at intrinsic scale \(r\). The proof combines the half-space blow-up and differentiability theory at regular points with boundary Harnack estimates in asymptotically cylindrical intrinsic Lipschitz domains. A principal difficulty is that the diffusion derivatives determining the normal do not satisfy the homogeneous Kolmogorov equation. We introduce a \(\mathcal K\)-harmonic-replacement argument that converts the commutator sources into an \(O(r)\) error under intrinsic blow-up and yields a contraction estimate for the oscillations of derivative quotients. This gives Hölder continuity of the non-degenerate normal. In addition, a boundary decay estimate for the transport derivative $Yu=(x\cdot\nabla_y-\partial_t)u$ provides the required improvement in the coupled time-transport direction. Combining these estimates with the available control in the degree-three variables yields a full intrinsic improvement of flatness.

For $0<a<1$, we study the linearization $L_a$ about a smooth odd self-similar collapse profile $(\Omega,c_l)$ of the generalized Constantin-Lax-Majda equation, where $c_l$ is the focusing exponent. On the origin-$H^2$ realization $X=\{\varphi\in L^2(0,\infty):\varphi(0)=0,\varphi''\in L^2(0,\infty)\}$, under explicit transport, regularity, weighted-derivative, and far-field hypotheses, the first three Edmunds-Evans essential spectra are exactly the union of two possibly coincident vertical lines: $\operatorname{Re}\lambda=F=-1+c_l/2$ at infinity and $\operatorname{Re}\lambda=O=-\tilde c/2$ at the origin, where $\tilde c=c_l+a(H\Omega)(0)$ and $H$ is the Hilbert transform. Off these lines, $L_a-\lambda$ is Fredholm, with index zero on the exterior components, $+1$ in the open strip when $F<O$, and $-1$ when $O<F$. The fourth and fifth Edmunds-Evans spectra and the Browder essential spectrum equal the closed strip. Every point of the open strip with index $+1$ is an eigenvalue. Every fixed point of the Huang-Qin-Wang-Wei positive-advection construction with $0<a<400/(848-9\pi^2)$, after normalization, satisfies the full hypothesis package. For all sufficiently small positive $a$, every such fixed point has $F<O$. Thus actual collapse profiles realize a nonempty Browder band. An additional kernel-nondegeneracy condition (ND) yields exact individual kernel and cokernel dimensions. We prove it outside an explicit vertical energy slab and, at sufficiently high frequency, on every closed vertical substrip of the open band separated from the far-field line. Failures in the open band are locally finite and can accumulate at high frequency only toward that line.

Let $K \subset \mathbb{R}^{m \times n}$ be a compact $C^1$-submanifold with boundary, $p \in (1,\infty)$ and $Q := (0,1)^n$. We prove that $K$ satisfies a rigidity estimate of the form $\|Du - (Du)_{Q}\|_{L^p} \leq C \|\mathrm{dist}_K(Du)\|_{L^p}$, $u \in W^{1,p}(Q,\mathbb{R}^m)$, if and only if $K$ satisfies sequential rigidity and for each $A \in K$, the tangent space to $K$ at $A$ satisfies exact rigidity. We further prove that this rigidity estimate is stable under small graphical perturbations of $K$. The key technical ingredient is proving that outside some small bad set where the maximal function of $\mathrm{dist}_K^p(Du)$ is large, the size of the superlevel sets of $\lvert Du - (Du)_Q\rvert$ decays exponentially. This is achieved by an adaptation of the John-Nirenberg inequality for $BMO$-functions.

The presented work investigates the Cauchy problems for parabolic equations in both non-divergence and divergence forms with rough diffusion coefficients, which commonly arise in composite media, financial pricing, and viscous fluids. Under critical regularity settings, we establish the unique solvability in optimal fractional Sobolev spaces. The first key technical step lies in constructing an effective approximation scheme with truncated and mollified diffusion coefficients $a^\epsilon(t,x)$, for which we rigorously prove the uniform preservation of high-frequency smallness. By incorporating this approximation scheme with paraproduct decomposition, Fefferman-Stein maximal inequalities, Coifman-Meyer bilinear estimates, and refined Sobolev embeddings, we close the uniform a priori estimates and then pass to the limit. Furthermore, we prove that the threshold $s < \frac{1}{2}$ is sharp by constructing explicit counterexamples. These theoretical results provide a rigorous mathematical framework in the study of Cauchy problems for parabolic equations with rough coefficients.

Let $u$ be the torsion function for the Ornstein-Uhlenbeck operator on a bounded domain $\Omega \subset \mathbb{R}^n$, i.e., the solution of $\Delta u - x \cdot \nabla u = -1$ in $\Omega$ with $u = 0$ on $\partial\Omega$. Let $T_\gamma(\Omega) = \int_\Omega u \, d\gamma$ be the Gaussian torsional rigidity. We prove that the Brunn-Minkowski-type inequality $T_\gamma((1-t)\Omega_0 + t\Omega_1)^{\alpha} \le (1-t) T_\gamma(\Omega_0)^{\alpha} + t T_\gamma(\Omega_1)^{\alpha}$ fails for every exponent $\alpha > 0$ and in every dimension $n \ge 2$, for a pair of convex bodies centrally symmetric with respect to the origin, which may be taken smooth with positive curvature. This answers Conjecture 1.4 for any $n \geq 2$, and Question~(Q), of Marín Sola and Salerno in the negative. The mechanism is a first-order lower bound for $T_\gamma$ at a ball $\Omega_0$ under Minkowski perturbations. When the perturbing body $\Omega_1$ has small torsion and large mean width, $T_\gamma((1-t)\Omega_0 + t\Omega_1)$ increases to first order. Since the Minkowski combination has larger torsion than both endpoints, no exponent can repair the inequality. For $n=1$, convexity with the optimal exponent $1/3$ holds on symmetric intervals by results of the same authors, \cite{MSS26}. We prove that the logarithm of the torsion is neither convex nor concave along Minkowski combinations of symmetric intervals, that no non-zero exponent yields concavity, and that convexity fails for every positive exponent when one set is a union of two intervals or when the sets are reflected off-center intervals.

This manuscript deals with a coefficient identification inverse problem for a semilinear damped wave operator in a bounded domain of $\mathbb{R}^{1+d}\ (d\geq 2)$. We establish the unique recovery of the damping coefficient, zeroth-order linear term, and the coefficient of the power-type nonlinearity from the partial Dirichlet-to-Neumann map. We investigate the corresponding uniqueness problem under the assumption that the coefficients are known in a neighborhood of the boundary, while the Neumann boundary data are prescribed only on an arbitrarily small open subset of the boundary. The analysis is largely based on the unique continuation principle, Fourier Analysis and the higher-order linearization technique.

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain whose boundary is a finite union of pairwise disjoint Jordan curves. We prove the planar Khavinson–Shapiro conjecture in this setting: if every polynomial on $\mathbb{R}^2$ agrees on $\partial\Omega$ with a harmonic polynomial, then $\partial\Omega$ is an ellipse and $\Omega$ is its bounded interior.

Let $X$ be a time-homogeneous one-dimensional diffusion whose coefficients satisfy the Engelbert–Schmidt conditions, and let $q$ be a bounded potential. We show that the associated Feynman–Kac semigroup is smooth in time and $C^1$ in space, with a locally absolutely continuous first spatial derivative; the Kolmogorov equation holds almost everywhere. If the drift, second-order coefficient, and potential are continuous, the Feynman–Kac value is $C^{1,2}$ in the interior. Thus, in the time-homogeneous one-dimensional setting, continuity suffices for interior $C^{1,2}$ regularity. We identify the corresponding semigroups with the killed and reflected Feynman–Kac functionals, treat nonzero, time-independent Dirichlet data, and give examples showing the sharpness of the continuity assumptions and the failure of the corresponding statement in two dimensions.

We prove two sharp monotonicity formulae for the natural Green's function of a closed three-manifold with positive scalar curvature, analogous to the Munteanu-Wang theorem for non-negative scalar curvature. We also show that both results may be interpreted as the monotonicity of a suitably defined "enclosed mass" of the sublevel sets of Green's function in the corresponding "conformal" blow-up of the manifold.

In this paper, we study the long-time behavior of the viscous conservation law equation on the whole real line, subject to a white-in-time force and arbitrarily small damping and viscosity constants. We establish polynomial mixing rates in two settings: (a) a general super-quadratic flux with hyperviscous dissipation, where the growth of the nonlinearity is controlled by the order of hyperviscosity, and the force is sufficiently non-degenerate on a generic L2-basis; and (b) a general quadratic flux with standard second-order dissipation, and the force is non-degenerate on some unconditional basis. Both settings include the classical Burgers-type nonlinearity. The proof is based on an abstract polynomial coupling criterion developed in arXiv:2408.00592 and a Foiaş–Prodi estimate.

We prove uniform-in-time quantitative relative entropy estimates for Kac's particle approximation of the three-dimensional spatially homogeneous Landau equation for Maxwellian molecules. Using the relative entropy method and logarithmic derivative estimates, we obtain a finite-time normalized entropy bound of order $N^{-1/2}$. The proof relies on explicit covariance identities and a new law-of-large-numbers estimate for the particle system using a new duality method. The uniform-in-time propagation of chaos is derived from an interpolation with an algebraic relaxation estimate toward equilibrium.

We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation \[ \mathcal Ku=\Delta_xu+x\cdot\nabla_yu-\partial_tu=0 \] in non-characteristic intrinsic Lipschitz graph domains whose defining functions may depend on the higher-order variable \(y_m\). Earlier theory required independence of \(y_m\). We replace this symmetry by quantitative asymptotic cylindricality, measured by a scale-invariant \(C^{0,1/3}_{y_m}\) defect. The perturbative comparison theorem applies whenever this defect is sufficiently small, and hence at all sufficiently small scales for \(C^{0,\alpha}_{y_m}\) graphs with \(\alpha>1/3\), as well as under a vanishing critical cylindrical modulus. We obtain local comparability and intrinsic Hölder continuity of quotients of positive solutions, together with a Hölder estimate for their logarithmic quotient. The proof combines local boundary estimates, valid without \(y_m\)-independence, with a global rigidity argument for cylindrical blow-up limits. The central issue is uniform balance of forward and backward reference values. If this balance fails at collapsing scales, maximal-scale selection and a second rescaling produce a non-negative polynomial-growth Dirichlet solution in an unbounded cylindrical domain that vanishes at the backward reference point but not at the forward one. We exclude this configuration using boundary-energy and mean-value estimates, finite-dimensionality of spaces of polynomial-growth solutions, and translation invariance in \(y_m\). The resulting finite-dimensional translation representation yields real-analytic dependence on \(y_m\). Propagation of zeros, analytic continuation along invariant fibres, and a Tikhonov uniqueness argument then force the limiting solution to vanish identically. The exponent \(1/3\) is critical because \(y_m\) has homogeneous degree three.

In this paper, we consider the Cauchy problem for the incompressible magnetohydrodynamic equations with the Coriolis force in the three-dimensional whole space. For large initial data, it is known that the Cauchy problem admits a unique global solution in critical Sobolev space $\dot{H}^{1/2}(\mathbb{R}^3)$ provided that the speed of rotation is fast enough. By using delicate dispersive estimates, we show refined decay estimates of the velocity field and magnetic field. These decay rates significantly improve the ones obtained by Kim [J. Differential Equations., 2022] in the subcritical Sobolev framework $H^s(\mathbb{R}^3)$ with $1/2<s<3/2$.

We study the periodic fractional stochastic Burgers equation $(\partial_t+\Lambda^\gamma)u=\partial_x(u^2)+|\partial_x|^{1-\alpha}\xi$, where $1<\gamma\leq 2$ and $\xi$ is space-time white noise. Under the condition $\alpha>\max{(7-4\gamma)/2,(15-8\gamma)/6}$, we establish pathwise $L^1$, energy, and Besov estimates for a transformed remainder. The argument combines a response decomposition with a conservative Zvonkin transformation. For $1<\gamma<2$, this produces a compensated nonlocal diffusion operator whose coercivity is coupled to a cubic-increment estimate adapted from the modified Karman–Howarth–Monin argument.

We study the uniqueness of the vortex wave system, i.e.\ the 2D incompressible Euler equations with vorticity consisting of a point vortex and a bounded part. We show that if the initial vorticity and point vortex location $(w_0,X_0)$ satisfy a constraint of the type $|w_0(x)-w_0(X_0)|=O(|x-X_0|^\alpha)$ for $\alpha> 2$ and the vorticity is in $L^\infty$, then solutions are unique, globally in time, even though the velocity is not log-Lipschitz.

In this paper, we will construct periodic solutions to the viscous steady Burgers equation with an external force $f(x)$, based on the following formal expansion $u^\varepsilon(x)=u_0(x)+\varepsilon u_1(x)+\cdots+\varepsilon^n u_n(x)+\cdots$, where $\varepsilon \ge 0$ represents the viscosity and $u_0(x)$ is a solution to non-viscous steady Burgers equation with the external force $f(x)$. We will focus on the solutions which are uniformly bounded with respect to the viscosity. In our previous work, starting from $u_0=-(2+\cos x)$, the authors constructed the periodic solutions to the viscous steady Burgers equation with the external force $f=u_0(x)u_{0x}=-2\sin x-\sin x\cos x$. In this paper, we will extend the main result obtained in our previous work and construct the solutions starting from general $u_0$ and $f$ satisfying the non-viscous steady Burgers equation $u_0(x)u_{0x}=f$. It will be shown that there exists a $\varepsilon_0>0$, which depends on $n$, such that for any $0<\varepsilon<\varepsilon_0$, the viscous steady Burgers equation with the external force $f$ has a periodic solution $u^\varepsilon(x) \in C^2([0,2\pi])$, satisfying $|u^\varepsilon(x)-u_0(x)-\varepsilon u_1(x)-\cdots-\varepsilon^n u_n(x)| \leq C\varepsilon^{n+1}$, where $C>0$ is a constant which depends on $n$, but is independent of $\varepsilon$. Compared with Jauslin-Kreiss-Moser's result, we present a new approach to construct periodic solutions to the viscous steady Burgers with a general external force. The constructed solutions will tend to the ones of the non-viscous Burgers equation with a sharper convergence rate (up to higher order) when the viscosity vanishes.

Planar cell polarity is preceded by the clustering of polarity proteins into discrete, low-turnover membrane subdomains (puncta), yet the minimal interactions that nucleate puncta, set their number, and segregate opposite orientations remain unclear. We address these questions with a mass-conserving reaction-diffusion model in which two diffusible monomers bind reversibly across a cell-cell junction into trans-complexes of two orientations, with feedback entering only through concentration-dependent rates. Above a critical density, the uniform state undergoes a long-wavelength mass-redistribution instability rather than a finite-wavelength Turing bifurcation. The form of the feedback then selects between two morphologies: broad mesas fixed by a Maxwell construction under saturating feedback, and narrow mass-limited spikes under unbounded feedback. For spikes we obtain closed-form expressions exhibiting a clean separation of amplitude (mass and feedback), width (the complex diffusion length), and spacing (the monomer screening length). Within a fast-monomer reduction we prove, for any number and arrangement of puncta, that like-oriented arrays coarsen, so multiplicity is metastable and kinetically determined. The second monomer reservoir introduces a second screening length that rate-limits competition by the harmonic mean of the monomer diffusivities, and the spectrum of a punctum remains free of oscillatory ("blinking") instabilities throughout. Finally, sign-definite cross-modulation of turnover converts clustering into orientation sorting - the mutual exclusion of orientations along a single contact: mass redistribution sets puncta number, and the sign of the cross-coupling determines whether orientations segregate. Puncta number and orientation sorting are thus governed by mathematically separable ingredients.

In this paper, we consider energy identity for intrinsic stationary biharmonic maps into homogeneous spaces in supercritical dimensions, extending the corresponding result of Hornung-Moser [Anal. PDE. 2012] in critical dimension. The proof follows a similar strategy as that of Lin-Rivière [Duke Math. J. 2002]. A key ingredient is a conservation law for intrinsic biharmonic maps into homogeneous spaces, which allow us to derive higher regularity of the map.

We study the time-dependent Hartree–Fock equation in $\R^3$ with physical Coulomb interactions in both direct and exchange channels, with exchange coupling $\eta$, linearized about homogeneous equilibria $\gamma_f=g(-i\nabla)$ having compact momentum support. Nguyen and You showed that, without exchange, two purely imaginary plasmon poles $\pm i\tau_0(|k|)$ exist below a survival threshold $\kappa_0>0$. Existing nonlinear Hartree–Fock theory requires a short-range direct interaction and a smooth small exchange kernel, excluding the present Coulomb setting. On every compact band $0<k_-\le|k|\le k_+<\kappa_0$, we prove unconditionally for Coulomb exchange that, for small $|\eta|$, the poles persist, remain simple and purely imaginary, and depend real-analytically on $\eta$. Their first-order shift $\tau_X$ splits into explicit static self-energy and dynamic vertex corrections. Each pole yields an exact undamped mode with a closed-form unit-density eigenfunction, embedded in the fiber generator's essential spectrum. The causal density Green function splits into an explicit undamped sine wave and a remainder whose Laplace transform is holomorphic near the poles, while the plasmon channel satisfies uniform $t^{-3/2}$ dispersive decay and endpoint Strichartz estimates. A first-order Ward-type cancellation gives $|\tau_X(r)|\le Cr^2$ down to $r=0$; hence exchange leaves the plasma gap unchanged to first order, and we obtain a closed formula for the stiffness correction. Numerics for a $C^9$ smoothed Fermi ball support the analysis: self-energy and vertex terms cancel within $1.2\%$ on the computed range, and exchange softens the dispersion most strongly near $0.7\,\kappa_0$.

We prove a critical Morrey rigidity theorem for the five-dimensional stationary Navier–Stokes equations. More precisely, every smooth solution on $\mathbb R^5\setminus\{0\}$ satisfying \[ \sup_{R>0}R^{-2}\int_{B_R}|u|^3\,dx<\infty \] is identically zero, up to an additive constant in the pressure. This replaces the pointwise Type-I control in the known higher-dimensional rigidity theory by a velocity-only, scale-invariant averaged condition that allows spatial concentration. The proof develops a weak head-pressure mechanism that does not rely on pointwise pressure estimates or classical normal traces. We reconstruct a canonical pressure from the velocity, derive a renormalized inequality for the positive head pressure, and introduce two monotone radial fluxes. Annular energy estimates, suitable-weak compactness, and blow-up and blow-down limits are then used to identify the endpoint fluxes and force rigidity. As an application, we obtain a removable-singularity criterion in dimension five: if a suitable weak solution is smooth away from one point and either its scale-invariant Dirichlet energy or its cubic velocity Morrey quantity remains bounded near that point, then the singularity is removable. Thus, within the isolated-singularity class, the smallness assumption in the classical stationary regularity criterion is replaced by boundedness. We also prove the corresponding velocity-only cubic Morrey rigidity theorem in dimension four by a different finite-energy argument.

In an inhomogeneous dissipative medium, different rays experience different round-trip attenuation, so an instantaneous time mirror may retrace rays without producing a balanced point focus. We solve this problem for a microlocally selected back-propagating transverse-magnetic component generated by a short pulsating modulation of a thin shell surrounding an inhomogeneous Debye medium with Ohmic conductivity. An exact decomposition of the Maxwell–Debye field isolates the pulse-generated return, and a semiclassical mode reduction yields a scalar returned operator whose principal symbol separates the temporal pulse response, shell modulation, and accumulated material loss. Our analysis shows that geometric distortion and amplitude attenuation are both involved in wave propagation, but the geometric distortion cancels between the outgoing and returning branches, whereas physical attenuation remains. A normalized directional energy, computed from pre-modulation measurements using the known lossless backward propagator, contains the same leading attenuation factor as the returned symbol. Its reciprocal defines a smooth, nonnegative, physically admissible modulation that equalizes the selected source-side gain up to a relative $O(h)$ error without requiring a complete attenuation model. With such a modulation, the field generated by a point source has a focal-scale leading profile independent of directional attenuation, with a unique maximum at the source and coherent return time. In every fixed focal window, the maximum is displaced by at most $O(h^2)$ in space and time, and the spatial half-amplitude area is $O(h^2)$.

We consider the two-dimensional Cahn-Hilliard equation forced by divergence of space-time white noise that represents Kawasaki dynamics in conservative form. The standard heuristic argument shows that the solution is a distribution and thus the product within the nonlinear term is ill-defined, making this stochastic partial differential equation challenging. We prove the global-in-time unique solution theory after multiple Da Prato-Debussche tricks where the last equation is solved in a weak formulation. Besides a minimum amount of the standard renormalization procedure to deal with ill-defined products, our proof consists of applications of deterministic analysis tools and taking advantage of the unique structure of the equation, which is crucial. We also prove the local-in-time unique solution theory of a mild solution, demonstrating that our improved estimate gets us arbitrarily close, although not quite, to being able to prove the global-in-time solution theory with the last equation in not only a weak formulation but also a mild formulation. Additionally, our approach can be readily extended to prove the global-in-time existence, although not path-wise uniqueness, of solution to the singular surface quasi-geostrophic equations with the resulting equation after multiple Da Prato-Debussche tricks in weak formulation.

We study the simplest parabolic-elliptic model of chemotaxis in ${\bf R^N}$, and show the optimal criteria for the existence of infinite time blow-up of solutions with the initial data below the Chandrasekhar singular solution when $N \geq 10$. In particular, we exhibit the differences of the criteria between the cases $N \geq 11$ and $N = 10$. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution and the asymptotic expansions of solutions to the Liouville equation.

We present an alternative proof of the global well-posedness of nonlinear wave equations in three spatial dimensions under the null condition, in the regularity regime of the classical local existence theory, assuming smallness of the angular derivatives. Unlike previous methods, which rely on decay in time, our approach is based solely on spatial decay. This provides a new perspective on the problem and offers potential applications to the study of Einstein's equations, which we intend to explore in future work.

We establish interior $C^{2,\alpha}$ regularity for admissible solutions of the quadratic Hessian equation with positive $C^\alpha$ right-hand side on the full positive branch. The main ingredients are a quantitative large-trace propagation argument and an adaptive Dirichlet comparison method.

We study a quadrature surface free boundary problem on a smooth compact finite-dimensional Riemannian manifold $(M,g)$. The problem is formulated as a shape optimization problem involving a Dirichlet problem for the Laplace–Beltrami operator, with a geometric condition on the free boundary. Under suitable uniform geometric assumptions on the admissible class, we establish its compactness and prove the stability of the corresponding Dirichlet problems, including strong convergence of the associated states in $H_0^1(M)$. We derive the first-order optimality condition for the associated shape functional. Finally, we establish a Riemannian comparison principle for the second fundamental forms and the mean curvatures of tangent boundaries at a contact point. These results provide a rigorous extension of the quadrature surface free boundary framework from the Euclidean setting to compact Riemannian manifolds.

We investigate the three dimensional Boussinesq equations in the planetary geostrophic scale, where the buoyancy/gravity driven flow is near geostrophic balance in the large horizontal spatial scale. By establishing the uniform (with respect to the characteristic number) estimates, we are able to obtain the global weak solutions and the unique global strong solutions to the Boussinesq equations without restriction on the initial data. Furthermore, sending the characteristic number to zero with the established compactness rigorously justifies the planetary geostrophic approximation of the Boussinesq equations. We employ the novel buoyancy flow-free flow decomposition to overcome the singularity caused by the buoyancy/gravity.