Papers

For the degenerate parabolic equation $u_t=\nabla\cdot (u^{m-1}\nabla u)$, the homogeneous Neumann problem in smoothly bounded domains $\Omega\subset {\mathbb{R}}^n$ is considered in the (super-fast) diffusion range when $m\in (-\infty,1)$. Upper and lower estimates for solutions are inferred by means of two Moser-type iterations, and optimality properties of these bounds are derived using comparison arguments. On the basis of the accordingly obtained information on regularization into spaces of bounded and uniformly positive functions, a result on global solvability in an associated initial-boundary value problem is achieved for all nonnegative and nontrivial initial data belonging to $L^1(\Omega) \cap \bigcup_{s>n(1-m)/2} L^s(\Omega)$.

This paper is a continuation of Part~I \cite{QTWPartI}, where blow-up analysis and compactness results for the CR Nirenberg problem were established. We prove some existence and degree results for the CR Nirenberg problem on the standard sphere. More precisely, we first prove existence results under antipodal and, more generally, finite-group symmetries. Without symmetry, we derive a Leray–Schauder degree formula under natural flatness conditions, yielding existence results for positive prescribed curvatures with certain homogeneous expansions at critical points. We also obtain Aubin-type inequalities under a vanishing first-moment condition, uniformly near the critical exponent.

We study the Cauchy problem in $\R^{3}$ for the repulsive compressible Navier-Stokes-Riesz system with Riesz exponent $0<s<1$ and viscosity $0<\varepsilon\leq1$, where the Riesz interaction $\nabla(-\Delta)^{-s}(\rho-\bar{\rho})$ is a generalization of the Coulomb interaction for electrons. For small perturbations of a constant equilibrium, with the solenoidal component of the initial velocity of order $\mathcal{O}(\varepsilon)$, we prove the global existence and uniqueness of smooth solutions. We derive time-decay estimates in $L^{2}$ norms and $L^{\infty}$ norms that capture both uniform-in-$\varepsilon$ dispersive behavior and viscosity-dependent dissipation. We further establish a global-in-time inviscid limit to the irrotational global solution of the compressible Euler-Riesz system whose initial data consist of the same density and the curl-free component of the velocity, with an explicit convergence rate in $W^{k,p}$ norms. The proof combines viscosity-adapted dispersive estimates, normal-form analysis and nonlinear energy estimates with control of both negative and positive Sobolev norms.

We study the initial boundary value problem for the compressible Navier-Stokes-Poisson equations without damping in the three-dimensional half-space. The velocity satisfies the no-slip boundary condition, and the electric potential satisfies the homogeneous Neumann boundary condition. For small perturbations of a constant state in $H^3$, with a small initial electric field in $L^2$, we establish global existence and uniqueness of strong solutions. The uniform estimates combine density dissipation, tangential and normal derivative estimates, and a modified energy that accounts for the electric boundary contribution through the Helmholtz projection. We also prove algebraic decay under an additional low-frequency assumption on the initial data. If the density perturbation, velocity and electric field have finite tangential negative Sobolev norms of order $-s$, where $1/2\leq s<1$, then the solution decays at the rate $(1+t)^{-s/2}$ in the full energy norm. The decay estimate follows from the formulation in terms of the momentum $\rho u$, uniform negative-order estimates and interpolation with the dissipation.

We study the homogenization of an optimal control problem governed by a Navier-Stokes-Cahn-Hilliard system in a periodically perforated three-dimensional domain. The obstacles are Lipschitz sets of size $\varepsilon^\alpha$, $\alpha>3$, where $\varepsilon$ is the period. The fluid satisfies the no-slip condition on the outer boundary and the free-slip condition on the obstacles, the phase-field equation contains a source term, and the convective term carries a pore-scale Reynolds number that tends to zero as $\varepsilon\to0$. The control is a distributed body force, and the cost functional tracks the velocity in $L^2$ and the order parameter in $H^1$. Since uniqueness of weak solutions of the state system is not known for fixed $\varepsilon$, the control-to-state relation is multivalued, and we pose the microscopic problems over control-state pairs. We prove the existence of microscopic optimal pairs and the sequential $\Gamma$-convergence of the cost functionals to the cost functional of a control problem governed by the unsteady Stokes-Cahn-Hilliard system on the full domain. As a consequence, the optimal values converge, every cluster point of microscopic optimal pairs is optimal for the homogenized problem, and the optimal controls converge strongly along the corresponding subsequences. The $\liminf$ inequality follows from the homogenization of the state system, including the strong convergence of the gradients of the order parameter. Only the recovery sequence requires uniqueness of weak solutions of the homogenized system, which we prove when the outer boundary is of class $C^{2,1}$.

We investigate a viscoelastic wave equation with nonlinear variable-exponent damping and a variable-exponent logarithmic source, allowing supercritical growth. The main difficulties arise from the failure of the source to map the energy space into $L^2(\Omega)$, the sign-changing logarithmic potential, and the lack of Sobolev control of the displacement for supercritical damping. Under an exponent condition linking the source and damping exponents, we prove local existence, an energy identity, and a continuation criterion. We prove finite-time blow-up for nontrivial solutions with nonpositive shifted initial energy. We also establish a blow-up criterion based on a sufficiently large initial displacement–velocity pairing, with a threshold depending on the positive part of the shifted initial energy, and derive an explicit upper bound for the lifespan. This criterion allows us to construct blow-up data at every prescribed level of the original energy. For initial data in a suitable stable region, solutions exist globally. For damping within the Sobolev range, a general relaxation condition yields energy decay estimates that include exponential and polynomial rates. Beyond this range, a time-weighted displacement estimate and a splitting of the memory integral give logarithmic decay.

We show that, in ambient dimension at most seven, varifold limits of the minimizing movements scheme introduced by Almgren–Taylor–Wang and Luckhaus–Sturzenhecker are integral Brakke flows. As a consequence, we show that $L^1$-limit points of the scheme, usually known as flat flows, are generalized BV flows. The result is unconditional and, in particular, remains valid even in the absence of energy convergence and in the presence of fattening.

In this paper, we consider the orbital stability of solitary waves for the derivative nonlinear Schrödinger equation $$iq_t+q_{xx}+i\partial_x\bigl(|q|^2q\bigr)=0, \quad (t,x)\in\mathbb{R}\times\mathbb{R}.$$ For the endpoint solitary waves satisfying $c>0$ and $c^2=4\omega$, we prove the orbital stability in $H^1(\mathbb{R})$ without any additional assumptions. The proof uses conservation of mass and the second Hamiltonian $\mathcal{H}_2$ to keep the Bäcklund-transformed solution small in $H^1(\mathbb{R})$ uniformly in time. We establish an inverse Bäcklund compactness result to reconstruct the algebraic solitary wave as the spectral zero approaches the negative real axis.

We study boundary vanishing orders and asymptotic expansions of planar harmonic functions satisfying the sublinear Robin condition $\partial_\nu u=\lambda u-|u|^{p-1}u,$ with $\lambda\geq0$ and $0<p<1$. For nontrivial global solutions on bounded connected Lipschitz domains, we bound the number of nodal domains by the number of Steklov eigenvalues strictly below $\lambda$ and establish finite vanishing order at every boundary point. The boundary scaling gives the critical exponent $\frac{1}{1-p}$. On $C^1$ boundaries, every positive subcritical vanishing order is an integer, and the leading term is a unique homogeneous Neumann profile in conformal coordinates. At a noninteger critical order, the rescaled solution converges in normalized graph coordinates to an explicit homogeneous profile determined by $p$ up to sign. Every supercritical bulk vanishing order $m$ is an integer, with a unique homogeneous Dirichlet profile in conformal coordinates. The boundary trace has vanishing order $\frac{m-1}{p}>m$. Integer critical vanishing orders are excluded if the boundary is $C^1$ with Dini mean oscillation.

In this paper, we study the focusing mass-critical Hartree equation \[ iu_t+\Delta u=-\bigl(|x|^{-2}*|u|^2\bigr)u \] at the ground-state mass $M(Q)$. We prove that, up to the symmetries of the equation, every global minimal-mass non-scattering solution is a solitary wave, while every finite-time minimal-mass blowup solution is a pseudo-conformal image of a solitary wave. This extends the results of \cite{LZ28,Miao-Xu-Zhao3,Wang-Gao} to the scaling-critical space $L^2$ and removes the radial assumption in the global case.

We give new, elementary proofs of interior second derivative estimates for the Monge-Ampère equation and the constant quadratic Hessian equation, based on the two-point maximum principle. The arguments require only one differentiation of the equation. For the Monge-Ampère equation, the inhomogeneous term is assumed to be positive and continuously differentiable. For the quadratic Hessian equation, the estimate holds in every dimension $n\ge2$.

We study spectral Gårding elliptic equations on compact Kähler manifolds. Smooth solvability is equivalent to the existence of a smooth representative in the admissible domain. The concavity theorem for ideal Gårding polynomials gives a fixed concave formulation to which Székelyhidi's second-order argument applies in every degree $2\le d\le n$, extending Lin's top-degree solvability theorem. Combined with the numerical criteria of Fu–Li–Zhang–Zhang and Chen–Nie–Xu, this yields a Nakai–Moishezon type criterion for solvability on compact Kähler manifolds. A solution converts the local ideal Gårding structure into a global Lorentzian structure. Its mixed derivatives give a strict nonlinear Hodge–Riemann hierarchy from the degree of the equation down to degree two, and the homogenization of the intersection polynomial is Lorentzian in nef class directions. Numerical criteria then produce explicit Hodge–Riemann classes, Lefschetz isomorphisms in bidegree $(1,1)$, and weighted Hodge-index inequalities.

We study radial bound states of \[ \Delta u-\abs{u}^{q-1}u+\abs{u}^{p-1}u=0 \quad\text{in }\R^n, \qquad n\ge3,\qquad 1<q<p<\frac{n+2}{n-2}. \] For every integer $k\ge1$, we prove that there is exactly one radial bound state with $k$ simple zeros and $u(0)>0$. Its absolute maximum is attained at the origin. This confirms the zero-mass, Sobolev-subcritical case of Tang's double-power uniqueness conjecture. The proof adapts Tang's phase-transition analysis and combines it with a finite-level separation argument inspired by Cortázar–García-Huidobro–Yarur. Three main difficulties arise in the zero-mass double-power setting: the sign changes of key comparison quantities in the low-amplitude region, the possible failure of pointwise positivity of an auxiliary quantity when $n=3$ and $p+q<3$, and the absence of exponential decay at infinity. We overcome these difficulties by introducing bridge functionals to propagate the required sign information through low amplitudes, establishing a weighted integral positivity estimate in the exceptional three-dimensional range, and combining a radial-flux tail estimate with finite-level inverse-branch comparison. We also establish the strict ordering of the bound-state shooting heights and show that the shooting variation along the \(k\)-node state has exactly \(k+1\) simple zeros.

In this paper, we consider the following reaction-diffusion system \begin{equation*} \begin{cases} u_t=\Delta u+F\bigl(x,t,u,v,V(t)\bigr), & (x,t)\in\Om\times(0,T), v_t=\Delta v+G\bigl(x,t,u,v,U(t)\bigr), & (x,t)\in\Om\times(0,T), \mathcal B u=\mathcal B v=0, & (x,t)\in\partial\Om\times(0,T), u(x,0)=u_0(x),\quad v(x,0)=v_0(x), & x\in\Om, \end{cases} \end{equation*} where \begin{equation*} U(t)=\int_{\Om}u(x,t)\dd x,\qquad V(t)=\int_{\Om}v(x,t)\dd x, \end{equation*} and the boundary operator $\mathcal B$ is either \[ \mathcal B_D w=w \qquad\text{or}\qquad \mathcal B_N w=\partial_\nu w, \] $\Om\subset\R^N$ be a bounded domain with sufficiently smooth boundary, $N\ge 1$, and $\nu$ is the outward unit normal on $\partial\Om$. Under certain conditions, we obtain the upper and lower bounds for the blowup time of the solution to the system.

In this paper, we investigate global solutions to the fully nonlinear obstacle problem. We prove that, after factoring out invariant directions, the coincidence set of any global solution belongs to one of the following four types: a single point, a compact convex set with nonempty interior, a ray for $n=1$, or a slender body for $n\geq 2$, where $n$ denotes the spatial dimension. Furthermore, we establish a bijection between the collection of all global solutions with compact coincidence sets and a family of functions describing the asymptotic behavior of such solutions at infinity.

We study stable solutions of the Lane–Emden equation $$-\Delta u=|u|^{p-1}u$$ on graphs with polynomial volume growth. Due to the lack of a chain rule in the discrete setting, we establish suitable algebraic inequalities which serve as substitutes for the usual differential identities. Combining these inequalities with the stability condition and cutoff arguments, we obtain nonexistence results for stable solutions in a range of exponents analogous to the Joseph–Lundgren range. As an application, we derive a lower bound for the Sobolev best constant in the discrete setting.

We study the uniqueness of bound states for the following scalar field equation $$ -\Delta u+u=-\varepsilon |u|^{q-1}u+|u|^{p-1}u \,\ \ \mbox{in}\,\ \mathbb R^N, \,\ \varepsilon>0,\tag*{$(*)$} $$ where $ N\ge3,\ 1<p<(N+2)/(N-2)$ and $q>1$ satisfies $q \neq p$. For each integer $k \geq 1$, we prove that the equation $(*)$ has, up to translation and reflection, a unique bound state with precisely $k$ nodes in a natural coupling regime, where $\varepsilon>0$ is sufficiently small. Our results address a conjecture of [H. Berestycki and P. L. Lions, ARMA, 1983][M. X. Tang, Invent. Math., 2026] on the uniqueness of fixed-node bound states for the equation $(*)$.

We study the long-time behavior of the infinite-energy solutions to the white-forced Navier–Stokes equation in an infinite pipe $\mathbb{R}\times (-1,1)$ with no-slip boundary conditions. The equation is globally well-posed in uniformly local spaces satisfying the divergence-free condition. Under suitable degeneracy and spatial decay assumptions on the noise, we prove the uniqueness of the invariant measure and establish polynomial mixing in the dual-Lipschitz metric. This provides an affirmative answer to Open Problem 1 posed by Kuksin and Shirikyan \cite{KS17} in the case of zero cross-sectional flux.

We study the effect of boundary conditions on the selection of wavenumbers in pattern-forming systems. Prior work established that, for a large class of boundary conditions, equilibria only within a narrow band of wavenumbers are compatible with the boundary. We demonstrate here that for a large class of initial conditions, solutions evolve towards an extremal member of this narrow band of equilibria. In fact, this extremal member is neutrally stable, nonlinearly unstable, and the convergence is realized through a series of snapping events at the boundary. We observe this behavior in simulations of the Swift-Hohenberg equation when the pattern forming region is confined to a large spatial interval through a parameter quench. We prove the convergence in a phase-diffusion approximation using comparison principles and astutely chosen subsolutions that mimic the snapping behavior.

The $N$-dimensional Euclidean Onofri inequality was proved by Del Pino and Dolbeault for smooth compactly supported functions in $\mathbb{R}^N$ with $N \ge 2$. In this paper, we consider the singular version introduced by Lam and Lu, and extend it to a natural weighted Sobolev space. Next, we establish a sharp singular Carleson-Chang type inequality and prove its equivalence with the singular $N$-dimensional Euclidean Onofri inequality. Furthermore, we prove the existence of nontrivial, non-constant optimizers for the functional associated with the singular Euclidean Onofri inequality, answering open questions left by Del Pino and Dolbeault in the non-singular case, and by Lam and Lu in the case with singularities. To this end, we exploit the classification result obtained for a singular Liouville-type equation involving the $N$-Laplacian. We also prove the conformal invariance of the functional, establish the sharpness of the singular $N$-dimensional Euclidean Onofri inequality in terms of optimal constants, and state several consequences, including a sharp Poincaré inequality.

We continue the study of weak universality for the fractional hyperbolic $\Phi^4_3$-model initiated by Tzvetkov, Wang, and the author (2023). In this work, we establish weak universality in a regime where the associated Gibbs measure is mutually singular with respect to the underlying Gaussian measure. We first investigate the weak convergence of the Gibbs measures associated with macroscopic models to the fractional $\Phi^4_3$-measure constructed by Tzvetkov, Wang, and the author (2023). We then obtain a structured representation for these Gibbs measures via the stochastic quantization method. Finally, we prove dynamical weak universality for the fractional hyperbolic $\Phi^4_3$-model by using an extended second order expansion and exploiting the invariance of the Gibbs measures.

We provide a new rearrangement in Gauss space transforming symmetric sets while decreasing their Gaussian perimeter. We use this rearrangement to analyze the minimizers of the Gaussian perimeter among symmetric sets.

Motivated by three-dimensional wave diffraction by conical shapes, we consider the two-dimensional problem of wave scattering on the real sphere $\mathcal{S}^2$ resulting from a point-source incidence on a scatterer $\mathbb{S}$. Using the recently introduced notion of plane waves on $\mathcal{S}^2$ as well as recent results on the Laplace–Beltrami Green's function $G$ on $\mathcal{S}^2$, we derive three plane-wave representations for the scattered field $u$. We then explain how $u$ can be analytically continued to the complexified sphere $\mathcal{S}^2_{\mathbb{c}}$. This allows us to formulate a complexified version of Green's third identity, and to provide a characterisation of complex sources. All our results are then used to draw strong analogies between planar and spherical scattering problems, notably through the introduction of directivities.

We study the large-time behavior of the $L^2$-norm of solutions to the free wave equation on $\mathbb{R}^N$ and, more generally, their homogeneous Sobolev norms of negative order. We refine and extend the results of R. Ikehata [J. Hyperbolic Differ. Equ. 20 (2023)] who identified and quantified the infinite-time $L^2$-growth of free waves in dimensions $N=1,2$. For finite-energy data with $u_1\in L^1(\mathbb{R}^N)$, we derive exact first-order asymptotics: when $P=\int_{\mathbb{R}^N}u_1(x)\,dx\neq0$, the squared $L^2$-norm grows linearly for $N=1$ and logarithmically for $N=2$, with a leading coefficient depending only on $P$. In particular, no weighted moment assumption on $u_1$ is required; $u_1\in L^1\cap L^2$ suffices. More generally, we prove that the $L^2$-norm is bounded, and hence convergent, if and only if $u_1\in\dot H^{-1}(\mathbb{R}^N)$. We further establish first-order asymptotic profiles in negative-order homogeneous Sobolev spaces, covering polynomial growth, logarithmic growth, and convergence to a finite limit according to the Sobolev index. These results reveal a unified low-frequency mechanism underlying the low-dimensional $L^2$-growth. We also derive second-order asymptotic expansions in dimension one and at the critical Sobolev index in dimensions $N\ge2$. Finally, in dimension two, the exact $L^2$-asymptotics yield a sharper local energy decay estimate with an explicit upper asymptotic constant independent of the localization radius.

We study the recovery of a spatial coefficient in the principal memory flux of a three-dimensional wave equation with the prescribed exponential relaxation kernel $\beta e^{-\beta t}$, where $\beta>0$ is known. From a single finite-time lateral Cauchy measurement generated by one fixed, coefficient-independent initial displacement that is affine near the unknown region, with zero initial velocity, we establish global uniqueness and conditional $L^2$-Hölder stability. Two difficulties arise in this setting. The affine initial displacement makes the initial acceleration vanish in the unknown region, while the coefficient-difference equation contains a divergence-form source whose direct $L^2$ estimate loses a spatial derivative. We establish an acoustic Carleman estimate that controls the vector source and its time derivative without spatially differentiating the source. The affine condition identifies the relevant initial trace with a directional derivative of the coefficient difference, and the quadratic temporal vanishing of the source permits absorption in the weighted estimate. A transport estimate then yields stability uniformly over a bounded, uniformly coercive $W^{5,\infty}$ coefficient class, without an ordering condition or a smallness assumption near a reference coefficient.

We prove that, in every dimension, a bounded, non-constant entire solution of the Allen–Cahn equation whose zero set is contained in a half-space is a one-dimensional heteroclinic solution.

In this article, we prove a stability result for the identification of the coefficients appearing in a coupled relativistic Schrödinger system posed on a bounded domain of $\mathbb{R}^{1+n}$, where $n \geq 3$. More precisely, by utilizing input-output operator, where Neumann boundary data are available only on a subset of the boundary, and constructing an appropriate geometric optics solution, we derive a stability estimate of $\log$-$\log$ type for the matrix-valued first-order coefficients. This result is subsequently used to obtain a $\log$-$\log$-$\log$ type stability estimate for the zeroth-order matrix-valued potential.

This paper is concerned with the inverse scattering problem of determining a real-valued potential for the three-dimensional Schrödinger equation from multi-wavenumber data. Under suitable a priori bounds, we establish local increasing stability from near-field Dirichlet data on a fixed sphere for polynomially decaying potentials, and global increasing stability from full far-field amplitudes for exponentially decaying potentials. The stability estimates consist of a noise term and a remainder term, with the latter decreasing as the upper measured wavenumber increases. We also obtain global uniqueness from exact near-field data under weaker regularity assumptions. The analysis combines analytic continuation, weighted resolvent estimates, and half-ray identities whose differences at chord endpoints cancel the unknown exterior contribution.

Discretization can change a sharp stability threshold even when the continuum threshold is recovered at infinity. We study this phenomenon for the Hardy inequality on the integer lattice in dimensions at least three: its best coefficient agrees with the continuum value through dimension eight and is smaller from dimension nine onwards. We characterize this transition by the total critical defect, the summable error obtained by testing the lattice operator against the critical continuum power. A theta-function representation proves that the defect is negative and strictly decreasing from dimension nine; explicit positive Hardy remainders establish its positivity in dimensions three through eight. An exact energy identity relates this defect to the energy saved by optimizing the profile outside the neighbours of the origin. This exterior remains stable in dimensions eight and nine, locating the first loss of positivity in its coupling to those neighbours. We give self-contained proofs of both dimensional ranges. We then describe how local attraction creates a negative eigenvalue in a general class of lattice operators with critical inverse-square tails. Its magnitude is exponentially small near the threshold, with both the rate and the multiplicative prefactor determined by zero-energy solutions. The result allows connected finite modifications of the lattice and nonradial corrections to the tail. The same analysis controls the spread of the emerging state and gives a two-dimensional return scale for the associated Markov chain in every ambient dimension.

It is a classical fact in spectral geometry that the eigenvalues of the Dirichlet Laplacian depend continuously on the geometry of the underlying domain, under suitable geometric restrictions. We show that this theoretical property is reflected in a concrete numerical situation: the convergence rate of iterative solvers for convection-diffusion problems remains stable under deformations of the domain. The connection is a superlinear estimate for preconditioned GMRES/GCR in which the relative residual errors are bounded by an explicit expression of the eigenvalues of the Laplacian, or of a symmetric positive elliptic operator in divergence form. We prove that this bound depends continuously on the domain, uniformly in the iteration index, under either regularity hypotheses on the boundary or a topological constraint in dimension two. We also establish corresponding statements on the discrete level. Two examples delimit the theory: domains bounded by a fractal curve, where continuity still holds, and perforated domains, where it fails. The theoretical results, including both examples with rough domains, are confirmed by numerical experiments.