Papers

In this note, we prove a Liouville type rigidity result for smooth entire solutions of the $k$-Hessian equation. The proof makes use of a concavity inequality for the $k$-Hessian operator.

We prove that the full Dirichlet-to-Neumann map on a finite time interval globally determines the Lorentzian metric \(\mathbf G_{g,c}=-c(x,t)^2\mathrm{d}t^2+g(x)\) encoded by the principal symbol of the wave operator. The result holds in spatial dimensions \(n\geq3\) for multiplicatively separable wave speeds \(c(x,t)=a(x)b(t)\), provided the accumulated effective time during the experiment exceeds the maximal travel time from the boundary to the interior and back. The data determine both the unknown Riemannian metric \(g(x)\) and the wave speed \(c(x,t)\) throughout the observation cylinder, up to a spatial change of coordinates that fixes the boundary and leaves the measured time unchanged. In particular, the temporal factor \(b\) is not prescribed but is recovered from the same measurements. We also construct several examples demonstrating that the dimensional and visibility conditions for uniqueness are sharp.

For the 3D semilinear Euler-Poisson-Darboux equation $\square u+\frac{\mu}{t}\partial_tu=|u|^p$, where $t\geq1$, $\mu>0$ and $p>1$, it is conjectured that there is a critical exponent $p_{crit}(3,\mu)=\max\{p_s(3+\mu), p_f(3)\}$ with the Strauss exponent $p_s(3+\mu)=\frac{\mu+4+\sqrt{\mu^2+16\mu+32}}{2(\mu+2)}$ and the Fujita exponent $p_f(3)=\frac{5}{3}$ such that when $p>p_{crit}(3,\mu)$, the small data solution $u$ exists globally, otherwise, when $1<p\le p_{crit}(3,\mu)$, the solution $u$ can blow up in finite time. It is pointed out that for $1<p\le p_{crit}(3,\mu)$, the blowup of solution $u$ has been shown. However, it is still open for the global existence of small solution $u$ when $p>p_{crit}(3,\mu)$. Note that $p_{crit}(3,\mu)=p_s(3+\mu)$ for $0<\mu<\frac{14}{5}$ and $p_{crit}(3,\mu)=p_f(3)$ for $\mu\geq\frac{14}{5}$. In the recent paper [16], the authors have obtained the global small solution $u$ for $p>\max\{\frac{5}{3},1+\frac{2}{\mu}\}$ and $\mu\geq\frac{14}{5}$. In this paper, by utilizing the hypergeometric Riemann representation and establishing some delicate pointwise spacetime weighted estimates, we prove the global existence of small data radial solution $u$ in the remaining range of $\frac{5}{3}<p\leq1+\frac{2}{\mu}$ and $\frac{14}{5}\leq\mu<3$. Therefore, for the radially symmetric case and $\mu\geq\frac{14}{5}$, the global existence problem of small data solution $u$ is solved when $p>p_{crit}(3,\mu)=\frac{5}{3}$.

This paper studies global existence for the Cauchy problem of nonlinear Klein-Gordon equations in three space dimensions, strictly within the regularity regime of classical local existence. We prove it via higher-order and lower-order energy estimates. The proof relies on two key ingredients. The first is due to the work of Georgiev and Popivanov, which reduces quadratic nonlinearities to cubic terms plus ghost-energy-controllable terms, securing lower-order estimates. The second is a sharp Klainerman-Sobolev-type inequality without the scaling operator, established herein, which yields enough time decay for derivatives up to second order; integration by parts then controls derivative loss terms in the higher-order estimates.

We establish stable cusp formation for the Novikov equation, a cubically nonlinear Camassa–Holm-type equation. We identify an open set of smooth initial data for which the first gradient blow-up produces a cusp with sharp Hölder regularity $C^{7/9}$. This result shows that, in nonlocal wave-breaking problems, the sharp regularity of the cusp is not determined by the nonlocal or nonlinear structure alone. While the conserved $H^1$-type quantity excludes the $C^{1/3}$ cusp associated with Burgers-type gradient blow-up, the precise Hölder exponent is selected by the coupling between the nonlocal term and the algebraic structure of the nonlinearity. In the Novikov equation, this coupling yields the exponent $7/9$, rather than the $3/5$ exponent known for the Camassa–Holm and Hunter–Saxton equations. The main difficulty is that the naive high-frequency limit retains the cubic character of the equation and therefore does not exhibit a self-similar leading flow. We overcome this by introducing a Galilean-type change of variables around a nonzero background, which reveals a quadratic Hunter–Saxton-type leading equation. Its self-similar profiles determine the $C^{7/9}$ cusp, while the nonlocal and cubic remainders are controlled perturbatively in modulated similarity variables.

Let \(u>0\) be the first Dirichlet \(p\)-eigenfunction on a bounded convex domain \(\Omega\subset\mathbb R^N\), and set \[ X_\Omega:=|\nabla\log u|^{p-2}\nabla\log u. \] We study whether the sharp one-dimensional two-point modulus for \(X_\Omega\), which for \(p=2\) reduces to the logarithmic-gradient estimate of Andrews and Clutterbuck, persists for \(p\neq2\). We prove sharp estimates on intervals and balls for every \(p>1\), with the radial modulus on balls strictly larger than the one-dimensional one. In dimensions \(N\ge2\), however, the one-dimensional modulus fails on general convex domains for every \(p\neq2\). For \(1<p<2\), at every sufficiently small fixed scale there are smooth uniformly convex domains for which the corresponding two-point flux tends to zero. For \(p>2\), on thin domains \(\Omega_\varepsilon=D\times(-\varepsilon,\varepsilon)\), with \(D\subset\mathbb R^{N-1}\) bounded and convex, the normalized first eigenfunctions satisfy \[ u_\varepsilon(x,\varepsilon z)\longrightarrow \frac{\phi(z)}{\phi(0)} \left(\frac{G_D(x)}{G_D(x_0)}\right)^{2/p} \] locally uniformly in \(D\times(-1,1)\), where \(\phi\) and \(G_D\) are the first Dirichlet eigenfunctions of the \(p\)-Laplacian on \((-1,1)\) and of the Laplacian on \(D\), respectively. This yields the failure for \(p>2\). Finally, for arbitrary \(C^2\) functions, positivity of the symmetric differential of the \(p\)-gradient implies convexity, and the converse holds universally if and only if \(p=2\).

We develop a high-order tensor neural network (TNN) method for locally periodic elliptic multiscale problems of the form $-\nabla\cdot(A(x,x/\varepsilon)\nabla u_\varepsilon)=f$. Because the coefficient depends on both the slow variable $x$ and the fast periodic variable $y=x/\varepsilon$, the high-order cell problems and macroscopic corrector equations are more involved than in the classical case $A=A(y)$, and the correctors depend parametrically on $x$. We derive a computable high-order two-scale expansion and prove an $H^1$ convergence estimate for the partial expansion in boundary-layer-free settings, including periodic domains and ideal boundary-matching configurations. The proof uses the recursive compatibility structure of the corrector hierarchy and a zero-mean oscillation estimate in $H^{-1}$. We then construct a TNN framework for the high-dimensional corrector problems. Its tensor-product structure permits deterministic one-dimensional quadrature for the cell problems, homogenized coefficients, macroscopic source terms, and loss functions, avoiding Monte Carlo integration error. The numerical realization assumes that the coefficient entries and assembled data admit finite or controlled tensor-product representations; this computational assumption is separate from the general matrix-valued coefficient class used in the analysis. Experiments with scalar locally periodic coefficients show accurate high-order correctors. The $H^1$ semi-norm errors are consistent with the proved estimate, while point-normalized $L^2$ errors display the nominal high-order behavior predicted by the formal expansion.

We obtain apriori estimates for positive solutions of some quasilinear elliptic problems in a half-space. Using these estimates and other techniques, we derive new nonexistence results for the respective Dirichlet problems.

We introduce an exact Rayleigh reduction for the direct construction and detection of turning bifurcations in nonlinear parameter-dependent equations. A generalized Rayleigh functional ordinarily provides only a scalar necessary constraint on solutions. We show that, along a suitable one-parameter transformation orbit $u=\mathcal G_s\phi$, it can instead become the exact parameter law $\lambda=\mathcal R(\mathcal G_s\phi)$ of a genuine solution branch. Turning points are then found from nondegenerate critical points of this one-dimensional Rayleigh profile. The branch tangent automatically yields a kernel direction of the state linearization, and under the usual Fredholm, kernel-simplicity, and parameter-transversality assumptions the detected point is a simple fold. For Kirchhoff equations, amplitude scaling on bounded domains and spatial dilation on $\mathbb R^N$ yield explicit exact branches, global parameter thresholds, and countable hierarchies of turning values. For the generalized Kirchhoff law $M_b(A)=a+bA^\theta$, the branch geometry is governed by the dimension–homogeneity index $\theta(N-2)-2$: an interior turning bifurcation occurs exactly when $\theta(N-2)>2$. This gives a universal bifurcation trichotomy that is independent of the particular Berestycki–Lions nonlinearity.

In this paper, we investigate the uniqueness and asymptotic behavior of the stationary Navier-Stokes equations in a slab domain with no-slip boundary conditions. Specifically, under the given external force and the small Poiseuille flow assumptions, we prove that there is an H1 solution, and obtain the pointwise decay around the Poiseuille flow at far field. Furthermore, if the force is small, the solution is shown to be unique. The key point of the proof is the estimate of the Dirichlet integral of the solutions in the truncated domains and the Stokes regularity estimates.

We prove a Bernstein theorem for $\Phi$-anisotropic minimal hypersurfaces in dimensions $1\leq n\leq 7$ that the only entire smooth solutions $u$ of $\Phi$-anisotropic minimal hypersurfaces equation are affine functions provided the anisotropic area functional integrand $\Phi$ is sufficiently $C^{2}$–close to the Euclidean area integrand. This settles the $C^2$ entire–graph version of anisotropic Bernstein problem posed by Mooney and Yang \cite{MooneyYang2024}, and the proof uses a compactness–rigidity argument combined with Figalli's regularity theorem established in \cite{Figalli2017}.

In this paper, we prove the absence of the Lavrentiev phenomenon for a general class of parabolic double phase functionals with Orlicz growth. The energy density is given by $$ G(|Dw|)+a(x,t)H(|Dw|), $$ where $G$ and $H$ are Young functions satisfying the $\Delta_2$ and $\nabla_2$ conditions with $G\prec H$, and $a(\cdot)$ is a continuous nonnegative coefficient. Under suitable balance conditions between the growth gap of $G$ and $H$ and the modulus of continuity of $a(\cdot)$, we show that every finite-energy map can be approximated locally by smooth functions without loss of energy. The result extends the known parabolic double phase theory from power type growth to a broad Young function framework and identifies the natural space-time Orlicz energy class for the problem.

We consider the Stokes operator $A$ on smooth exterior domains $\Omega$ of $\mathbb{R}^n$ in homogeneous Sobolev spaces $\widehat H^{\kappa,q}_w(\Omega)$ with radially symmetric Muckenhoupt weights $w\in \mathscr A_q$. A fundamental property is the existence of a bounded $\mathscr H^\infty$-calculus of the Stokes operator on weighted nonhomogeneous and homogeneous $L^q$ Sobolev spaces. This property implies the existence of uniformly bounded purely imaginary powers $A^{it}$, $t\in\mathbb{R}$, and the characterization of domains of fractional powers $A^\theta$ equipped with nonhomogeneous ($\| u\|_{L^q_w} + \|A^\theta u\|_{L^q_w}$) as well as homogeneous norm ($\|A^\theta u\|_{L^q_w}$) as complex interpolation spaces. The final aim is the identification with homogeneous spaces $\widehat{\mathcal D}((-\Delta_{q,w})^\theta) = [L^q_{w},\widehat{\mathcal D}(-\Delta_{q,w})]_\theta$ intersected by a space of solenoidal vector fields. Moreover, we obtain weighted variational inequalities for weak solutions of the Stokes equations, weighted $L^q$-$L^r$ decay estimates of the Stokes semigroup and $L^p$-maximal regularity on $L^q_{\sigma,w}(\Omega)$.

In this paper we consider the Navier-Stokes equations in exterior domains of $\mathbb{R}^n$, $n\geq 3$, with a periodically in time moving boundary $\partial\Omega(t)$ and external force $f(t)$. For this case we prove the existence of a locally unique mild time periodic solution in weighted function spaces with radially symmetric Muckenhoupt weights. The solutions split into a stationary part controlled by potential theoretic estimates and a purely oscillatory part constructed as mild solution via analytic semigroup theory. To deal with perturbation terms of even second order - coming from a coordinate transform and the moving boundary - in weighted, homogeneous Sobolev spaces a maximal $L^1$ type regularity estimate will be used in weighted Lorentz spaces. To control the convective term an $\mathcal H^\infty$-calculus in weighted spaces of the Stokes operator, its $BIP$ property and embedding estimates of fractional powers are exploited, see a recent paper by the authors: The Stokes operator on exterior domains in homogeneous weighted function spaces: From weak theory to $\mathscr H^\infty$-calculus to fractional domains (2025).

We prove the optimal global regularity of admissible solutions to a transformed very fast diffusion equation in the range $-1<p<0$, posed on smooth bounded domains with zero Dirichlet boundary data and initial data comparable to the distance function. More precisely, we establish existence and uniqueness and show that solutions belong to $C^{1,p+1}(\overline\Omega)$ in space for every positive time and are $C^\infty$ in time uniformly up to the boundary. Moreover, all their time derivatives belong to $C^{1,p+1}(\overline\Omega)$, and the exponent $p+1$ is optimal. These regularity estimates further yield fine long-time asymptotics toward the friendly giant solution, including a first-order expansion in the $C^{1,p+1}(\overline{\Omega})$ topology and an improved convergence rate for the relative error in $C^{p+1}(\overline\Omega)$.

We establish Schauder estimates for linearized very fast diffusion equations with Dirichlet boundary conditions in bounded smooth domains. These estimates provide an important ingredient for deriving optimal boundary regularity and long-time dynamics of solutions to very fast diffusion equations.

This paper studies the 2D Navier–Stokes equation on a bounded domain with Navier-slip boundary conditions, driven by spatially localized stationary forcing generated by a stochastic heat equation. We prove exponential mixing for the associated Navier–Stokes–heat system, and consequently exponential convergence of the velocity law under stationary forcing. The main difficulty is that the white noise is confined to a subdomain and reaches the velocity indirectly through the heat component. To address the degeneracy, the proof combines Malliavin calculus with PDE control theory. The key ingredient is a stabilization scheme that converts localized Navier–Stokes controls into time-regular controls compatible with the heat dynamics.

We introduce a new class of algebras with variable operator-valued structural constants and develop elements of the theory of monogenic functions in such algebras. Compatibility conditions for the structural operators are established. It is shown that the components of monogenic functions satisfy second-order operator-differential equations with variable coefficients, and converse results are proved. The proposed approach provides a constructive method for obtaining solutions of various differential, integro-differential, and functional-differential equations. Several illustrative examples are presented.

We introduce a new class of algebras with variable structural constants and develop elements of the theory of monogenic functions in such algebras. Compatibility conditions for the structural functions are established. It is shown that the components of monogenic functions satisfy certain second-order partial differential equations with variable coefficients, and converse results are proved. This approach provides a constructive method for obtaining explicit solutions of such equations. Several illustrative examples are presented.

We present a numerical formulation for the resolution of finite systems of interacting edge dislocations and wedge disclinations. The approach is based on a finite element discretization of a fourth-order elliptic boundary value problem arising from the mechanical equilibrium equations of plane-strain linear elasticity in the presence of kinematic incompatibilities. The numerical implementation follows a continuous interior penalty discontinuous Galerkin framework and relies on the solution of a finite set of local cell problems. Our method is validated against analytical benchmarks and used to explore several non-trivial dislocation-disclination configurations, illustrating how translational and rotational incompatibilities shape the stress state of the body.

To describe the evolution of grain boundaries based on the underlying microscopic mechanisms of line defects (disconnections) and the integrated effects of a diverse range of thermodynamic driving forces, Zhang, et al. in 2017 formulated a continuum equation. We prove the global-in-time existence, uniqueness, and regularity of the weak solution to an initial-boundary value problem for this model. The existence, uniqueness of the stationary solution are also established. Finally, we investigate the large-time behavior of the weak solution of the evolution problem, and show that the solution converges to the stationary solution in a suitable sense. Numerical simulations are carried out to validate the analysis results. The main difficulties in the proof of main theorems are due to a non-local term with singularity, a non-smooth coefficient of the highest derivative associated with the gradient of the unknown, and the special form of free energy which is not uniformly bounded from below. The key ingredients in the proof are the energy method, an estimate for a singular integral of the Hilbert type, Fourier transform of Hilbert transformations, and an estimate with a weight, for time-derivative of the unknown.

We prove a sharp nonlinear analogue of the Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation \[ u_t = \Delta u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u),\ \ \ 1<p<\infty, \] on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpreted in the viscosity sense on the critical set of the solution. We also derive a sharp global Harnack inequality from the gradient estimate. The constant is best possible. Equality holds identically for an explicit self-similar solution in $\Rn$, and sharpness on closed manifolds follows by taking the limit along the flat tori $\mathbb R^n/(L\mathbb Z)^n$ as $L \to \infty$. The proof rests on an exact Bochner-type identity for a family of uniformly parabolic approximating flows. We choose the first-order term separately from the second-order regularization so that the maximum principle yields the sharp constant, uniformly in the regularization parameter and without any assumption on the critical set of the solution. Using this Bochner identity with a modified Li-Yau functional, we extend the estimate to closed manifolds with $\operatorname{Ric} \ge -\kappa$.

We introduce a notion of continuity equation on metric spaces that is capable of describing curves of probability measures which are absolutely continuous, and more generally of bounded variation (BV), with respect to the 1-Wasserstein distance. This continuity equation is based on a notion of measure-valued derivations, whose basic theory is also developed in this paper. On $\mathbb{R}^n$, our formulation is consistent with the continuity equation with singular flux introduced by Almi–Rossi–Savaré (arXiv:2506.15333), including the corresponding notion of minimal solutions. In this work, we characterize BV-curves in the space of probability measures equipped with the (extended) 1-Wasserstein distance as those curves satisfying the continuity equation with a measure-valued derivation of finite mass. To this aim, we extend our previous work (Calc.Var.(2024)63:16) on probabilistic representations on BV-curves and construct from them measure-valued derivations (resp. flux measures) on geodesic metric spaces (resp. on $\mathbb{R}^n$).

We derive necessary conditions for instantaneous dynamo action for rotating convection. A magnetohydrodynamic model is considered in two settings: the rapidly rotating plane layer where inertia and viscosity are absent, and at an arbitrary rotation rate where viscosity is finite. In contrast to kinematic dynamo bounds, the evolution of the magnetic field is coupled via an inertialess force balance. The buoyancy-driven part of the flow $\mathbf{u}^{\mathrm{A}}$ in the event of dynamo action must in fact satisfy, for $3\leq p \leq \infty$ $$ Rm\, A_p\| \mathbf{u}^{\mathrm{A}}\|_{L^p} \geq 1 $$ where $A_p$ is an explicit constant, and $Rm$ is the magnetic Reynolds number. In the inviscid model, $\mathbf{u}^{\mathrm{A}}$ depends only on the horizontal gradients of the vertical primitive of temperature. A refinement via the poloidal-toroidal decomposition allows us to replace $L^p$ in our constraint with an anisotropic norm for $L^{\infty}_z \dot{H}^1_{x,y}$. For the viscous model, we also derive necessary conditions for the growth of magnetic enstrophy and a combined thermo-magnetic energy. One branch of our constraints implies that the scaling $Ra_\nu \gtrsim Ek^{-3/2}$ is necessary for dynamo action, where $Ra_\nu$ is the classical Rayleigh number and $Ek$ is the Ekman number.

We prove strong existence for Itô SDEs with diffusion coefficients that can introduce strong attraction to a submanifold. We extend and strengthen the Röckner-Zhao approach, which uses Malliavin calculus to establish compactness of the approximating solutions in Wiener-Sobolev space. At least when the diffusion coefficients are sufficiently regular in time, this provides an alternative to Krylov's recent proof of strong existence via analysis of the Itô-Duhamel series.

We study a coupled Stokes-reaction-diffusion-advection system in a three dimensional domain perforated by a large number of evolving spherical inclusions with radii of order $\varepsilon^\alpha$ for $\alpha\in(1,3]$. Their centers are separated on the order of $\varepsilon$ but are not assumed to form a periodic lattice. The inclusions grow or shrink through an interfacial adsorption-desorption mechanism, so that the evolution of the geometry is coupled to the Stokes-reaction-diffusion-advection system. We first establish well-posedness on arbitrary large time intervals for sufficiently small $\varepsilon$. We then derive quantitative homogenization limits in terms of the limiting empirical measure describing the spatial distribution and size of the inclusions. Its evolution is governed by a continuity equation in the radius variable, and the convergence of the empirical measures is controlled in the $2$-Wasserstein distance. In the critical case, $\alpha=3$, the Stokes system converges to a Brinkman system, while non-vanishing concentration boundary layers modify the microscopic exchange law for the reaction-diffusion-advection equation. For $\alpha\in(1,3)$, the limit is of Darcy type and the original exchange law is retained. The homogenization results are quantitative and provide explicit error estimates.

We establish a weighted Wasserstein spectral gap for the three-dimensional damped cubic wave equation with genuinely finite rank Brownian forcing. Under a saturation condition, the gap holds with respect to the negative phase topology $\mathcal E_s=H^{-s}\times H^{-1-s}$ for every $0<s<1/2$, from which we deduce unique ergodicity and exponential mixing in the energy topology. Sharp geometric characterizations of the saturation condition are also obtained. The method we develop is a stable–compact asymptotic coupling mechanism for hypoelliptic dissipative SPDEs beyond the parabolic setting. Instead of relying on positive time smoothing or asymptotic gradient estimates, it reduces the infinite-dimensional obstruction to contraction to a compact defect in a weaker coupling topology. Dense Malliavin range then permits this defect to be compensated by a finite dimensional Cameron–Martin shift, producing a finite distance contraction on bounded Lyapunov cores. Together with a separate high energy contraction from dissipation, this yields a global weighted Wasserstein spectral gap.

This paper provides sharp rates of invasion in nonlinear integro-differential equations of ignition type. The framework covers both integrable convolution kernels and singular fractional-type kernels. While the emergence of a threshold on the decay $1+2s$ of the jump kernel is expected from earlier works, far fewer quantitative results were available in such a general framework. In particular, in addition to the spreading rates, our study provides information on the shape of the invasion profiles. Importantly, we obtain the first sharp spreading estimate in the critical case $s=\frac12$, which separates linear-in-time and accelerated regimes.

We prove the uniqueness, asymptotics, symmetry, and orbital stability of energy ground states for 3D magnetic Gross-Pitaevskii equations under mild conditions on the electric potential and magnetic field. In particular, both the electric potential and the magnetic field are allowed to have singularities, and we cover in a unified approach the physically relevant Aharonov-Bohm magnetic field as well as the constant magnetic field. Since we are in the 3D case, the unique energy ground state (up to a phase factor) is obtained as a local minimizer, rather than a global one, by restricting the kinetic energy of candidate critical points within a suitable range. The qualitative analysis of the energy ground state we carry on is mainly based on a related Pohozaev identity, the implicit function theorem, and variational methods.

On doubling metric measure spaces, we study nonlocal operators with non-standard $(p,q)$-Orlicz growth ($1<p\leq q<\infty$), where the interaction kernel is given by a general, non-translation-invariant measure. Under natural assumptions on the interaction measure - namely symmetry, a suitable nonlocal Poincaré inequality, and a tail bound - we prove that every weak solution to the corresponding homogeneous nonlocal equation admits a locally Hölder continuous representative. These regularity results are new even in the Euclidean setting.