Papers

This paper intends to understand the regularity and stability problem on the 3D incompressible magneto-micropolar equations with zero magnetic and angular viscosities in a strip domain. The magneto-micropolar system models the electrically conducting micropolar fluid in the presence of a magnetic field. The lack of magnetic diffusion and angular dissipation makes it impossible to prove even small data global well-posedness result, let alone general large data global regularity. This paper presents a steady-state setup around which any perturbations can be shown to be globally regular and stable. More precisely, any small perturbation near a steady magnetic field perpendicular to the horizontal boundary leads to a unique global classical solution. In addition, the solution is shown to converge to the steady state at an almost exponential rate as time goes to infinity. These appear to be the very first rigorous global results on the magneto-micropolar equations concerned here.

We consider the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data. We establish an improved decay estimate of the global solutions under the zero-mass condition. Moreover, we derive a self-similar asymptotic profile of the global solutions. This result shows that the improved decay rate is optimal and provides a necessary and sufficient condition for attaining this decay rate.

We provide a classification result of positive solutions with a non-removable singularity at the origin to the subcritical Lane-Emden equation $-\Delta_p u = u^q$ in $\mathbb{R}^N \setminus \{0\}$, where $p\in(1,N)$, $q\in (p-1,p^*-1)$ and $u^{q-p+1} \in L^{\frac{N}{p}}(\mathbb{R}^N \setminus B_R)$.

The dynamics of polynomial roots under repeated differentiation has recently been conjectured to converge to a limiting measure governed by specific nonlinear PDEs, the conjectures being shown in some particular settings. For rotationally invariant initial distributions, a deterministic structured sampling model placing roots on concentric circles was recently introduced by Galligo, Najnudel, and Vu. In this paper, the authors proved convergence under the technical growth condition $m_n / (n \log n) \to \infty$, where $n$ is the number of circles and $m_n$ is the number of points per circle. In this paper, we significantly improve this result by relaxing the growth condition to $m_n / \log n \to \infty$, thus allowing for regimes where the number of points per circle grows proportionally to the number of circles. The key innovation is a refined upper bound on the root magnitudes after differentiation. This sharper estimate prevents the rapid accumulation of errors over multiple differentiations, fully validating a recent conjecture regarding the robustness of the sampling scheme.

Spectral and transport properties of high-contrast resonator systems can be described in the subwavelength regime in terms of the so-called capacitance operator. In this paper, we consider an infinitely periodic chain of high-contrast resonators in three dimensions. The first result is a precise estimate of the off-diagonal decay rate of the capacitance operator $C$. Importantly, we demonstrate that the decay rate is long-range and critical: as $|n-m|\to\infty$, \begin{equation*} C(n,m)\sim \frac{1}{|n-m|\log^2|n-m|}, \end{equation*} which is $\ell^1$ summable but slower than quadratic. This borderline decay of the off-diagonal entries makes the present proof of Anderson localisation with arbitrary disorder, which is observed numerically in this paper, out of reach; we hope that this physical example of classical wave systems with critical long-range interactions provides new insight in the field of Anderson localisation. As the second main result, based on the off-diagonal decay estimate, we prove a strong convergence of the finite capacitance operator, which corresponds to a truncated chain, to the capacitance operator as the size of the truncated chain grows to infinity. Using this strong convergence, we improve the results of [Ammari et al., SIAM J. Math. Anal., 2023 and Bull. London Math Soc., 2025] by presenting a rigorous estimate of the convergence rate of the spectrum.

We study $k$-equivariant energy critical wave maps $\mathbb{R}^{1+2} \to \mathbb{S}^2$, for any equivariance degree $k\ge 2$. We prove that the radiation associated with any finite-energy blow-up solution cannot satisfy a certain regularity condition; in particular, it cannot be smooth. The assumption $k \geq 2$ is necessary, since for $k = 1$ such solutions are known to exist. The starting point of our analysis is the soliton resolution theorem. The key ingredient is a novel application of the modulation method, in which we compare the effects of the radiation and inner bubbles to study the dynamic behavior of the widest bubble.

We study integrability properties of the Wolff potential in context of Choquet integrals with respect to the Hausdorff content. As an application we give integrability results to the solutions of the $p$-Laplace equation in this context.

In this paper, we consider the Cauchy problem for the $3$D incompressible Navier–Stokes equations and prove the existence of unique global solutions in the framework that the horizontal component of the velocity field is small in some critical Besov spaces including the classical Fujita–Kato class, while the vertical component is large in the wide class $\dot{B}_{\infty,\sigma}^{-1}(\mathbb{R}^3)$ ($1 \leq \sigma < \infty$) where the Navier–Stokes equations are known to be ill-posed in.

This paper is mainly concerned with the large-time dynamics of bounded nonnegative solutions of reaction-diffusion equations on the real line with nonlinearities mainly of the bistable-type. We first consider initial data of the type $\alpha \mathbf{1}_I$, namely scalar multiples of indicator functions of intervals $I$. For each amplitude $\alpha$, the existence of a threshold length $L^*(\alpha)$ separating the extinction and the persistence of the solutions at large time is known. We here address the question as to whether the limit of the threshold sizes $L^*(\alpha)$ as $\alpha\to+\infty$ is positive or zero. We provide sufficient conditions under which this limit is positive, and others under which it is zero. Secondly, when the limit is positive, we show that some fragmented initial data, which are equally distributed as $\alpha \mathbf{1}_I$, give rise to solutions persisting at large time, whereas the solutions emanating from $\alpha \mathbf{1}_I$ go to extinction at large time. This result shows an unexpected favourable effect of the fragmentation of the initial datum on the large-time dynamics. Last, we consider the mass concentration principle for parabolic equations, which states that nonnegative solutions can be controlled from above in an integral sense by the solutions emanating from the Schwarz symmetrically decreasing rearrangements of the initial data. Since the pioneering results of [Bandle, 1976] and [Alvino, Trombetti, Lions, 1990], this principle is known to hold in bounded domains with Dirichlet boundary conditions under different types of assumptions of the coefficients of the equation. We show that this mass concentration principle is not valid in general, even for simple equations of the type $\partial_t u = \partial_{xx} u + f(u)$.

We investigate steady compressible Euler flows in two-dimensional infinitely long nozzles, where the piecewise smooth upstream data at infinity admits a characteristic discontinuity. Except the subsonicity condition, no additional constraints are imposed on the data. We establish the existence and uniqueness of subsonic weak solutions associated with a smooth contact discontinuity curve. The original problem is reformulated into an elliptic equation in divergence form with discontinuous coefficients, such that the contact discontinuity conditions are inherently preserved in the solution of the elliptic problem. We further investigate the downstream asymptotic behavior and show that the convergence rate of the flow matches that of the nozzle walls.

We study the local Lipschitz regularity of local minimizers for a class of degenerate orthotropic functionals with $\varphi$-growth, where $\varphi$ is a general N-function. Unlike standard isotropic functionals, the ellipticity of the associated Euler-Lagrange equation degenerates separately in each coordinate direction, presenting significant anisotropic difficulties. Furthermore, the general N-function setting lacks the algebraic scale invariance available in the classical orthotropic $p$-Laplacian case. Despite these structural difficulties, we prove that local minimizers are locally Lipschitz continuous. Our approach relies on a regularized approximation scheme, mixed-direction Caccioppoli inequalities, and a carefully designed Moser-type iteration that incorporates an interpolation argument to bridge the gaps between consecutive integrability exponents.

We study the Cauchy problem for biwave type equations with coefficients depending only on time. Assuming $\mathcal{C}^\kappa$ regularity of the principal symbol and suitable Levi conditions on the lower order terms, we establish well-posedness in Gevrey spaces.

We consider the interaction of a general viscous compressible and heat-conducting fluid with an elastic shell located at the boundary of the fluid's domain. The fluid is described by the compressible Navier-Stokes-Fourier equations and the shell evolves in accordance with a viscoelastic beam equation. Both are coupled through kinematic boundary conditions and the balance of forces. Our first result is the local-in-time well-posedness of the underlying coupled system of nonlinear PDEs in smooth function spaces. Eventually, we prove a conditional regularity criterion which is in the spirit of the Beale-Kato-Majda condition for the incompressible Euler equations combined with the boundedness of density and temperature. It rests upon a weak-strong uniqueness result for the underlying system.

We study the obstacle problem for the parabolic fractional $p-$Laplace equation \[\partial_t u+(-\Delta_p)^su = 0\] in the degenerate range $2<p<2/(1-s)$. We prove that viscosity solutions are locally Lipschitz continuous in space and Hölder continuous in time. If, in addition, $p>1/(1-s)$, the time regularity improves to Lipschitz continuity.

We establish quantitative unique continuation estimates for solutions of the complex Ginzburg-Landau equation in the framework of two-sphere and one-cylinder inequalities. While prior work [Dou et al. SIAM J. Control Optim. (2023)] relied on Carleman estimates, the present study develops a novel logarithmic convexity approach to prove the quantitative unique continuation property. We derive an explicit quantitative unique continuation constant and fully characterize its dependence on the parameters.

We study complex $m$-Hessian equations on bounded hyperconvex domains with right-hand side in $L^1(\Omega)$. The main contribution of this paper is a strong stability result for weak solutions in the Hessian energy sense. More precisely, if $f_j \to f$ in $L^1(\Omega)$ and $u_j, u$ are the corresponding solutions, then \[ \int_\Omega |u_j-u|\, H_m(u_j) \longrightarrow 0. \] This provides convergence in the natural energy topology associated to the Hessian operator, which is significantly stronger than convergence in capacity. For completeness, we also recall the existence of solutions and stability in capacity, which follow from known results in the literature.

We consider an inverse source problem for the $k$-Hessian equation \begin{equation*} \sigma_k(D^2u)=f(x) \end{equation*} on the $k$-admissible branch in a smooth uniformly convex domain in $\mathbb R^n$, where $2\le k\le n$. We prove that the nonlinear Dirichlet-to-Neumann map determines the positive smooth source from its values on the restricted large-data rays $t\phi_E|_{\partial\Omega}$, where $E\in Gr(k-1,n)$, $\phi_E(x)=|P_Ex|^2/2$, and $t$ is sufficiently large. The Hessian of each boundary profile has exactly $k-1$ large directions and is flat on $V=E^\perp$. Thus, the leading profile lies on a rank $k-1$ face of the $k$-Hessian structure, and the first source-dependent correction is governed by the missing directions. More precisely, this correction solves fiberwise Poisson equations on the affine sections $\Omega\cap(y+V)$ of dimension $q=n-k+1$. We prove that these sectionwise solutions patch smoothly through glancing points where the sections collapse, and we obtain boundary normal derivative asymptotics by local barriers. The boundary flux of the correction gives the section integrals $\int_{\Omega\cap(y+V)}f\,d\mathcal H^q$. Varying $E$ yields the affine $q$-plane Radon transform of the zero extension of $f$. We give an explicit reconstruction formula through the Fourier slice identity, and the injectivity of the affine Radon transform gives uniqueness. The endpoint $k=n$ recovers the Monge–Ampère chord/X-ray geometry, while the range $n\ge3$ and $2\le k<n$ gives inverse source results for genuinely non-determinant Hessian equations.

We prove uniqueness for an inverse boundary problem for the prescribed Gaussian curvature equation $\det D^2u=K(x)(1+|\nabla u|^2)^2$ on a smooth bounded simply connected planar domain. We compare smooth positive curvatures on a common open class of Dirichlet values for which the convex problems are well posed. Equality of the nonlinear Dirichlet-to-Neumann maps, together with the prescribed first boundary jet of the curvature, implies equality of the curvatures throughout the domain. The first linearized equation has conductivity $\Sigma=K(D^2u)^{-1}$. The planar anisotropic conductivity theorem determines this tensor up to a boundary fixing diffeomorphism. We prove that the Hessian and gradient structure forces this diffeomorphism to be the identity. The difference between the pulled-back background gradient and the original gradient satisfies a local elliptic system whose components have the same scalar principal part and zero boundary Cauchy data. A first-order Carleman estimate gives unique continuation for this system. The argument uses the nonlinear boundary response at one admissible Dirichlet value and its first derivative there.

We consider the initial-boundary value problem for a variant of the Keller-Segel chemotaxis system with doubly degenerate diffusion, i.e. we study \[ \left\{ \begin{array}{ll} u_t = \nabla \cdot (uv\nabla u) - \nabla \cdot (u^2v\nabla v),\\ v_t = \Delta v + u - v,\\ (uv\nabla u-u^2v\nabla v)\cdot\nu=\nabla v\cdot\nu=0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \end{array} \right. \] in a smoothly bounded domain $\Omega\subset\mathbb{R}^2$. Crucially, we only assume the sufficiently regular initial data to be nonnegative, but allow those functions to be zero at non-trivial parts of the domain. We show that, despite possibly starting from a degenerate state, the system admits global solutions in a framework of generalized energy solutions, whenever the initial mass is below the threshold number $m_0=4\pi$. Moreover, in a radial setting the threshold number can be increased to $m_0=8\pi$.

We establish the existence of time-periodic solutions to the Boltzmann equation for the full range of soft potentials and prove their global stability. The proof relies on refined energy estimates combining Besov and Sobolev regularity with velocity-weighted energy methods. The analysis is complicated by the lack of a spectral gap in the soft potential regime together with the absence of time integrability of the periodic forcing over $\mathbb{R}^+$. This provides a framework for the study of time-periodic behavior for the Boltzmann equation.

We consider the 2D incompressible Euler equations on a periodic channel $\mathbb{T}\times (0,1)$ with inflow-outflow boundary condition $u=(0,1)$ on $\mathbb{T} \times \{0,1 \}$. We also impose the incoming vorticity boundary condition $\omega =\eta $ on $\mathbb{T}\times \{ 0 \}$, where $\eta$ is prescribed. We show that the problem is globally well-posed in Gevrey spaces (for any value of the Gevrey exponent $s>1$) as long as $\eta$ remains Gevrey. This proves that the inflow-outflow velocity boundary condition determines the solution locally in time if and only if the solution is considered in an analytic class. In particular, leaving the analytic class, nonuniquness of solutions occurs already in any Gevrey class, by prescribing $\eta$. Hence, prescribing an analytic inflow-outflow velocity leads to precisely one analytic and continuum $s$-Gevrey solutions for every $s>1$. Furthermore, the result implies that if $\eta$ is analytic, then the unique global solution can lose analyticity in space for all $t>0$, but remain $s$-Gevrey regular for all $s$.

There is an interesting open question: for $n$-D ($n\ge 1$) semilinear Euler-Poisson-Darboux equation $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $\mu>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,\mu)=\max\{p_s(n+\mu), p_f(n)\}$ with the Strauss exponent $p_{s}(n+\mu)=\frac{n+\mu+1+\sqrt{(n+\mu)^2+10(n+\mu)-7}}{2(n+\mu-1)}$ and the Fujita exponent $p_f(n)=1+\frac{2}{n}$. The blowup of weak solution $u$ has been shown when $1<p\le p_{crit}(n,\mu)$ meanwhile this open question has been solved for $n=1,2$. In the present paper, we focus on this open question for $n=3$ and establish the global existence of small data solution $u$ for $\mu\geq\frac{14}{5}$ (equivalent to $p_{crit}(3,\mu)=p_f(3)=\frac53$) and $p>\max\{\frac53, 1+\frac{2}{\mu}\}$.

We study irregular double-phase parabolic equations with variable exponents and non-divergence data, \[ u_t-\operatorname{div} \left(\mathcal{F}(z,\nabla u)\nabla u \right)=f(z),\quad z=(x,t)\in Q_T:=\Omega\times (0,T), \] under the homogeneous Dirichlet boundary conditions. Here, $\Omega \subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain, $T>0$, \[ \mathcal{F}(z,\nabla u)=a(z)|\nabla u|^{p(z)-2} + b(z) |\nabla u |^{q(z)-2} \] with given Lipschitz-continuous exponents $p,q$ that satisfy a suitable balance condition. The nonnegative coefficients $a(z), b(z)$ satisfy the inequality $a(z)+b(z)>0$ in $Q_T$, the space and time derivatives of $a$ and $b$ belong to $L^d(Q_T)$ with some $d$ depending on the data. If \[ f\in L^\sigma(Q_T) \quad \text{for} \ \sigma \in (2, N+2] \quad \text{and} \quad \mathcal{F}((\cdot,0),\nabla u_0)\,|\nabla u_0|^{r+2}\in L^1(\Omega), \] where \(0\le r\le K(N,\sigma,p,q)\) if \(\sigma<N+2\), while \(r\ge0\) is arbitrary if \(\sigma=N+2\), then the problem has a unique strong solution, for which we prove the global transfer of integrability from the initial data and the forcing term to the double-phase flux in the spirit of Calderón-Zygmund theory, higher integrability of the gradient, and the second-order space regularity: \[ \begin{split} & \text{$\mathcal{F}((\cdot,t),\nabla u(\cdot,t))|\nabla u(\cdot,t)|^{r+2}\in L^1(\Omega)$ for a.e. $t\in (0,T)$}, \\ & \text{$|\nabla u|^{2(\min\{p(z),q(z)\}-1)+r+s}\in L^1(Q_T)$ for every $s\in\left(0,\frac{4}{N+2}\right)$}, \\ & \mathcal{F}(z,\nabla u)|\nabla u|^{\frac{r+2}{2}} \in L^2(0,T;W^{1,2}(\Omega)). \end{split} \] The results improve and complement the results in \cite{Arora-Shmarev-JGA-2026} and extend them to the full range $r \geq 0$.

This paper concerns a conformally invariant elliptic problem on $\mathbb R^n$ driven by $(-\Delta)^{n/2}$ that has a nonlocal exponential nonlinearity of Choquard type. The problem under consideration is a nonlocal generalization of the constant $Q$-curvature problem on $\mathbb R^n$. We classify the asymptotic behavior at infinity of all solutions that satisfy a suitable integrability assumption. Under a growth restriction at infinity and a lower bound on the energy we provide an explicit classification for solutions. The classification is heuristically consistent with the classification of the corresponding local problem.

We show convergence of a continuous data assimilation method for the Oberbeck-Boussinesq system in the dimension $d=2,3$. Our working hypothesis is boundedness of the reference solution, while the synchronized solution satisfies the equations in a weak sense. The main tool is the relative energy inequality for stochastic problems.

The Vlasov-Hartree system is a mean-field model for a mixture of infinitely many interacting bosons and fermions where the bosons are described quantum mechanically and the fermions are described classically. This paper studies the well-posedness and dispersive properties of the Vlasov-Hartree system with initial data of arbitrary size. We prove that the Vlasov-Hartree system is globally well-posed in a low-regularity functional framework where the particle trajectories are meaningfully defined, but which includes discontinuous fermion densities. Moreover, when the interaction between the bosons and fermions is repulsive, we prove that the system exhibits dispersion in the form of time decay estimates for the particle densities and fields. When the interaction is attractive, we show that, at worst, the fields exhibit very mild growth in time.

We study a nonlocal continuity equation on $\mathbb{R}^d$ in which a probability density is driven by the competition between attraction toward a prescribed background measure $\omega$ and self-repulsion among particles, governed respectively by the power-law kernels $\psi_a(x) = |x|^{1+a}$ and $\psi_r(x) = |x|^{1+r}$ with exponents $a, r \in [0,1)$. We establish global Lagrangian well-posedness via a squared-radius regularization, obtaining {finite-time $L^\infty$ and moment bounds}, $W^{n,\infty}$ regularity, and uniqueness in the Lagrangian class. When the initial data is compactly supported and attraction dominates ($a > r$, or $a = r$ with $\omega(\mathbb{R}^d) > 1$), we prove that the support remains uniformly bounded at all time; a counterexample shows this fails for $a = r > 1$. For the attractive-dominant nonquadratic range $0 \leq r \leq a < 1$, we characterize zero-flux stationary states via a free-boundary problem involving a fractional Laplacian operator, reducing the stationarity condition to a fractional exterior Dirichlet problem. This characterization allows us to exhibit explicit examples of stationary measures in dimensions $d \in \{1,2,3\}$. Numerical particle simulations {are consistent} with the theoretical stationary profiles. Finally, for $0 \le r \le a < 1$, we prove that every global solution whose energy is bounded from below and whose moments are bounded uniformly in time converges, along a sequence of times, to a zero-flux stationary state. Full narrow convergence follows whenever the omega-limit set contains a single stationary state.

We study one-centre point interactions for the Dirichlet Laplacian on unbounded domains in dimensions two and three, with emphasis on exterior domains and special Lipschitz domains. These operators are singular perturbations constructed as self-adjoint extensions of the Dirichlet Laplacian restricted to functions vanishing at the interaction centre, and their resolvents are given by an explicit Kreĭn formula with a single extension parameter $\alpha$. The negative spectrum is completely characterized by a scalar equation and the critical coupling $\alpha$ separating binding from non-binding is the threshold limit of the Weyl function appearing in the Kreĭn formula. We establish domain monotonicity of the Weyl function, of the critical coupling, and of the unique negative eigenvalue when existing, and we derive sharp near-boundary asymptotics of the critical coupling in uniformly $C^{1,1}$ geometries. These estimates imply that, for every fixed coupling, nonpositive spectrum disappears when the interaction centre approaches the Dirichlet boundary. We also prove limiting absorption principles and purely absolutely continuous positive spectrum for a point-interaction in exterior domains case and in classes of special Lipschitz domains. We also analyze in depth several threshold phenomena. We show that the critical coupling is governed by the far-field behavior of the zero-energy Green function: exterior domains give threshold resonances, domains contained in a three-dimensional half-space give threshold eigenvalues, the half-plane gives a $p$-wave resonance, and planar wedges exhibit types of threshold states that are aperture-dependent. Finally, low-energy resolvent expansions are computed in the model cases and persistence or disappearance of eigenvalues at threshold are studied. The present paper seems to be the first systematic work on the subject.

We consider an elliptic problem with slightly subcritical nonlinearities at the interior and on the boundary; the nonlinearity at the interior is also depending on a gradient term. For any $u$ weak solution, we provide explicit $L^\infty$ a priori estimates depending only on both nonlinearities, on the $H^1(\Omega)$ norm of $u,$ and on the domain $\Omega$. To obtain our results, we combine De Giorgi-Nash-Moser iteration procedure, elliptic regularity when the gradient term appears, Lebesgue interpolation and the Gagliardo-Nirenberg interpolation inequalities.

We investigate the existence and (in)stability of standing waves that are discontinuous at the vertex for the one-dimensional nonlinear Schrödinger equation (NLS) with a power nonlinearity, posed on a star graph consisting of a finite number $N$ of half-lines and endowed with a delta-prime interaction. Our non-variational approach characterizes all possible configurations of such standing wave profiles and shows, rather surprisingly, that the components of any discontinuous-at-the-vertex stationary solution split into two groups, each determined by one of exactly two distinct types of shifted soliton for the NLS on a half-line. More precisely, for every $N\geqq 2$ and every $n\in\{1,\dots,N-1\}$, there are two groups consisting of exactly $n$ and $N-n$ components, with each group sharing a common shift. The stability properties of these discontinuous-at-the-vertex stationary states are analyzed via the Grillakis-Shatah-Strauss framework and the Grillakis-Jones Instability Theorem, where non-standard techniques are required to establish the necessary spectral properties of the linearization operators, in particular their Morse and deficiency indices. Our approach yields a complete description of these indices. The results of this manuscript optimally extend previous findings in the literature concerning delta-prime type interactions on star graphs. Moreover, the methods developed here have the prospect of being adapted to study the stability of other discontinuous-at-the-vertex stationary solutions of the NLS on non-compact metric graphs, such as looping-edge graphs.