Papers

We prove interior $C^{1,\alpha}$ estimates for weak solutions of the fractional $p$-Poisson equation, with $p\geq2$. For right-hand sides in $L^q$, we obtain the estimate when $p<(1-d/q)/(1-s)$. For $\gamma$-Hölder continuous right-hand sides, the range is $p<(1+\gamma)/(1-s)$. Both ranges of parameters are optimal.

We prove instantaneous loss of Sobolev regularity across a fixed gap for the inviscid surface quasi-geostrophic equation on $\mathbb R^2$. More precisely, for every $1<s<2$, $T>0$ and $\varepsilon>0$, we construct initial data $\theta_0\in H^s(\mathbb R^2)$ with $\|\theta_0\|_{H^s}\leq\varepsilon$ that generate a solution $\theta$ on $[0,T]$ such that \[ \theta\in L^\infty([0,T];H^{1+\gamma}(\mathbb R^2)), \quad \theta(t)\notin H^\sigma(\mathbb R^2) \quad\text{for every }t\in(0,T],\ \sigma>\sigma_*(s), \] where $\gamma>0$ and $\sigma_*(s)$ satisfy \[ 1+\gamma<\sigma_*(s):= \frac{s(3-s)}{1+2s-s^2}<s. \] The solution is unique in a determined family of classical solutions with initial datum $\theta_0$.

We study the exact controllability of the cubic nonlinear Schrödinger equation in the exterior $\Omega=\mathbb{R}^n\setminus\Theta$ of a non-trapping obstacle $\Theta$, with internal controls supported outside a large ball, and the corresponding boundary control problem on $\Omega_0=B_{R_0}\setminus\Theta$ with Dirichlet controls acting only on the outer sphere $\partial B_{R_0}$. Using the local smoothing effect of Burq, Gérard and Tzvetkov [9], we prove observability inequalities for the linear Schrödinger equation in the whole scale of Sobolev spaces $H^\sigma_D(\Omega)$, $\sigma\in[-2,2]$, associated with the Dirichlet Laplacian; they require only the non-trapping assumption (no star-shapedness of $\Theta$), and their proofs use no normal boundary traces. Combined with Strichartz-type estimates and a perturbation argument, this yields local exact controllability in $H^1_0(\Omega)$ and in $H^2(\Omega)\cap H^1_0(\Omega)$, in dimensions two and three, around the zero solution and, for internal controls and under a unique continuation assumption, around nontrivial trajectories. The same method applies to the quintic equation, which is energy-critical in dimension three. Several open problems are discussed.

We study minimizers of the Pekar functional with a transverse harmonic potential, which models a polaron harmonically trapped in the transverse plane but free along the longitudinal direction. For any confinement strength $\Omega>0$, minimizers exist and, up to a translation along the $x_3$-direction, are radially decreasing in $(x_1,x_2)$ and symmetric decreasing in $x_3$. For small $\Omega>0$ we prove uniqueness up to these symmetries. For sufficiently large $\Omega>0$ we derive a three-term asymptotic expansion of the ground state energy and show that, after a suitable rescaling, the minimizers converge in $H^1(\mathbb{R}^3)\cap L^\infty(\mathbb{R}^3)$ to the product of the normalized ground state of a two-dimensional linear harmonic oscillator and that of an effective one-dimensional nonlinear local problem, displaying asymptotic variable separation. This gives, for the transversely confined Pekar model, a rigorous ground-state counterpart of the three-dimensional-to-one-dimensional dimension-reduction phenomenon numerically observed for Coulomb-type Schrödinger equations with anisotropic confining potentials in [W. Z. Bao, H. Y. Jian, N. J. Mauser and Y. Zhang, SIAM J. Appl. Math., 2013]. Moreover, uniqueness up to the same symmetries is also established for all sufficiently large $\Omega>0$.

We consider the pressureless Euler equations in one spatial dimension. These equations model the dynamics of particles on a line that interact only through perfectly inelastic collisions. The flow was first shown to be unique for given initial conditions by Huang and Wang, assuming the entropy and the strong initial continuity of energy conditions hold. In this paper, we prove an analogous uniqueness theorem using a Lagrangian interpretation. The key ingredients are showing that each solution satisfying the entropy condition and an initial kinetic energy bound has this particular type of Lagrangian interpretation and a monotonicity inequality involving conditional expectation.

We extend the theory of De Giorgi-Nash-Moser to elliptic equations with symmetric coefficients $\mathbf{a}(x)$ which are possibly degenerate and unbounded but satisfy a coarse ellipticity condition. We prove local upper bounds for weak subsolutions, a weak Harnack inequality for nonnegative supersolutions, and a Harnack inequality for nonnegative solutions. The coarse ellipticity hypothesis requires spatial moments of coarse-grained matrices, suitably discounted and summed across scales, to be finite. In particular, it holds if, for some $\alpha,\beta\geq0$ and $1<p,q<\infty$, $$\mathbf{a}\in W^{-\alpha,p}\cap L^1, \quad \mathbf{a}^{-1}\in W^{-\beta,q}\cap L^1 \quad\text{and}\quad \frac{\alpha+\beta}{2}+\frac{d-1}{2}\Big(\frac1p+\frac1q\Big)<1.$$ We show that the coefficient range is sharp, including its boundary, in every dimension $d\geq3$, for $\alpha=\beta=0$ and for the corresponding Besov-type quasi-norms of negative order when $\alpha,\beta>0$. For $\alpha=\beta=0$, it corresponds to the results of Bella and Schäffner [8].

We prove the existence and uniqueness of mild solutions to the ellipsoidal BGK (ES-BGK) model with the correct Prandtl number, under suitable weighted bounds and a free-transport non-vacuum condition. At the correct Prandtl number, the temperature tensor is no longer uniformly comparable to the scalar temperature, so classical scalar moment estimates do not directly yield the required Gaussian bounds. We derive anisotropic velocity moment estimates by splitting velocity space along the longest principal direction of the stress tensor and using the geometry of transverse elliptical slices. These estimates yield a linear weighted $L^{\infty}$ bound for the ellipsoidal Gaussian. Together with uniform ellipticity and weighted $L^1$-Lipschitz continuity under the stated assumptions, this bound allows us to construct unique mild solutions.

In this paper, we establish global-in-time Strichartz estimates for Klein-Gordon equations on non-trapping asymptotically hyperbolic manifolds. Our result is a positive-mass counterpart of the global Strichartz estimates of Sire, Sogge, Wang, and Zhang for shifted wave equations, and recovers the same range of admissible pairs. The main new feature is the low-frequency analysis, where the positive mass changes the spectral phase and leads to a different dispersive mechanism. Combining the resulting low-frequency estimates with microlocalized high-frequency estimates yields the full global Strichartz estimates. As applications, we obtain small-data global existence and scattering for power-type and certain derivative semilinear equations, as well as small-data global existence for a class of smooth semilinear equations without imposing a null condition.

In this paper, we study a family of weighted Gagliardo–Nirenberg interpolation inequalities associated with Grushin–type operators on $\mathbb{R}^n \times \mathbb{R}^m$. Specifically, we examine how to control the weighted norm $\||x|^\beta u \|_{L^B}$ by balancing fractional derivatives $D_x^s u$ with weighted derivatives $|x|^b D_y^r u$. Using two–parameter scaling, spatial translations, and Littlewood–Paley decompositions, we map out exact parameter conditions and demonstrate that the scaling exponents are both necessary and optimal. For multi-parameter weighted extensions, we leverage Khintchine-type random sums to show why $Q \ge 2$ is strictly required. When $x$ and $y$ are radially symmetric, two key advantages emerge: the allowable parameter range for the weights expands, and compact embeddings of the energy space into $L^p$, $2 < p < B$, are completely restored. We establish this compactness through two complementary tools, geometric covering arguments and a generalized three–norm version of Strauss's lemma, that yields explicit pointwise weighted decay estimates. Finally, in the scale–invariant case $\beta=0$, a concentration–compactness analysis proves that the inequality's sharp constant is achieved by an optimal function, with minimizing sequences remaining precompact up to the natural three-parameter group of symmetries.

Inverse source problems in divergence form consist in finding a vector field with prescribed support S, whose divergence is the Laplacian of some observed potential. In this paper, we assume the unknown vector field is a vector-valued measure, and we study the corresponding least square inversion problems, regularized by penalizing the total variation, without discretizing the criterion nor the unknown. We prove that this problem has a unique minimizer in the case where S is a slender set; i.e., it has zero Lebesgue measure and each connected component of its complement has infinite Lebesgue measure. The proof relies on a refinement of Smirnov's decomposition of divergence-free measures [62] which is of independent interest.

We investigate weighted perimeters of convex bodies in $\mathbb{R}^n$. Our main result states that the weighted perimeter is continuous with respect to the Hausdorff metric for every radial weight function. We also establish monotonicity under inclusion under suitable assumptions on the radial weight, and show that these assumptions are sharp. These results are applied to show the existence of minimizers for a shape optimization problem.

In this paper, both the pointwise space-time structures of the solution for a three-dimensional model describing gas dynamics in thermal non-equilibrium and the corresponding background stabilities within the $L^p$ framework are investigated under the small initial perturbation assumption and the irrotationality condition. Our main tools are the modified Green's function according to the irrotationality condition, a priori energy method and a refined decay estimates of products composed of the entropy function and high-order derivatives.

In this note we study the regularity properties of the \emph{Effort Function $E(\beta)$}. This can be defined as the minimal $L^1$-norm of the control $\tilde \alpha(x) \geq 0$ so that the PDE $$U_t = \Delta U + f(U) - \tilde \alpha(t,x) U, \quad U \in [0,1], %\ \tilde \alpha \geq 0,$$ admits a traveling profile $U(x - \beta t)$ connecting $0$ to $1$. The above PDE models the evolution of an invasive species, where the control $\tilde \alpha(t,x)$ describes the percentage amount of species removal. In the literature some conditions are assumed on the properties of $f(U)$ and (consequently) on the effort function $E(\beta)$, the main ones being that the traveling profile has a simple structure and that the effort function can be used as an anisotropic perimeter to generated a cost function over sets of finite perimeter. This note gives some examples showing that in the interesting cases, i.e. when the PDE has a non trivial dynamics, these assumptions are in general not satisfied. We also show that $E(\beta)$ is in general only $C^1$.

In this paper, we establish a quantitative finite-time approximation of scattering fields for nonlinear Alfvén waves in three-dimensional ideal incompressible magnetohydrodynamics near a strong constant magnetic background. Using the global weighted estimates, we prove that truncating the scattering profiles at time $T$ introduces an error in both the $L^\infty$ norm and the $H^{N_*+1}$ norm of order $$ O\bigl(\varepsilon^2(R+T)^{-\delta}\bigr), $$ where $\varepsilon\in (0,1)$ measures the size of the initial perturbation, $R\ge100$ is the scale parameter in the weighted energy, $\delta\in(0,2/3)$ is the exponent related to weights, and $N_*+1$ is the Sobolev regularity order assumed for the initial data. The pointwise estimate follows from the decay of the nonlocal pressure along nonlinear characteristics, while the Sobolev estimate combines higher-order weighted pressure bounds with estimates for derivatives of the characteristic flows. Numerical experiments measure pointwise and Sobolev pressure-integral tails to verify the quadratic dependence of the initial perturbation amplitude.

We study the energy-subcritical and energy-critical complex Ginzburg-Landau equations on three-dimensional Riemannian manifolds. We establish the asymptotic behavior of solutions via a purely energy-based approach and subsequently prove a fast/slow dichotomy of the solutions. Moreover, for the slow solution, we show that the solution converges to an explicit asymptotic profile. In particular, we obtain a precise description of both the amplitude and the phase of the leading-order dynamics. As a by-product, the complex Ginzburg-Landau equation approximation yields global weak solutions to the energy-subcritical and energy-critical defocusing nonlinear Schrödinger equations on compact three-dimensional Riemannian manifolds. We also investigate the long time behavior of the solutions to the energy-subcritical and energy-critical complex Ginzburg-Landau equations on a class of complete noncompact three-dimensional Riemannian manifolds and prove decay of global solutions under suitable geometric assumptions.

In this article, we investigate the boundary controllability of finite point-vortex systems in bounded multiply connected planar Lipschitz domains. The controls are applied locally on an arbitrary nonempty relatively open subset of the boundary. Zero incoming vorticity is assumed at the inflow part of the boundary. For time-dependent controls, we prove exact trajectory controllability along arbitrary prescribed collision-free vortex paths remaining away from the boundary. For stationary controls, we prove exact endpoint controllability associated with mutually disjoint real-analytic arcs. The proofs rely on harmonic interpolation, localized Runge approximation, and an implicit-function argument. The optimal control minimizing the length of trajectories is also studied. We prove the nonexistence of minimizers within classes of slightly smooth controls, and establish existence when the control time and the size of the control are penalized.

In this article, we establish uniqueness in the inverse problem of time-harmonic elastic scattering from a bounded, traction-free obstacle in two dimensions. For a single incident plane wave, it is proved that a connected polygonal obstacle is uniquely determined by the measured far-field pattern. The analysis is based on a reflection principle for the Navier equation under traction-free boundary conditions and unique continuation of elastic fields.

In this paper, we prove the existence of stationary pulse solutions for a bulk-surface reaction-diffusion system modeling cell polarization in two-dimensional domains. Using a singular perturbation method, we construct pulse solutions whose centers lie near non-degenerate critical points of a geometry-dependent potential expressed in terms of the Neumann Green's function, as formally predicted in \cite{watanabe2026}. The main analytical difficulty arises from the half-Laplacian induced by the logarithmic singularity of the Green's function, resulting in algebraically rather than exponentially decaying inner solutions. Our analysis may also serve as a useful guide for applying singular perturbation methods to other bulk-surface type systems.

We study preservation of spatial convexity for viscosity solutions of first order Hamilton-Jacobi evolution equations with Neumann-type boundary conditions in bounded convex domains. Under a monotonicity assumption on the Hamiltonian in the outward normal direction, every viscosity supersolution of the Neumann problem is also a state-constraint supersolution. This reduces the boundary difficulty to a state-constraint problem and enables us to prove convexity preservation via the convex-envelope argument established by Alvarez, Lasry and Lions (1997). We also obtain propagation estimates for semiconvexity and strong convexity under similar structural assumptions.

We establish local volume doubling and a local $L^2$-Poincaré inequality for metric measure spaces tamed by a measure in the extended Kato class that satisfy the bounded interpolation property. To this end, we construct a bi-Lipschitz time change to a metric measure space satisfying a uniform lower Ricci curvature bound. As an application, we obtain pointed measured Gromov-Hausdorff precompactness for such spaces.

We show global existence of weak solutions to the following system of partial differential equations with nonlinear diffusion in both equations and anisotropic signal production \begin{equation} \begin{cases} u_t=\nabla\cdot((u+1)^{m_1-1}\nabla u)-\nabla\cdot(u\nabla v), \\ v_t=\nabla\cdot((v+1)^{m_2-1}\nabla v)+\nabla\cdot(u\nabla v)-v+u, \end{cases} \end{equation} in a smooth bounded domain $\Omega\subseteq\mathbb{R}^n$, $n\in\mathbb{N}$ with nonnegative initial data and homogenuous Neumann boundary conditions. Moreover, we show boundedness of $u$ in $L^\infty((0,\infty);\:L^{m_1}(\Omega))$ and $v$ in $L^\infty(\Omega\times(0,\infty))$. The strategy will be to show global existence and boundedness of classical solutions $(u_\varepsilon,v_\varepsilon)$ to the approximating system \begin{equation} \begin{cases} u_{\varepsilon t}=\nabla\cdot((u_\varepsilon+1)^{m_1-1}\nabla u_\varepsilon) -\nabla\cdot(\frac{u_\varepsilon\nabla v_\varepsilon}{1+\varepsilon\vert\nabla v_\varepsilon\vert}), \\ v_{\varepsilon t}=\nabla\cdot((v_\varepsilon+1)^{m_2-1}\nabla v_\varepsilon)+\nabla\cdot(u_\varepsilon\nabla v_\varepsilon)-v_\varepsilon+u_\varepsilon. \end{cases} \end{equation} Passing to the limit $\varepsilon\searrow 0$, we obtain convergence of weak solutions of the second to weak solutions of the first system. One of the main challenges lie in the proof of weak convergence of $\frac{\nabla v_\varepsilon}{1+\varepsilon\vert\nabla v_\varepsilon\vert}$ to $\nabla v$ in $L^2(\Omega\times(0,T))$ for $T\in(0,\infty)$.

In this paper, we study the global regularity of oblique boundary value problems for n-1 type augmented Hessian equations on a bounded domain, without imposing any convexity condition either on the domain or on the matrix-valued function in the n-1 type augmented Hessian equations. We establish a global existence and uniqueness theory for classical elliptic solutions by deriving global a priori estimates up to second-order derivatives. Besides the known applications for n-1 type Monge-Ampere operators in optimal transportation and geometric optics, the general theory here embraces equations raised by Li-Sheng, Some Dirichlet problems arising from conformal geometry. Pac. J. Math. 251, 2011, 337-359 and Guan-Qiu-Yuan, Fully nonlinear elliptic equations for conformal deformations of Chern-Ricci forms. Adv. Math. 343, 2019, 538-566.

We consider entire solutions $u$: $\mathbb{R}^n \rightarrow \mathbb{R}$, $n \ge 2$, of the equation $\operatorname{div}(\nabla f(\nabla u)) = 0$ with convex density $f$: $\mathbb{R}^n \rightarrow \mathbb{R}$ given for example by $f(\nabla u) = \sum_{i=1}^n |\partial_i u|^{p_i}$ with exponents $p_i \ge 2$ including Giaquinta's example for the choice $p_1 = \dots = p_{n-1} = 2$, $p_n = 4$. If $n \le 2 + \min\{p_i\}$ holds, then we have the Liouville property: the boundedness of $u$ implies its constancy.

Global existence of Leray-Hopf (type) weak solutions to the viscous primitive equations in two or three dimensions was established by Lions–Temam–Wang \cite{Lions1,Lions2,Lions3,Lions4} in the 1990s; however, the corresponding uniqueness, no matter in two or three dimensions, has been a long standing open problem since then. In this paper, we solve this open problem in the two-dimensional case. Precisely, we prove the uniqueness of Leray-Hopf weak solutions, with arbitrary $L^2$ initial data, to the viscous primitive equations in a periodic channel, subject to impermeability and stress-free boundary conditions on the solid boundaries.

We prove that for every $N\geq 3$, every solution to the $Q$-curvature-type problem $$ \alpha P_N u + (N-1)!\left(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw}\right)=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ \alpha\ge\frac{1}{2}$ and $\alpha \not =1$. The proof consists of a spectral gap estimate and pointwise estimates of the normalized density $\left(\int_{\mathbb{S}^N} e^{Nu}dw\right)^{-1}e^{Nu}$.

For every integer $n\ge4$, we prove the existence of a $(-1)$-homogeneous solution of the stationary incompressible Navier-Stokes equations on $\mathbb{R}^n\setminus\{0\}$ with an arbitrary locally Lipschitz $(-3)$-homogeneous force. The pressure is $(-2)$-homogeneous, and the solution is regular away from the origin. This extends the homogeneous existence theorem of Bang, Gui, Liu, Wang, and Xie beyond dimension sixteen. The principal estimate combines a subcritical localized pressure identity with an elementary dyadic estimate for an auxiliary tangential vector field in a suitable finite Lebesgue space. Testing the Bernoulli equation at the homogeneous cancellation exponent then controls its positive part in $L^{\frac{(n-2)(n-1)}{2(n-3)}}(S^{n-1})$. The resulting estimate closes with the energy inequality in every fixed dimension. We also establish uniqueness throughout the classical homogeneous class for sufficiently small force. Using the same ideas, we also give a simplified proof of the known existence theorem for bounded compactly supported forces on $\mathbb{R}^n$, $n\ge5$. A general finite-exponent estimate for a scalar equation with divergence-free drift, together with a Euclidean version of the dyadic kernel estimate, bounds the positive Bernoulli function without BMO estimates or an iteration to obtain its supremum norm.

We prove a sharp Hessian inequality for the natural comparison Green's function of an Einstein manifold with positive sectional curvature, which may be viewed as a Green's function analogue of the Hessian comparison theorem and as a positively-curved, elliptic counterpart to the matrix Li-Yau-Hamilton inequality for the heat equation. We also show that this inequality is closely related to a family of monotonicity formulae, extending previous results of Park to the setting of positive curvature. We further present several geometric applications, such as a sharp comparison inequality for the sum of two distinct Green's distance functions.

We study quantitative stability for the fractional Schrödinger equation $(-\Delta)^s u+u-|u|^\alpha u=0$ near finite sums of widely separated positive ground states. For every $n\ge1$, $0<s<1$, and Sobolev-subcritical exponent $\alpha>0$, we estimate the $H^s$ distance to the family of such sums in terms of the $H^{-s}$ norm of the equation's residual. The optimal rate changes at $\alpha=n/[2(n+2s)]$, with a logarithmic correction at the threshold. Above the threshold the rate is $t^{(n+2s)/(n+2s+1)}$; below it the exponent is $[(1+\alpha)(n+2s)-n/2]/(n+2s+1)$. The proof combines uniform invertibility away from the translation modes with precise interaction estimates and a weighted correction of the approximate configuration. This correction resolves the translation interactions even when the nonlinearity has a small exponent. We also construct nonnegative configurations that attain these rates. The estimates quantify the effect of the algebraic decay of fractional ground states on stability.

We derive the pressureless Euler-Maxwell system as a semiclassical limit from the Maxwell-Schrödinger equation. The method relies on studying a quantum modulated energy accounting for the self-consistent magnetic effects. Crucially, a correction is incorporated in the quantum modulated energy as to enable to close the Grönwall estimate provided a smallness assumption on the initial data is imposed. This is the first result on the semiclassical limit of the Maxwell-Schrödinger equation avoiding any regularization or conditional uniform in $\hbar$ bounds on the wave

We establish local existence and uniqueness of strong solutions to the compressible Navier-Stokes/Cahn-Hilliard system in a bounded smooth domain in $\mathbb R^3$, allowing the initial density to vanish. The initial data satisfy suitable regularity and compatibility conditions for the momentum and phase equations. The main difficulty arises from the simultaneous degeneracy of the momentum equation and the density-weighted phase equations, which complicates the construction of compatible positive-density approximations and uniform initial estimates. We overcome this difficulty by introducing a time-independent residual into the chemical relation. The residual preserves the phase compatibility exactly, keeps the initial phase field unchanged, and vanishes in $H^1$ as the approximation parameter tends to zero. For each positive-density approximation, we construct a local strong solution using a semi-discrete Galerkin scheme that discretizes only the phase variables and retains the full transport and momentum equations. We then derive a priori estimates independent of the density lower bound, which yield a common lifespan and allow us to pass to the vacuum limit. Uniqueness is proved using a modified difference energy that accounts for the coupling between the density and the chemical potential. To our knowledge, this is the first local well-posedness result for the three-dimensional compressible Navier-Stokes/Cahn-Hilliard system with initial vacuum.