Papers

In this paper, we establish local existence of strong solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes equations with initial data $(\rho_0,u_0)$ lying in $C^1 \times (L^2 \cap VMO^{-1})$, where $\rho_0$ has a positive lower bound. Furthermore, if $\rho_0 \in C^2$ and $||\rho_0-1||_{L^\infty}+||u_0||_{BMO^{-1}}$ is sufficiently small, we prove global existence of the solution. To achieve this, we employ an estimate for the transport equation to obtain regularity for the density and apply a new freezing-coefficient method for the momentum equation.

This paper explores the relationship between weak and viscosity solutions to a nonhomogeneous mixed local and non-local $p$-Laplace equation in a bounded Lipschitz domain in $\mathbb{R}^N$. Under certain conditions, we derive the comparison principle for weak subsolutions and weak supersolutions to the problem. For $1<p<\infty$, we establish that continuous weak supersolutions to the problem are viscosity supersolutions, using the comparison principle. Furthermore, we show that bounded viscosity supersolutions are weak supersolutions for $p \geq 2$.

We employ the vector field method to establish a Liouville-type theorem for a class of \((p,q)\)-Laplace equations in the Euclidean space \(\mathbb{R}^n\). By modifying the exponents in the differential identity, we prove nonexistence in the subcritical range \(p-1<\alpha<q^*-1\), where \(q^*=nq/(n-q)\). The approach relies on constructing a suitable differential identity, carrying out precise integral estimates with cutoff functions, and combining sign control and decay of the cutoff errors.

In this paper, we investigate the orbital stability of vortex patches for the two-dimensional incompressible Euler equations in both a class of domains that satisfy the “weak finite volume condition" and a strip of arbitrary width. We establish that for suitable parameters $(\mu,\lambda)$, the penalized kinetic energy functional admits a minimizer, and that every such minimizer satisfies the elliptic equation $\omega = \lambda(\psi - W x_2 - \gamma)_+$. Furthermore, we demonstrate that the set of minimizers is orbitally stable under the Eulerian dynamics. This work extends the variational framework developed by Abe and Choi to domains that lack both spatial scaling invariance and horizontal translation invariance. The absence of these properties introduces substantial difficulties in the proof, as classical rearrangement and scaling arguments are no longer applicable. We overcome these obstacles by comparing the Green's function with that of the half-plane and exploiting the decay condition to formulate a concentration-compactness argument that ultimately yields the desired stability result.

In this paper, we consider the classical overdetermined capillary problem: \begin{equation*} \begin{cases} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) - bu =0 &~~\mbox{in}~~ \Omega, \partial_{\nu} u=\kappa &~~\mbox{on}~~\partial\Omega, u=c &~~\mbox{on}~~\partial\Omega, \end{cases} \end{equation*} where $b$, $c$ and $\kappa$ are positive constants, and $\Omega\subset \mathbb{R}^2$. When $\Omega$ is an infinite strip, i.e., a domain bounded by two parallel straight lines, there exists a unique one-dimensional solution (called the trivial solution) to this problem. By means of a bifurcation argument, we establish the existence of a critical period $T_*$ at which a branch of non-trivial solutions bifurcates from the trivial one. These solutions are genuinely two-dimensional and are defined in unbounded periodic domains $\Omega$ that are diffeomorphic to an infinite strip, yet whose boundaries are no longer straight lines. This result offers a significant physical interpretation in the context of capillary phenomena.

We study the homogenization of the compressible isentropic Navier-Stokes equations in periodically perforated domains where the size of the obstacles is of the same order as the distance between neighboring obstacles. Using the two-scale convergence method, which can be characterized via the unfolding operator, we derive the corresponding macroscopic model determined by Darcy's law. In particular, the macroscopic density satisfies the porous medium equation. The main challenge lies in identifying the pressure term in the limit. We overcome this by establishing the strong two-scale convergence of the densities, which is achieved by controlling the oscillation defect measure of the unfolded densities. A crucial contribution of our work is the development of a methodological framework applicable to more complex compressible fluid models. Furthermore, regarding conservative forces, we extend existing results from the literature to adiabatic constants $\gamma > \frac95$.

In this paper, we are concerned with the optimal dimension-dependent $\ell^p$ norm of the discrete Riesz Transforms $R_{\text{dis}}^{(k)}$ on $\mathbb{Z}^d$ given by the singular convolution kernel $K_k(m)=c_d m_k/|m|^{d+1}$, where $c_d=\Gamma(\frac{d+1}{2})/\pi^{(d+1)/2}$ . We show that for fixed $1<p<\infty$, when $d\to \infty$ $$\|R_{dis}^{\left( k \right)}\|_{\ell ^p\left( \mathbb{Z}^d \right) \rightarrow \ell ^p\left( \mathbb{Z}^d \right)}=2c_d\left( 1+\frac{\left( \sqrt{2}+o\left( 1 \right) \right) d}{2^{\frac{d}{2}}} \right) .$$ The operator norm of $R_{\text{dis}}^{(k)}$ grows super-exponentially as $d\to\infty$ since $c_d\sim(\frac{d-1}{2e\pi})^{\frac{d-1}{2}}\sqrt{\frac{d-1}{\pi}}$ by Stirling's formula, which gives a negative answer to the conjecture proposed by Bañuelos, Kim and Kwaśnicki in \cite{BKK}. The optimal dimension-dependent $\ell^{1,\infty}$ estimate of $R_{\text{dis}}^{(k)}$ is also established.

We study the Cauchy–Dirichlet problem for parabolic complex Hessian equations on Hermitian manifolds and on bounded strictly m-pseudoconvex domains. In the smooth setting, we prove global existence and uniqueness of classical solutions under the presence of an admissible parabolic subsolution, by establishing a priori estimates up to the parabolic boundary. The estimates combine parabolic boundary techniques for complex Hessian equations with interior second order estimates and a blow-up argument. We then develop a general pluripotential framework for degenerate right-hand sides with L^p densities, p>n/m, and bounded Cauchy–Dirichlet data. Since the usual automorphism and Walsh-type arguments do not directly apply in a variable Hermitian background, we use approximation by smooth data, balayage, parabolic Perron envelopes, and a continuous obstacle approximation based on Harvey–Lawson–Plis subequation theory. The resulting solution is continuous for positive time, locally uniformly Lipschitz and semi-concave in time, and continuous up to the initial slice when the initial datum is continuous. We also prove a parabolic comparison principle via time regularization, Riemann sum approximations, and mixed Hessian inequalities.

In this work, we study a coupled nonlinear thermoelastic system in which the heat capacity, thermal conductivity, and elastic modulus are all temperature-dependent. The thermal conduction equation is formulated in a generalized radial geometry governed by a parameter $\nu$, which encompasses planar, cylindrical, and spherical configurations as special cases. A complete Lie symmetry classification of the admissible constitutive functions is carried out, yielding three principal classes: exponential laws, power laws, and a unified class that contains both as limiting cases. For each class, the admitted symmetry algebras are determined, optimal systems of one-dimensional subalgebras are constructed, and the governing partial differential equations are reduced to coupled ordinary differential equation systems via similarity transformations. Exact closed-form solutions are obtained for each class using a combination of integrating factor methods, flux-variable substitutions, and power-law and logarithmic ansätze. For the exponential class, the thermal field exhibits a logarithmic similarity profile while the displacement field decays algebraically, with the geometry parameter absorbed into the leading-order transformation. For the power-law class, the thermal field follows a self-similar power-law profile and the displacement field decomposes into distinct elastic, self-similar, and thermoelastically driven modes. Travelling-wave solutions are also constructed, with elementary forms recovered in limiting cases. The influence of geometry, particularly the structural distinction between cylindrical and other configurations, is identified and physically interpreted. All solutions are discussed in terms of the interplay between thermal diffusion, elastic stiffness, and thermoelastic coupling.

We develop a capacitary approach to semilinear elliptic inequalities on weighted graphs with a potential. More precisely, we study the nonexistence of nontrivial nonnegative solutions of \[ \Delta u+w(x)u+v(x)u^\sigma\le0 \qquad\text{in }V, \] where \((V,\omega,\mu)\) is a connected, locally finite weighted graph, \(\Delta\) is the associated graph Laplacian, \(\sigma>1\), \(v>0\), and \(w\) is a real-valued potential. The potential term is handled by means of a positive solution \(H\) of \(\Delta H+wH=0\), which transforms the operator \(\Delta+w\) into the \(H\)-Laplacian associated with a new weighted graph. Our main nonexistence criterion is formulated directly in terms of cut-off functions and the regions where their \(H\)-Laplacian is controlled. Unlike metric criteria based on pseudo-metric annuli, our formulation determines the capacitary sets from the support of the \(H\)-Laplacian estimates for the cut-off functions. We provide an example showing that our result applies in situations not covered by previous nonexistence criteria based on structural lower bounds or pseudo-metric annular volume estimates. We also show that the growth exponent in our capacitary condition is sharp by constructing an example for which the condition fails by an arbitrary power \(R^\varepsilon\), while a positive nontrivial solution exists.

For the Zakharov system in four space dimensions, we prove that all solutions inside the potential well of the ground states are global and scattering in the energy space, with no other restriction such as symmetry. The proof has already been reduced by [3] to ruling out the existence of a minimal non-scattering solution that is precompact along some trajectory. This paper carries out the final step in the proof, namely we exclude the possibility of precompact solutions inside the potential well by combining two distinct arguments depending on the motion of trajectory.

We extend the optimal transport theory for signed measures supported on Ahlfors-regular fractal sets (Bwo'Nyahre et al., 2026) to allow a controlled dimensional distortion between source and target. A penalization term $\varepsilon \Phi(d_s(x) - d_t(y))$ – where $\Phi$ is a fixed smooth strictly convex function and $d_s, d_t$ are the local Hausdorff dimensions of the fractal supports – is added to the transport cost on inter-sign regions, with~$\varepsilon \ge 0$ controlling the tolerance for distortion. Under hypotheses H1–H7, we prove: the existence and uniqueness of an optimal transport map~$T^{\varepsilon}$ for every~$\varepsilon > 0$; coupled Monge–Ampère equations with a distortion correction term, generalizing the classical Brenier–Caffarelli equation; a double Legendre–Fenchel characterization of the optimal potentials, giving a complete variational description of the transport in each of the four sign regimes. The double Legendre–Fenchel system (Theorem~4.2) is the central contribution: it shows that the optimal potentials are the unique fixed points of a system of conjugacy equations, one per transport regime, and it provides the foundation for numerical algorithms and asymptotic analysis.

We study the Monge–Kantorovich optimal transport problem between two signed measures~$\mu$ and~$\nu$ on convex compact subsets of~$\mathbb{R}^d$, with a positional penalization function~$\lambda(x, y)$ that modulates the cost of inter-sign transport. Using four independent positive measures~$(\pi^{++}, \pi^{+-}, \pi^{-+}, \pi^{--})$ as decision variables, we prove that the admissible set~$\mathcal{A}(\mu, \nu)$ is weakly-$*$ compact and non-empty if and only if $\mu^+(X) = \nu^+(Y)$ and~$\mu^-(X) = \nu^-(Y)$. Strong duality is established via the Kantorovich minimax theorem, yielding a new compatibility condition on~$\lambda$ at the intersection of inter-sign supports. The penalization~$\lambda$ is shown to be Lipschitz and to admit Alexandrov second derivatives almost everywhere. Modified Monge–Ampère equations governing inter-sign transport maps are derived in the Alexandrov sense, with well-posedness characterized by $\sigma \det(D^2_{yx}\Lambda) e > 0$. The classical Brenier equation is recovered in the limit~$\lambda \to 0$.

We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form $H(x,p,m)=H_0(x,p)-g(m)$, with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order $\bar\gamma$-Laplacian regularization of the stationary MFG system as a variational inequality on $L^{\bar\beta}(\mathbb T^d)\times W^{1,\bar\gamma}(\mathbb T^d)$. To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue-Sobolev exponents of the problem and analyze the exact constrained one-step mirror iteration and a frozen two-step variant. Under quantitative monotonicity of the coupling, both exact iterations converge strongly to the unique solution of the regularized variational inequality for each fixed $\varepsilon>0$, under natural summability conditions on the step sizes. Along weakly convergent vanishing-regularization sequences, we also prove strong convergence in the full product space. Numerical experiments in one and two dimensions show first-order accuracy on manufactured solutions and weak dependence of outer iteration counts on the tested grids. At matched effective step sizes, the frozen two-step and one-step implementations agree up to inner-solve accuracy, while the latter requires fewer mirror solves.

Considered and analyzed below are fully non-homogeneous initial-boundary-value problems for the celebrated Dirac system, formulated on the spatial half-line. Analytical solution formulae are derived formally via suitable implementation of the well-known Fokas' unified transform methodology, and rigorously verified a posteriori. The latter substantial task relies on complex-analytic tools and careful interpretation of the obtained integral representations. These valid solutions are then used for investigating qualitative properties. These include boundary behavior near the axes of the domain as well as long-range asymptotics and long-time (eventual) periodicity. Notably, smoothness of the solution, both within and upto the boundary of the domain, depends heavily on certain compatibility conditions between initial, boundary and forcing data. Further results pertaining to solution's regularity and uniqueness are thence established based on the qualitative theory. The closed-form expressions reported here are also useful in the study of non-linear counterparts.

We prove norm inflation for the cubic hyperbolic nonlinear Schrödinger equation in $H^s(\mathbb T^2)$ for every $s\in(-\infty,0)\cup(0,\frac12]$. The scaling-critical point $s=0$ is excluded by conservation of the $L^2$ norm. The strong ill-posedness below and above the scaling-critical point arises from two completely different mechanisms. Particularly in the scaling-subcritical regime, this dynamical instability stems from the hyperbolic nature. Together with the local well-posedness result in \cite{WangHNLS}, this gives a sharp dichotomy away from the mass space $L^2(\mathbb T^2)$: local well-posedness holds for $s>\frac12$, whereas norm inflation occurs for all $s\le \frac12$ with $s\ne0$.

We establish the parabolic Harnack inequality on weighted Riemannian manifolds for a large class of parabolic differential operators building on an approach due to Alexander Grigor'yan.

We prove local and global two weight estimates in which we bound difference quotients of a function in terms of certain weighted $L^p$ norms of its gradient.

We study the well-posedness problem for the extended Schrödinger-Benjamin-Ono system (eSBO) on the real line. This system couples a Schrödinger field $u$ with a Benjamin-Ono type field $v$, including a term of the form $\partial_{x}(v^2)$. This latter term, just as in the case of the Benjamin-Ono equation, causes the system to become quasilinear and unsolvable via Picard iteration. We prove that eSBO is locally well-posed in $H^{s+\frac 12}(\mathbb{R})\times H^{s}(\mathbb{R})$ for any $s\geq 0$. In particular, this result covers the energy space at $s=\frac 12$, yielding global well-posedness in $H^{1}(\mathbb{R})\times H^{\frac 12}(\mathbb{R})$ with a small $L^2$-assumption on the Schrödinger part of the initial data.

We establish well-posedness of the stationary Kolmogorov equation on a bounded domain, with either spherical or Gaussian velocities, subject to either inflow boundary conditions or specular reflection. For the sake of completeness, we also include the problem on the torus, which has already been solved by Albritton, Armstrong, Mourrat, and Novack (2024). Although the natural trace estimate with weight $|n_x\cdot v|$ does not hold in general in either setting, we provide a natural functional framework in which the trace is defined via the transport operator. We prove a Poincaré-type inequality with trace, which is an essential step towards the well-posedness of the inflow problem without damping. Moreover, we obtain a one-sided energy estimate for the trace with weight $|n_x\cdot v|$, in which the outflow (resp. inflow) flux is bounded by the energy inside the domain and the inflow (resp. outflow) flux. Finally, for the bounded-velocity model, we complement the existing well-posedness theory for the stationary Kolmogorov equation with inflow conditions on the hypoelliptic boundary and Dirichlet conditions on the velocity boundary.

In this work we consider the pressureless Euler-Poisson system on $\mathbb{R}^3$ describing ion dynamics, which is a common model in the study of cold plasma. We prove global regularity and scattering for small, irrotational velocity fields around a constant density profile. At the heart of the result is a dispersive mechanism leading to amplitude decay at the full rate in $3$D. This fast decay feature and a favourable structure of the nonlinearity allow to control quadratic interactions globally.

This paper studies the uniqueness of bound states for the problem \Delta u + u\log u ^2=0, \quad u\in H^1(\RN), \quad n\geq 2, which arises from the logarithmic Schrödinger equation. We prove that for every integer $k\geq 1$, there exists a unique radial solution $u(r)=u(|x|)$ that has exactly $k$ simple zeros for $r>0$. This resolves an open problem posed by Troy [{Arch. Ration. Mech. Anal.} 222 (2016), 1581–1600] and confirms the Berestycki-Lions conjecture for the logarithmic nonlinearity. The proof combines the shooting method with suitable auxiliary functions introduced by Tang [{Invent. math.} 243 (2026), 245–291]. A major difficulty arises from the singular behavior of the nonlinearity $f(u)=u \log u^2$ at origin. We overcome it by establishing asymptotic convergence and sharp decay rates at infinity for any ground state or bound state. More precisely, every such solution satisfies \lim_{r\to\infty}\frac{ u'(r)}{u(r) \sqrt{\abs{ \log u^2(r)}}}=\lim_{r\to\infty}\frac{u'(r)}{ru(r)} = -1, \quad \limsup_{r\to\infty}|u(r)|e^{(\frac12-\epsilon)r^2}<\infty, ~~\forall \epsilon \in ( 0,\frac{1}{2}). These asymptotic behaviors are of independent interest and may be useful for other problems involving logarithmic nonlinearities.

We study random Schrödinger operators on closed Riemannian manifolds with Anderson-type potentials. We prove high-probability spectral inclusion bounds showing that eigenvalues remain close to those of the Laplacian, with deviations controlled by a norm of the potential coefficients. Compared with deterministic bounds, this yields a square-root cancellation gain. The proof is based on a general principle showing that randomisation improves operator norm bounds for multiplier-type operators, which we formulate in both discrete and continuous settings.

In this paper, we consider the quasilinear Schrödinger-Poisson system with concave and convex nonlinearities \begin{align*} \begin{cases} -\Delta_{p} u+\lambda V(x)|u|^{p-2}u + \mu \phi |u|^{p-2}u= a(x)|u|^{m-2}u + b(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3}, -\Delta \phi=|u|^{p} &\ \ \ \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{align*} where $\lambda >0, ~\mu >0$, $\frac{3}{2}<p<3$, $1< q<p < m < 2p$ and $\Delta_{p} u= \hbox{div}(|\nabla u|^{p-2}\nabla u)$. We assume that $V(x) \in C(\mathbb{R}^{3}, \mathbb{R})$ is a steep potential well, while $a(x)$ and $b(x)$ are allowed to be sign-changing and satisfy some suitable assumptions in $\mathbb{R}^3$. By using the Ekeland's variational principle and combining the constraint approach, we prove that the system admits two positive solutions.

We study an obliquely reflected optimal stopping problem in the nonnegative quadrant with nonsmooth max-type payoff \(G(x)=x_1\vee\alpha x_2\), and we develop a measure-valued potential-theoretic formulation of the associated obstacle problem. The kink of \(G\) on the diagonal \(x_1=\alpha x_2\) produces a singular surface measure in the distributional generator, while the oblique reflection directions generate boundary local-time contributions on the coordinate faces. Together with the absolutely continuous stopping gain, these terms define a total signed stopping measure \(\Gtot\). We derive the corresponding reflected Itô–Tanaka identity, prove a killed-resolvent representation of the value function in the continuation region, and show that the unrestricted reflected resolvent is generally incorrect because the process is not absorbed on the stopping set. The free boundary is formulated through a continuation-side trace condition for the killed potential. Under a vertical monotonicity hypothesis on \(V-G\), the stopping set is shown to have an epigraph form. We finally prove a verification theorem: any admissible epigraph candidate satisfying contact, strict continuation, reflected Neumann compatibility, growth, the trace condition, and measure-superharmonicity coincides with the value function, and its first entry time is optimal.

We study the vorticity formulation of the 3D Navier-Stokes equation driven by transport noise in a periodic channel with Navier-slip boundary conditions. We consider both non-degenerate transport noise and degenerate tangential transport noise. For any prescribed $T>0$ and $\epsilon>0$, we prove that, by choosing the noise intensity sufficiently large and concentrating the noise on sufficiently high modes, the solution exists up to $T$ with probability at least $1-\epsilon$. A main contribution of this work is to identify and analyze the interaction between enhanced dissipation induced by transport noise and physical boundary effects. The no-flux condition breaks the isotropy of the noise and changes the scaling limit of the Itô-Stratonovich corrector. In the non-degenerate case, a boundary feedback term appears in the limiting effective operator; in the degenerate case, the limiting operator is a nonlocal anisotropic tangential dissipation. The proof is based on a combination of a boundary correction operator, a Meyers-type estimate, a scaling-limit analysis of the Itô-Stratonovich corrector, and resolvent estimates for the deterministic limiting equations.

We consider stratified steady water waves in a two dimensional channel. Our subject is branches of Stokes waves, bifurcating from laminar flows. We assume that the mass flux and the Bernoulli constant are fixed and consider the period of the wave as a parameter, which can change its value along the branch. A new class of density and Bernoulli functions is presented, for which laminar flows generate global bifurcation branches. The laminar flows are not necessary unidirectional and we show that the bifurcation branch can bifurcate from the laminar flow with arbitrary large period.

We prove a local exact controllability result for controlled Landau–Lifshitz–Gilbert equations on $\mathbb T^2$: if the initial energy is sufficiently small, then for any terminal time $T>0$, there is a localised external magnetic field such that the system can be steered exactly to the terminal value of any nearby uncontrolled trajectory. We first transform the equation to a quasilinear parabolic system on $\mathbb R^2$ by a suitable stereographic chart. Then the Carleman estimate is established for the linearised system through a decomposition adapted to the self-adjoint and skew-adjoint structure of the conjugated adjoint operator. This yields observability and $L^\infty$-null controllability for the linearised system. The nonlinear projected equation is then recovered by a Kakutani fixed-point argument. We also obtain a semi-global controllability result under a hemisphere condition.

We review the probabilistic representation of solutions of wave equations with polynomial nonlinearities in spatial dimensions d=1,2,3 using stochastic branching processes. Under regularity assumptions on the initial data, we derive conditions ensuring the integrability of the corresponding Monte Carlo estimator, and the existence and smoothness of mild and classical solutions. We also present numerical results and comparisons with grid-based algorithms for the solution of nonlinear wave equations.

For compressible nonlinear isotropic elasticity it is well known that rank-one convexity, polyconvexity and the monotonicity of the Cauchy stress tensor with respect to the logarithmic stretch tensor (the true-stress-true-strain monotonicity, TSTS-M$^+\!$) are independent constitutive conditions which should, however, all together be satisfied for a physically meaningful description of idealized elastic materials. In the incompressible case, TSTS-M$^+\!$ turns into Hill's inequality since the Cauchy stress $\boldsymbol{\sigma}$ reduces to the Kirchhoff stress $\boldsymbol{\tau}$. Hill's inequality requires then monotonicity of the Kirchhoff stress in terms of the logarithmic stretch tensor evaluated for incompressible response. In this paper we clarify how the a priori independent notions of Legendre-Hadamard ellipticity (LH), polyconvexity and Hill's inequality are nevertheless intimately connected. More precisely, by providing several alternative proofs, we show that both LH-ellipticity (rank-one convexity) and polyconvexity imply the weak Hill inequality in the incompressible two-dimensional case.