arXiv:2401.02242v3 Announce Type: replace Abstract: In this paper we consider sequences $u_j:B_2\subseteq M\to N$ of stationary harmonic maps between smooth Riemannian manifolds with uniformly bounded energy $E[u_j]\equiv \int |\nabla u_j|^2\leq \Lambda$ . After passing to a subsequence it is known one can limit $u_j\to u:B_1\to N$ with the associated defect measure $|\nabla u_j|^2 dv_g \to |\nabla u|^2dv_g+\nu$, where $\nu = e(x)\, H^{m-2}_S$ is an $m-2$ rectifiable measure \cite{lin_stat}. For a.e. $x\in S=\operatorname{supp}(\nu)$ one can produce a finite number of bubble maps $b_j:S^2\to N$ by blowing up the sequence $u_j$ near $x$. We prove the energy identity in this paper. Namely, we have at a.e. $x\in S$ that $e(x)=\sum_j E[b_j]$ for a complete set of such bubbles. That is, the energy density of the defect measure $\nu$ is precisely the sum of the energies of the bubbling maps.

arXiv:2606.19841v1 Announce Type: cross Abstract: 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

<\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 Ba\~{n}uelos, Kim and Kwa\'{s}nicki in \cite{BKK}. The optimal dimension-dependent $\ell^{1,\infty}$ estimate of $R_{\text{dis}}^{(k)}$ is also established.

arXiv:2606.19611v1 Announce Type: cross Abstract: 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 a constrained two-step mirror method with frozen operator evaluation. For the exact constrained iteration and each fixed regularization parameter \(\epsi>0\), we derive a one-step Bregman inequality and use it to prove that the constrained iteration converges strongly to the unique solution of the regularized variational inequality under natural summability conditions on the step sizes. Numerical experiments on one- and two-dimensional models, validated against exact test solutions, illustrate residual decay under mesh refinement and suggest improved practical performance of the two-step implementation in the tested discretizations.

arXiv:2606.20484v1 Announce Type: new Abstract: We consider a family of optimization problems, based on a mean-field description of particles interacting through Coulomb forces in a quadratic trap. In addition, the particles are constrained to lie in a halfspace and we are interested in the way the particle distribution changes as the halfspace varies. In particular, we can prove the existence of a phase transition, thereby settling a recent conjecture by Byun, Forrester, Majumdar and Schehr.

arXiv:2606.20273v1 Announce Type: new Abstract: We investigate the spectral stability of small-amplitude, periodic, traveling-wave solutions of the modified Camassa-Holm equation with cubic nonlinearities. More precisely, we analyze the $L^2(\mr)$-spectrum of the associated linearized operator in a neighborhood of the origin in the spectral plane. Inspired by a recently novel method based on Kato's perturbation theory [Berti et al, Full description of Benjamin-Feir instability of Stokes waves in deep water, \textit{Invent. Math.}, 230 (2022), 651-711.], we provide a complete description of the spectrum near the origin of the linearized operator--an integro-differential operator with periodic coefficients--and thus prove that such waves are not subject to modulational instability. Moreover, a spectral analysis reveals a remarkable threshold phenomenon: such waves with wave number $k^2\leq 3$ exhibit spectral stability, while instability emerges when $k^2>3$.

arXiv:2606.20237v1 Announce Type: new Abstract: This work concerns regularity properties of weak solutions to elliptic equations in divergence form -div(a$\nabla$u) = div F , under low regularity assumptions on both the coefficient a and the source term F . We introduce generalized Morrey and Campanato spaces extending the classical definitions by replacing uniform boundedness requirements with suitable integrability conditions. Within this framework, we establish regularity estimates for the gradient of weak solutions in these generalized spaces. As applications, we recover classical H{\"o}lder and Lebesgue estimates and derive fractional Sobolev regularity results. In particular, the proposed approach yields fractional Sobolev estimates in situations where the coefficient may be discontinuous and the gradient of the solution is not expected to be locally bounded.

arXiv:2606.20217v1 Announce Type: new Abstract: In this paper, we study a class of quasilinear elliptic problems involving the $(1,q)-$Laplacian operator and a discontinuous superlinear nonlinearity governed by the Heaviside function. The main difficulty of the problem arises from the presence of the $1$-Laplacian operator, whose natural setting is the Space of Functions of Bounded Variation. Our approach is based on an approximation method involving $(p,q)-$Laplacian problems as $p\to1^+$. As a consequence, we prove the existence of a nontrivial and nonnegative solution belonging to $W^{1,p}_0(\Omega)$, in an appropriate weak sense. Moreover, we investigate the asymptotic behavior of the solutions as $\beta\to0^+$, showing that the family of solutions converges to a solution of the limit problem without discontinuity.

arXiv:2606.20207v1 Announce Type: new Abstract: 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.

arXiv:2606.20099v1 Announce Type: new Abstract: 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

<\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 $p \geq 2$.

arXiv:2606.20033v1 Announce Type: new Abstract: 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

arXiv:2606.19942v1 Announce Type: new Abstract: 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.

arXiv:2606.19885v1 Announce Type: new Abstract: 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.

arXiv:2606.19872v1 Announce Type: new Abstract: 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 $\gamma > \frac95$.

arXiv:2606.19713v1 Announce Type: new Abstract: 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.

arXiv:2606.19650v1 Announce Type: new Abstract: 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.

arXiv:2606.19634v1 Announce Type: new Abstract: 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.

arXiv:2606.19631v1 Announce Type: new Abstract: 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--Amp\`ere 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.

arXiv:2606.19621v1 Announce Type: new Abstract: 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--Amp\`ere 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~$\lambda \to 0$.

arXiv:2605.15031v3 Announce Type: replace-cross Abstract: Already in $\bf{R}^4$, there are many minimal hypersurfaces, yet few structural results. We show that minimal submanifolds, of any dimension and codimension, that are confined in space are very restricted. It is well-known that the half-space theorem fails already for hypersurfaces in $\bf{R}^4$, where there are many examples contained in a slab. In $\bf{R}^3$ the height of the catenoid grows at a logarithmic rate, whereas in higher dimensions the height of the catenoid remains bounded. We will see that even in high dimensions, minimal submanifolds that are confined in space must satisfy strong structural restrictions. We show that any proper minimal immersion whose height grows sublinearly must have Euclidean volume growth. A consequence is an optimal Bernstein theorem in any dimension for stable hypersurfaces with sublinearly growing height that generalizes results of Moser, Bombieri-De Giorgi-Miranda, Trudinger, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. Euclidean volume growth is a powerful property and there are many other consequences.

arXiv:2512.21423v2 Announce Type: replace-cross Abstract: We analyze the trajectories of a massive particle in one space dimension whose motion is guided by a spin-half wave function that evolves according to the free Dirac equation, with its initial wave function being a Gaussian wave packet with a nonzero expected value of momentum $k$. We prove that at large times, the wave function is approximately equal to the superposition of two wave packets traveling in opposite directions, which results in trajectories with approximately constant asymptotic momentum $k$ and asymptotic energy $\pm c^2\sqrt{m^2+k^2}$, with $m$ the rest mass of the particle and $c$ the speed of light. The sign of the asymptotic energy is determined by the initial position of the particle. Particles with negative energy will have an asymptotic velocity that is in the opposite direction of their momentum. The proof uses the stationary phase approximation method, for which we establish a rigorous error bound.

arXiv:2512.19647v4 Announce Type: replace-cross Abstract: This article studies the temporal approximation of hyperbolic semilinear stochastic evolution equations with multiplicative Gaussian noise by Milstein-type schemes. We take the term hyperbolic to mean that the leading operator generates a contractive, not necessarily analytic $C_0$-semigroup. Optimal convergence rates are derived for the pathwise uniform strong error \[ E_h^\infty := \Big(\mathbb{E}\Big[\max_{1\le j \le M}\|U_{t_j}-u_j\|_X^p\Big]\Big)^{1/p} \] on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $u_j$ its Milstein approximation at time $t_j=jh$ with step size $h>0$ and final time $T=Mh>0$. For sufficiently regular nonlinearity and noise, we establish strong convergence of order one, with the error satisfying $E_h^\infty\lesssim h\sqrt{\log(T/h)}$ for rational Milstein schemes and $E_h^\infty \lesssim h$ for exponential Milstein schemes. This extends previous results from parabolic to hyperbolic SPDEs and from exponential to rational Milstein schemes. Moreover, root-mean-square error estimates are strengthened to pathwise uniform estimates. Numerical experiments validate the convergence rates for the stochastic Schr\"odinger equation. Further applications to Maxwell's and transport equations are included.

arXiv:2606.07977v2 Announce Type: replace Abstract: In this paper, we prove the local boundedness of solutions to systems of partial differential equations in divergence form. More specifically, we consider systems that include the first variations of functionals depending on the spatial variable and exhibiting nonstandard growth with respect to the gradient, such as $$\int_{\Omega} \left( 1+ h(|Du|)\right) ^{\alpha(x)} \, d x,$$ where the convex function $h=h(t)$ does not satisfy the so-called $\Delta_2$ property and does not exhibit the conventional polynomial growth behavior.

arXiv:2601.17968v2 Announce Type: replace Abstract: This paper investigates density driven flow in porous media, focusing on the roles of viscosity contrast, density contrast, and linear adsorption. In this setup, the fluid on top is heavier and more viscous than the fluid below. Under the effect of gravity, this system becomes unstable, and finger-like structures appear. The phenomenon is described mathematically by coupling Darcy's law with a convection-diffusion reaction equation. The nonlinearity in this model arises mainly from the concentration dependence of viscosity and the convective transport term. The existence of a unique pair of weak solutions is shown using the Galerkin approximation method and truncation technique. Moreover, an application of the maximum principle shows non-negativity of the concentration. Additionally, we analyze the long-time behavior of the solution and prove that the concentration converges exponentially to zero in the $L^p$-norm for all $1 \le p \le \infty$ as $t \to \infty.$ To complement the theoretical analysis, we perform numerical simulations based on a pressure formulation. By tracking total kinetic energy and mixing measures over time, we discuss the instability and the mixing efficiency, respectively. The present study reveals that although increasing the density contrast amplifies the total kinetic energy, the marginal impact diminishes with successive increments of density contrast. Similarly, while adsorption acts to suppress mixing, its efficiency in doing so tends to saturate with further increases. These behavior are consistent with the numerical simulations.

arXiv:2512.21171v3 Announce Type: replace Abstract: We study a diffuse--interface model for a binary incompressible mixture in a periodically perforated porous medium, described by a time-dependent Navier--Stokes--Cahn--Hilliard (NSCH) system posed on the pore domain $\Omega_p^\varepsilon\subset\mathbb{R}^3$. The microscopic model involves a variable viscosity tensor, a non-conservative source term in the Cahn--Hilliard equation, and mixed boundary conditions: no-slip on the outer boundary and Navier slip with zero tangential stress on the surfaces of the solid inclusions. The capillarity strength $\lambda^\varepsilon>0$ depends on the microscopic scale $\varepsilon>0$. The analysis consists of two main parts. First, for each fixed $\varepsilon>0$ we prove existence of a weak solution on a finite time interval $(0,T)$ and derive a priori estimates that are uniform with respect to $\varepsilon$ (and $\lambda^\varepsilon$). Second, we perform the periodic homogenization for the perforated setting in the limit $\varepsilon\to0$. Depending on the limit value $\lambda$ of the capillarity strength $\lambda^\varepsilon$, we obtain two distinct effective models: (i) in the vanishing capillarity regime $\lambda=0$, the limit system decouples completely into a standalone linear Stokes system for the velocity--pressure pair and a standalone Cahn--Hilliard system with source term $G$ for the phase field and chemical potential, with no macroscopic convection, advection, or capillary coupling between the two; (ii) in the balanced regime $\lambda\in(0,+\infty)$, we derive a Navier--Stokes--Cahn--Hilliard system with nonlinear convection and advective transport of the phase field at the macroscopic scale, coupled through a capillary forcing term. Finally, we establish the convergence of the microscopic free energy to a homogenized energy functional satisfying an analogous dissipation law.

arXiv:2511.08104v2 Announce Type: replace Abstract: This article focuses on establishing a positive weak solution to a class of semipositone problems over the Heisenberg group $\mathbb{H}^N$. In particular, we are interested in the positive weak solution to the following problem: \begin{equation}\label{p1} -\Delta_{\mathbb{H}}u= g(\xi)f_a(u) \text{ in } \mathbb{H}^N \tag{$P_a$}, \end{equation} where $a>0$ is a real parameter and $g$ is a positive function. The function $f_a: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and of semipositone type which means it becomes negative on some parts of the domain. Due to this sign-changing nonlinearity, we can not directly apply the maximum principle to obtain the positivity of the solution to \eqref{p1}. For that purpose, we need some regularity results for our solutions. In this direction, we first prove the existence of weak solutions to \eqref{p1} via the mountain pass technique. Further, we establish some regularity properties of our solutions and using that we prove the $L^\infty$-norm convergence of the sequence of solutions $\{u_a\}$ to a positive function $u$ as $a \rightarrow 0$, which yields $u_a \geq 0$ for $a$ sufficiently small. Finally, we use the Riesz-representation formula to obtain the positivity of solutions under some extra hypothesis on $f_0$ and $g$. To the best of our knowledge, there is no article dealing with semipositone problems in Heisenberg group set up.

arXiv:2510.17629v3 Announce Type: replace Abstract: This paper studies the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

arXiv:2509.01268v2 Announce Type: replace Abstract: For any initial datum $\theta_0\in L^{\frac{4}{3}}_x$ it is proved the existence of a global-in-time weak solution $\theta \in L^\infty_t L^{\frac43}_x$ to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the $\dot{H}^{-\frac{1}{2}}_x$ norm, is constant in time. The solution is obtained as a vanishing viscosity limit. The main idea is to propagate in time the non-concentration of the $L^{\frac{4}{3}}_x$ norm of the initial data, from which the strong compactness in the Hamiltonian norm is deduced. Minimal Onsager supercritical conditions preventing anomalous dissipation are given.

arXiv:2508.07802v3 Announce Type: replace Abstract: In this paper, we would like to study the critical exponent for semilinear damped wave equations with the nonlinearity terms of Coulomb-type singularities $|x|^{-\alpha} |u(t,x)|^p$ and the initial data belonging to Sobolev spaces of negative order $\dot{H}^{-\beta}$. Precisely, we obtain a critical exponent $$p_{\rm c}(\alpha,\beta,n): = 1 + \frac{4-2\alpha}{n+2\beta} $$ for $1 \leq n \leq 4$ and $ 0 \leq \alpha, \beta < n/2,$ by proving the global (in time) existence of small data solutions when $p \geq p_{\rm c}(\alpha,\beta,n)$ and the blow-up result for weak solutions in finite time even for small data if $1 < p < p_{\rm c}(\alpha,\beta,n)$. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs.

arXiv:2504.21300v3 Announce Type: replace Abstract: In this paper, we prove a decomposition lemma for symmetric matrix fields on bounded domains: $D+\mathrm{Sym}\nabla\Phi=\sum_i a_i^2\xi_i\otimes\xi_i$ with uniform control on $\Phi$ and $a_i^2$, using fewer than the usual $n(n+1)/2$ rank-one symmetric terms. Except possibly in dimensions $n=8,16$, the decomposition is shown to be optimal through algebraic arguments. This reduces the number of steps in convex integration for a nonlinear PDE system, improving H\"older regularity of flexible solutions in dimension $n\ge3$. This PDE is a partial linearization of the codimension-one local isometric embedding equation in the Nash--Kuiper theorem, and also yields improved regularity for very weak solutions of related 2D Monge--Amp\'ere and $2$-Hessian systems. The improved H\"older exponent is any $\alpha<(n^2+1)^{-1}$ for $n=2,4,8,16$ and any $\alpha<(n^2+n-2\rho(n/2)-1)^{-1}$ otherwise, where $\rho$ is the Radon--Hurwitz number, related to Bott periodicity. The proof involves novel applications of algebraic geometry and topology that yield the optimality of decomposition, including Adams' theorem on vector fields on spheres, intersections of projective varieties, and projective duality, combined with an elliptic method that avoids loss of differentiability.

arXiv:2412.11176v4 Announce Type: replace Abstract: We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on $\mathbb{R}^n$. A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth.

arXiv:2412.08493v3 Announce Type: replace Abstract: We consider weak solutions to the incompressible Euler equations. It is shown that energy conservation holds in any Onsager critical class in which smooth functions are dense. The argument is independent of the specific critical regularity and the underlying PDE. This groups several energy conservation results and it suggests that critical spaces where smooth functions are dense are not at all different from subcritical ones, although possessing the "minimal" regularity index. Then, we study properties of the dissipation $D$ in the case of bounded solutions that are allowed to jump on $H^d$-rectifiable space-time sets $\Sigma$, which are the natural dissipative regions in the compressible setting. As soon as both the velocity and the pressure posses traces on $\Sigma$, it is shown that $\Sigma$ is $D$-negligible. The argument makes the role of the incompressibility very apparent, and it prevents dissipation on codimension 1 sets even if they happen to be densely distributed. As a corollary, we deduce energy conservation for bounded solutions of "special bounded deformation", providing the first energy conservation criterion in a critical class where only an assumption on the "longitudinal" increment is made, while the energy flux does not vanish for kinematic reasons.

arXiv:2405.11486v4 Announce Type: replace Abstract: In this work, we study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author in [22] in the context of the energy conservation for the Euler equations in Onsager-critical classes. Among other things, we prove that the normal Lebesgue trace satisfies the Gauss-Green identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measure-divergence vector fields and the strong one for $BV$ functions. These results are then applied to the study of the uniqueness of weak solutions for continuity equations on bounded domains, allowing to remove the assumption in [19] of global $BV$ regularity up to the boundary, at least around the portion of the boundary where the characteristics exit the domain or are tangent. The proof relies on an explicit renormalization formula completely characterized by the boundary datum and the positive part of the normal Lebesgue trace. In the case when the characteristics enter the domain, a counterexample shows that achieving the normal trace in the Lebesgue sense is not enough to prevent non-uniqueness, and thus a $BV$ assumption seems to be necessary to get uniqueness.

arXiv:2304.10964v3 Announce Type: replace Abstract: We discuss the Lugiato-Lefever equation and its variant with third-order dispersion, which are mathematical models used to describe how a light beam forms patterns within an optical cavity. It is mathematically demonstrated that the solutions of these equations follow the Talbot effect, which is a phenomenon of periodic self-imaging of an object under certain conditions of diffraction. The Talbot effect is regarded as the underlying cause of pattern formation in optical cavities.