Papers

We prove global reverse Hölder estimates in \(\mathrm L^2\) for the Poisson kernel of scalar divergence-form parabolic operators in the upper half-space. The coefficient matrix is real, symmetric, and periodic in every spatial variable \(X=(\lambda,x)\), while no periodicity in time is assumed. If its boundary trace \(A^0=A^0(x,t)\) has a quantitative uniform modulus of continuity in time and its convergence to \(A^0\) in the transverse variable is controlled by a square-Dini modulus, then the Poisson kernel satisfies a uniform \(\mathrm{RH}_2\) estimate at scales below the period. Full spatial periodicity is then used to propagate this estimate to every scale. The main new ingredient in the large-scale argument is a positive adjoint height coordinate \(\Phi^*=\lambda+O(1)\). Boundary comparison with \(\Phi^*\) yields the required large-scale decay of the Green function and hence the global \(\mathrm{RH}_2\) estimate. Consequently, the \(\mathrm L^2\) Dirichlet problem is uniquely solvable among weak solutions \(u\) satisfying \(N_\ast u\in\mathrm L^2\). We also establish, for fully spatially periodic coefficients, a nonsymmetric \(A_\infty\) theorem under square-Dini convergence to a transverse-independent boundary trace, with no temporal regularity beyond measurability. We explain how the results transfer to time-independent Lipschitz graph domains under the stated coefficient and periodicity-compatibility hypotheses, and we derive scale-uniform boundary estimates for the periodic homogenization families considered below. The condition \(D_t^{1/2}A^0\in\mathrm L^\infty_x(\mathrm{BMO}_t)\) provides a natural sufficient criterion for the temporal continuity required in the symmetric theorem.

We prove that the system of Favre-filtered compressible Navier–Stokes equations with subgrid-scale closure terms, written in the coordinates $(Lt,Lx_1,Lx_2,Lx_3)$ for an arbitrary parameter $L>0$, is obtained from the system written in standard coordinates $(t,x_1,x_2,x_3)$ by the action of a smooth diffeomorphism of the underlying spacetime manifold. The proof begins with a self-contained treatment of infinite-dimensional manifolds and the Fréchet manifold of smooth field configurations. We then define the one-parameter dilation group $\{\phi_L\}_{L>0}$, verify that each $\phi_L$ is a diffeomorphism, and compute its tangent and cotangent actions. The induced pullback action on sections of the configuration bundle is shown to commute with the total derivative operator, which implies that the first-jet prolongation of $\phi_L$ maps the equation submanifold of the system onto itself. Equation-by-equation verification—continuity, momentum, and energy—confirms that every residual is multiplied by the common nonzero factor $L^{-1}$ (or, equivalently, is identically zero in the transformed coordinates), so that the solution set is preserved. We conclude that the family of systems parameterised by $L>0$ satisfies the diffeomorphism transformation expression and that solution spaces are canonically isomorphic for all $L$.

We initiate a geometric theory of $p$-Brownian motion, the nonlinear Markov process associated with the $p$-Laplacian introduced by Barbu-Rehmeier-Röckner. More precisely, we analyze its radial processes thoroughly, relative to an arbitrary center and in every dimension. On the one hand, we show explicit Tanaka–Meyer semimartingale formulas; our consequential characterization of nontriviality of the associated local times reveals notable differences to classical Brownian motion. In parallel, we establish sharp exit time estimates as well as scaling estimates for the self-similarly rescaled radial process. We also prove isometry of the corresponding marginal laws in each Wasserstein distance. Our contributions equally cover the Leibenson process - the nonlinear Markov process associated with the "porous medium equation" with $p$-Laplacian - recently introduced by Barbu-Grube-Rehmeier-Röckner.

The Hausdorff dimension of the singular set of local minimizers of quasiconvex integrals is strictly less than the ambient dimension.

We provide a shorter and more direct proof of $L^p$ estimates for maximal Riesz transforms (of an arbitrary order) in terms of the corresponding Riesz transforms, with a constant independent of the dimension of the Euclidean space $\mathbb R^d$. This result was originally proved by Mateu, Orobitg, Pérez and Verdera with a constant depending on the dimension, and improved to a dimension-free inequality by Kucharski, Wróbel and Zienkiewicz.

In this paper, we study the first variational Steklov eigenvalue of the $ p$-Laplace equation on exterior domain $ \Omega^{\text{ext}}$ for $ 1< p <n$. If $ \Omega$ is convex and $ \partial \Omega\in C^{1,1}$, we prove a sharp lower bound in terms of $p$-logarithmic mean of the principal curvatures of $ \partial \Omega$. For the linear case $ p=2$, our estimate reduces to the logarithmic mean bound of Bundrock et al. (arXiv:2511.09490). We also derive an upper bound in terms of a boundary isocapacitary constant. The analysis relies on the established finite energy theory on exterior domains, together with a decay estimate for $p$-harmonic extensions.

In this paper, we study the coupled singular weighted elliptic system $$\left\{ \begin{aligned} -\operatorname{div}(|x|^{-2a}\nabla u)+\lambda_1 \frac{u}{|x|^{2a}}&=\beta p\frac{|v|^q|u|^{p-2}u}{|x|^{b(p+q)}}+\frac{|u|^{2^{\sharp}-2}u}{|x|^{b2^{\sharp}}},\quad\text{in}~\mathbb{R}^N, -\operatorname{div}(|x|^{-2a}\nabla v)+\lambda_2\frac{v}{|x|^{2a}}&=\beta q\frac{|u|^p|v|^{q-2}v}{|x|^{b(p+q)}}+\frac{|v|^{2^\sharp-2}v}{|x|^{b{2^\sharp}}},\quad\text{in}~\mathbb{R}^N, \displaystyle\int_{\mathbb{R}^N}\frac{|u|^2}{|x|^{2a}} dx=\rho_1^2,\quad &\int_{\mathbb{R}^N}\frac{|v|^2}{|x|^{2a}}dx=\rho_2^2. \end{aligned} \right. $$ where $N\ge3$, $\beta\in\mathbb{R}$, $\rho_1,\rho_2>0$, $\max\{0,\tfrac{N-4}{2}\}\le a<\tfrac{N-2}{2}$, $a<b<a+1$, $2^{\sharp}:=\frac{2N}{N-2(1+a-b)}$, $p,q>1$ and $2<p+q<2^{\sharp}$. Here $2^\sharp$ is the critical exponent of the Caffarelli–Kohn–Nirenberg inequality, and $\lambda_1,\lambda_2\in\mathbb{R}$ are unknown, since the two weighted masses are prescribed. We show that the ground state level is not attained for $\beta\le0$. For $\beta>0$ we obtain positive normalized ground states, with positive Lagrange multipliers, in the mass subcritical, mass critical, and mass supercritical regimes, for explicit ranges of $\beta$. In the mass supercritical regime, a threshold $\beta_0\ge0$ appears: a ground state exists for every $\beta>\beta_0$ and does not exist for $0<\beta<\beta_0$. We give conditions on $p,q$ under which $\beta_0=0$, and we prove that $\beta_0>0$ when $p,q\ge2$. Finally, we describe the ground states as $\beta\to0^+$ and as $\beta\to+\infty$: after a dilation, they converge to the ground states of the limit system without critical terms, or one component vanishes and the other concentrates on an extremal of the Caffarelli–Kohn–Nirenberg inequality.

We consider distributional weak solutions $u\in L^\infty_tL^2_x$ of the two-dimensional Navier–Stokes equations with horizontal viscosity on $\mathbb T^2$ or $\mathbb R^2$, without assuming $\partial_1u\in L^2_tL^2_x$. We show that the Lions integrability condition $u\in L^4_tL^4_x$ together with Onsager-critical regularity in the nondissipative direction, $u\in L^3_tB^{1/3,v}_{3,\infty}$ with the Besov regularity imposed only in the vertical direction, implies $\partial_1u\in L^2_tL^2_x$, with a quantitative bound on the horizontal dissipation. Thus the dissipative regularity is a consequence of the equation and need not be part of the definition of the solution. If moreover $u\in L^3_tB^{1/3,v}_{3,c_0}$, then $u$ belongs to the isotropic critical Onsager space $L^3_tB^{1/3}_{3,c_0}$, satisfies the energy equality, and is continuous in $L^2$. Here $c_0$ indicates that the corresponding dyadic Besov sequence tends to zero at high frequencies. The proof combines absorption of the horizontal energy flux into the dissipation with a one-dimensional commutator estimate for the vertical energy flux.

We develop a randomly-localized energy method to derive qualitative results on the support propagation of stochastic porous media equations with linear conservative noise. Unlike in previous works, where energies are localized by weighting them with a spatial bump function, we weight them with profiles which are solutions to stochastic transport equations. This modulates out the support propagation due to the conservative noise term, and energy arguments—which otherwise fail in this setting—are again applicable. As a result, finite speed of propagation along with sufficient and necessary conditions on the existence of waiting time phenomena (modulo stochastic transport) are proven. These methods and results demonstrate, on short time scales, that the support propagation may be disintegrated into two independent parts, one due to the evolution of the porous media equation and one due to stochastic transport.

Let $B=B(p, 1)\subset\mathbb{R}^n$ be a unit ball and $n\ge 3$. We prove that there are positive constants $c$ and $C$, depending only on $n$, such that every non-zero real-valued harmonic function $u: 4B\to \mathbb{R}$ with $u(p)=0$ satisfies $$ \mathcal{H}^{n-1}\bigl(\{u=0\}\cap 2B\bigr)\ge C\mathcal{N}, $$ and $$ \mathcal{H}^n\bigl(\{u>0\}\cap {\frac 12}B\bigr) \ge c\bigl(\log(1+\mathcal{N})\bigr)^{1-n}, $$ where $\mathcal{N}$ is the doubling index defined by $$\mathcal{N}=\log_2\frac{\sup_{B}|u|}{\sup_{\frac{1}{2}B}|u|}.$$ The first estimate confirms a folklore conjecture on the nodal volume of harmonic functions, and its linear dependence on $\mathcal{N}$ is optimal. The second estimate extends the planar result of Nazarov, Polterovich, and Sodin to higher dimensions, and the logarithmic order is optimal. As a further consequence of the ideas developed in the proof, we obtain an alternative proof of Nadirashvili's conjecture that does not rely on the multiscale analysis.

We stabilize stationary states of coupled multi-species McKean–Vlasov equations by feedback control, extending a spectral approach based on the Wasserstein Hessian and a Riccati equation from a single density to a product of Wasserstein spaces. We give necessary and sufficient conditions for local exponential stabilization at a prescribed rate when only one species is actuated, together with the minimum number of control inputs required. For two species, the condition reduces to a rank condition on the Fourier coefficients of the cross-interaction kernel over the eigenspaces of the unactuated species' linearized dynamics. A two-community noisy Kuramoto model and a cell-sorting model illustrate the results.

We establish sharp Sobolev-type geometric inequalities on $\mathbb{S}^n$ involving the total $\sigma_k$-curvatures $\int_{\mathbb{S}^n}\sigma_k(g)\,dv_g$. These results extend the optimal inequalities of Guan–Wang~\cite{GWDuke} from the cone $\mathcal{C}_k$ to the strictly larger cone $\mathcal{C}_{k-1}$, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in $\mathcal{C}_{k-1}$ introduces substantial analytic difficulties; in particular, one must obtain $C^2$ a priori estimates while simultaneously verifying that the flow remains parabolic. We resolve these issues via a carefully designed test function and by applying the maximum principle to the maximal eigenvalue of the Hessian matrix. As applications, we solve two open problems in dimensions 3 and 4. Finally, we give examples to show that these inequalities cannot be extended to $\mathcal{C}_{k-2}$.

We derive the $\Gamma$-limit of convolution-type and discrete energies subject to Dirichlet boundary conditions on periodically perforated domains at the critical exponent. We assume that the length-scale of the non-local interactions is much smaller than the side-length of the cubic perforations and prove that a separation of scales occurs. Exploiting the analogies between the variational frameworks of our interest, we employ a unified argument to overcome the technical difficulties that arise from the scaling invariance of the energies. Our multiscale analysis yields a novel observation that is not related to the non-local nature of the functionals, but rather to the analysis at the critical exponent: we prove that the energy density of the $\textit{strange term}$ is convex, even in the vector-valued setting.

For a wide class of Helmholtz scattering problems (namely, those fitting in the black-box framework of Sjöstrand–Zworski) we prove that the error incurred by approximating the radiation condition by a Cartesian perfectly-matched layer (PML) decreases exponentially in the width of the PML, the strength of the PML scaling, and the frequency.

We study quasilinear symmetrizable partially dissipative hyperbolic systems with non-autonomous relaxation coefficients in $\mathbb{R}^d$ ($d\geq1$). The existence of global strong solutions is established in a critical regularity setting for systems satisfying the so-called Shizuta-Kawashima (SK) and entropy conditions. When the initial data are additionally bounded in a lower-regularity norm, we prove that the corresponding solutions converge to equilibrium at optimal algebraic decay rates. Furthermore, we show that the conservative part of the solution behaves asymptotically as the solution of a non-autonomous parabolic equation. Our results apply to the compressible Euler system with the time-dependent damping coefficient $\frac{K}{(1+t)^{\alpha}}$ ($\alpha<1$, $K>0$ or $\alpha=1$, $K\gg 1$) in the velocity equation. The natural low/high-frequency splitting of the autonomous theory persists in the non-autonomous setting, but with a frequency-threshold that evolves in time. To handle this moving frequency structure, we introduce a new class of hybrid Besov spaces adapted to time-dependent thresholds and derive hypocoercive estimates in each frequency regime. Our results reveal the qualitative and quantitative effects of general time-dependent relaxation coefficients on dissipation and large-time dynamics.

This is the first part of a project concerning the regularity of the free boundary of a function, where the free boundary depends on the gradient of the function. We study the minimizer of the expression \begin{equation*} J(u) : = \int_{B_1} F(\nabla u) \ dx, \end{equation*} where $F(x)$ is a uniformly convex function whose second derivatives might jump at $|x|=1$. This results in an Euler-Lagrange equation that varies over the free boundary, which we define as $$ \Gamma = \partial \{x \in B_1: |\nabla u| >1 \} \cap \partial \{x \in B_1: |\nabla u| <1 \}. $$ We consider two-phase flat points for which the second derivatives of $F$ jump over $\Gamma$. In this paper, we show that under some regularity and non-degeneracy assumptions, a minimizer $u$ can be expressed as $$ u(x)= a + \nu \cdot x+ \delta p(x) + \delta \epsilon q(x), $$ where $a \in \mathbb{R}$, $\nu \in \mathbb{R}^n$, $0 < \delta, \epsilon << 1$. Here $p$ is a $C^1$ function, which consists of one polynomial in the upper half ball and another polynomial in the lower half ball. The function $q$ is a rest term with bounded $L^2$ norm. Furthermore, assuming that we have a sequence, $u^j$, of minimizers, we show that $q^j$ converges weakly to a function $q^0$ that satisfies a certain PDE. In addition, we show that $q^0$ is $C^3$ in $B_r^+$ and $B_r^-$, respectively, for $0<r<1$. This is the main result of this paper which is intended to be used to show regularity of the free boundary.

We study the evolutionary equation $$\frac{\partial u}{\partial t} = (\mathcal{L}_{\infty})^{\alpha}u \quad \text{in} \quad \Omega,$$ where $(\mathcal{L}_{\infty})^{\alpha}u$ denotes a nonlocal infinity Laplacian acting on the function $u$, $0<\alpha \leq 1$ and $\Omega$ is a bounded open set in $\mathbb{R}^{n+1}$. We prove existence and uniqueness using Perron's method for the Dirichlet problem when $\Omega$ is a cylinder and $0<\alpha<1$.

We establish a sharp criterion for global-in-time existence versus finite-time blow-up of strong solutions to the 3D spherically symmetric compressible Navier-Stokes equations on a solid ball, based solely on the initial sign of effective velocity. The viscosity coefficients are assumed to satisfy the Bresch-Desjardins structure $\mu=\rho^{\alpha}$, $\lambda=(\alpha-1)\rho^{\alpha}$, with the effective velocity given by $v=u+\alpha \rho^{\alpha-2}\rho_r$. Previously, global existence results for strong solutions in higher dimensions were restricted to the case $\alpha\le 1$ with the endpoint $\alpha=1$ corresponding to the viscous Saint-Venant (shallow water) system. In this paper, we extend the global existence theory beyond this threshold to the supercritical regime $\alpha>1$. Specifically, whenever the initial effective velocity is nonnegative on the boundary of the ball, we prove the global-in-time existence of strong solutions for arbitrarily large initial data. The main difficulty lies in deriving a uniform lower bound for the density, which requires handling singular estimates at the center of the ball. To overcome this, we exploit several novel quantities near the center and employ a refined maximum principle to establish, for the first time, the Lipschitz continuity of the velocity at the center. Subsequently, we derive Dini-Gronwall-type inequalities for suitably constructed functions, which close the density lower bound estimate. Conversely, we can construct a family of initial data for which the initial effective velocity is negative on the boundary, such that the corresponding strong solutions blow up in finite time, with vacuum appearing exactly on the boundary at the blow-up time. Our results establish a clean dichotomy between global regularity and singularity formation, governed purely by the initial sign of the effective velocity.

We study the continuity equation with reaction $\partial_t\mu + \operatorname{div}(v\mu) = w\mu$ on $[0,T]\times\mathbb{R}^d$, driven by Borel velocity and reaction fields $v$ and $w$. We provide the first version of a Lagrangian superposition principle for the equation in its natural generality, assuming only finite quadratic energy without imposing boundedness conditions on the reaction term neither any form of compactness for the support of the measure. More precisely, every solution $\mu \in \mathcal{C}([0,T];\mathscr{M}_+(\mathbb{R}^d))$ with finite quadratic energy is represented by a probability measure $\eta$ on absolutely continuous curves in the geometric cone over $\mathbb{R}^d$, concentrated on the solutions of the characteristic system $x' = v(x)$, $k' = w(x)\,k$, through the $2$-homogeneous marginal $h^2_t(\eta)=\mu_t$. We also establish the converse implication. The proof passes through a regularization of the triple $(v,w,\mu)$ which produces fields satisfying only local bounds; the core of the argument is therefore a representation theory under such local assumptions, whose key tool, of independent interest, is a two-time representation formula relating $\mu_s$ and $\mu_t$ along the flow of $v$ for arbitrary times $s,t$.

For \(\frac12\le s<1\), we study a fractional Choquard equation with prescribed mass and nonlinearities at the lower Hardy-Littlewood-Sobolev and \(L^2\)-critical exponents. The sharp Hardy-Littlewood-Sobolev and Choquard Gagliardo-Nirenberg inequalities determine an explicit critical mass \(a_*\). For \(0<a\le a_*\), we compute the exact infimum of the constrained energy and prove that it is not attained and that no normalized solution exists. For \(a>a_*\), the energy is unbounded from below on the mass sphere, while the Pohozaev set is nonempty.

Free boundary problems (FBPs) are differential equations where the domain depends on the solution, with applications in diverse fields such as flame propagation, financial mathematics, and tumor growth. Classical examples include the Bernoulli problem, the obstacle problem, and the Stefan problem. The field has advanced significantly through variational and geometric approaches, notably by Alt, Caffarelli, and collaborators, who developed monotonicity formulas and viscosity solutions to address free boundary regularity. Extensions have included nonlinear operators, inhomogeneous problems, and lower-dimensional cases like the thin Bernoulli problem. Evolution problems, such as the Hele-Shaw flow, also connect to FBPs. To celebrate these developments and honor Sandro Salsa's pioneering contributions, a special session on FBPs was held at the AMS-UMI joint meeting in Palermo (2024). This is the editorial of a volume dedicated to Sandro Salsa on the occasion of his 75th birthday.

In this work, we study the null controllability by one-sided boundary controls of one-dimensional hyperbolic balance laws with constant coefficients. Our first result shows that, when the system has only one negative or positive speed, the minimal null control time of such systems depends on some orthogonality conditions for a particular sequence. This sequence is explicit in function of the coefficients of the system but it is defined by a nonlinear recurrence relation. Our second result then completes the previous one by giving explicit bounds on the number of orthogonality conditions that have to be checked in two nontrivial situations. The proofs rely on a careful analysis of the so-called kernel equations associated with the system, including a new well-posedness result. Our results are also valid for the finite-time stabilization property.

Hyperbolic PDEs with time delay are essential for modeling a wide range of real-world applications. Since closed-form solutions are often not attainable, developing accurate and efficient numerical methods becomes crucial. In this work a novel substructuring waveform relaxation method namely Neumann-Neumann Waveform Relaxation (NNWR) has been implemented for numerically solving hyperbolic delay PDEs in the case of non-uniform subdomain partitioning. The convergence behavior is first analyzed using Fourier analysis to obtain an estimate, followed by a demonstration of finite-step convergence through Laplace transform techniques. Various test cases are presented as numerical illustrations.

We construct modified wave operators for the nonlinear Schrödinger equation $i\partial_tu+\frac12\Delta u=|u|^pu$ in the full subcritical long-range case $0<p<2/d$, with small, nonvanishing, analytic final data $W(x)$ with bounded logarithmic gradients. Previous results established large-time asymptotics for selected classes of Cauchy data. Moreover, the exact asymptotic expansion for $p<1/d$ remained unknown. When $1/d<p<2/d$, our result gives the approximation $$\frac{1}{(it)^{\frac{d}{2}}} e^{\frac{i|x|^2}{2t}} W\left(\frac{x}{t}\right) \exp\left[ -i\frac{t^{1-\frac{dp}{2}}-1}{1-\frac{dp}{2}} \left|W\left(\frac{x}{t}\right)\right|^p \right]$$ The wave operator is constructed by an iteration in the analytic spaces with decreasing radius. When $p\le 1/d$, we construct the profile from a finite truncation of a Fuchsian equation coupled with a transport equation. This profile still leaves a long-range triangular coupling whose terminal integral does not preserve the required fast decay class. The construction yields quantitative $L^q$ asymptotics for $2\le q\le\infty$ and uniqueness in the prescribed analytic asymptotic classes. The central new ingredients are a nonlinear final-state normal form that removes this long-range coupling and a mixed iteration in particular analytic spaces.

This is the second part of a two-part work concerning boundary layer solutions to the coupled Chemotaxis-Navier-Stokes system in the two-dimensional half-space. In the present work, we address the convergence of boundary layer solutions to singular chemotaxis-fluid equations under slip boundary conditions with respect to the chemical diffusion-viscosity parameter $\varepsilon$ in the two-dimensional half-plane. More precisely, we show that the boundary layer for $\varepsilon>0$ (viscous convection coefficient) converges to the superposition of the outer layer (solution with $\varepsilon=0$) and the inner layer as $\varepsilon\rightarrow0$. The outer and inner profiles are explicitly derived as in the first part\cite{WWZ}. Furthermore, the well-posedness results of the coupled Chemotaxis-Navier-Stokes system in conormal Sobolev spaces will be presented in Appendix. They answer the question mentioned in the first part of the two-part work. This study could help the understanding of the chemotactic movement of aerobic bacteria to the water-air surface observed experimentally in fluids, and enrich the theoretical results of boundary layer in chemotactic fluid models.

We study the Cauchy problem for a three-dimensional bipolar Euler–Poisson system in which one fluid is pressureless and undamped, while the other is subject to momentum relaxation. For sufficiently small smooth perturbations of a constant equilibrium, we prove the global existence and uniqueness of smooth solutions under an irrotationality assumption on the initial velocity of the pressureless fluid, together with algebraic time-decay estimates. The main difficulty is that the velocity of the pressureless fluid is dissipated only indirectly through the Poisson coupling, and this mechanism degenerates strongly at high frequencies, leading to a regularity-loss structure. We overcome this difficulty by combining refined Green-function estimates, a low–middle–high frequency decomposition, and high-order nonlinear energy estimates adapted to the asymmetric regularity hierarchy. The result establishes a global small-data theory for this asymmetric regime, in which pressure and damping are simultaneously absent from the same fluid.

This paper is devoted to the well-posedness of classical Prandtl equations in a finite order Sobolev space. For a initial data with degenerate critical points and general outflow, we obtain the local-in-time existence and uniqueness of the solution to the Prandtl equations in a Sobolev space, by introducing a new iteration scheme and linear cancelation. This result shows that Oleinik's monotonicity condition is not a necessary condition for the Prandtl equations to be well-posed in Sobolev spaces and provides evidence to demonstrate that zero shear stress does not necessarily lead to boundary layer separation in two-dimensional unsteady boundary layers.

Let $\Omega\subset\R^N$, $N\ge2$, be a bounded domain and let $1<p<\infty$. Inspired by the exponential case treated in \cite{BO2026}, we study the quasilinear Schrödinger–Maxwell system \[ \begin{cases} -\Delta_p u+\psi G'(u)=f &\text{in }\Omega,\\ -\Delta_p\psi=G(u) &\text{in }\Omega,\\ u=\psi=0 &\text{on }\partial\Omega, \end{cases} \] where $G\in C^1(\R)$ is even, convex, nonnegative and nontrivial, with $G(0)=0$, and satisfies \[ G(t)\lesssim 1+|G'(t)| \qquad\text{for large }|t|. \] Under the assumption \[ |f|\ln(1+|f|)\in L^1(\Omega), \] we prove the existence of a finite-energy weak solution and show that the solution is a saddle point of the associated functional.

We provide evidence of a finite-time singularity in the 3D Euler equations on the unbounded domain. Using a physics-informed neural network (PINN) with a self-similar ansatz, we find an approximate singular profile for the Euler system at the critical blowup rate of $0.5$ and certify it using a spline representation. The transport field associated with the obtained profile has local outgoing property throughout the domain that suggests linear damping, a key stabilizing mechanism for the candidate profile. We also establish a framework for proving nonlinear stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants.

In a recent numerical study, we found a high-precision singular profile for the Euler equations on the unbounded domain $\mathbb{R}^3$. The present manuscript complements that study by establishing in detail a preliminary framework for proving (nonlinear) stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants. Conditional on rigorous certification of the estimates and constants appearing in the argument, and on the candidate profile satisfying the required nonlinear stability conditions, the framework closes the stability proof and, crucially, allows the resulting stable rescaled profile to be reconstructed as an admissible solution in the original variables that becomes singular in finite physical time. With the overall stability and reconstruction mechanisms formulated, the remaining work within this approach is largely quantitative: determining whether the explicit constants and margins can be rigorously certified with sufficient positive margin and, where necessary, sharpening selected analytic estimates.