Papers

We study the inviscid limit of the one-dimensional viscous Burgers equation from smooth initial data through nondegenerate shock formation and subsequent shock development. Chaturvedi and Graham \cite{CG} obtained sharp estimates of order $\nu^{1/4}$ up to the shock-formation time. We extend their analysis beyond this time, comparing the viscous solution with the inviscid entropy solution evolving from the same initial data. With the shock-formation time normalized to $t=0$, we prove a pointwise error bound on a fixed interval $0\leq t \leq \bar{\varepsilon}$, where $\bar{\varepsilon}>0$ is independent of the viscosity $\nu$. The bound consists of a term of order $\nu^{1/4}$ and an exponentially localized contribution originating from the viscous shock layer. The proof is based on matching an outer expansion, an inner expansion near the formation point, and a viscous shock-profile expansion. Combined with the results of Chaturvedi and Graham, this yields a quantitative description of the inviscid limit from the initial time through the development of the young shock.

Classical harmonic analysis is built upon Lebesgue (or more general) measure. Such a framework is remarkably successful for sets of full ambient dimension, but becomes intrinsically insensitive to lower-dimensional geometry: curves, surfaces, and more generally fractal subsets of \(\mathbb R^n\) with zero \(n\)-dimensional Lebesgue measure. In contrast, Hausdorff content provides a dimension-sensitive notion of size for lower-dimensional sets or fractal sets, but harmonic analysis in this setting is still largely limited to maximal operators. In this paper, we establish a weighted two-parameter good-\(\lambda\) inequality under Hausdorff content setting, providing a route toward other general operators from maximal operators over Hausdorff content. To do this, we firstly identify an appropriate reverse Hölder class $\mathcal R\mathcal H_{q,\delta}$ associated with Hausdorff content, and then prove the self-improving property of $\mathcal R\mathcal H_{q,\delta}$ by establishing the fundamental equivalence between reverse Hölder weights and capacitary Muckenhoupt weights. As direct applications, we obtain weighted Fefferman–Stein inequality and some boundedness on general operators over Hausdorff content. The methodology developed in this paper, even when reduced to the Lebesgue measure case, also provide novel insights, which shows that the good-\(\lambda\) principle is not intrinsically tied to an underlying measure, but can extended to a non-additive framework.

We study the no-flux problem for a taxis-type migration-consumption system with logistic damping in a bounded planar domain, where the motility function \(\phi\) vanishes at zero and degenerates algebraically of order \(\alpha\ge1\). We prove uniform boundedness in \(L^\infty(\Omega)\times W^{1,\infty}(\Omega)\) for every fixed damping coefficient \(b>0\), without any smallness assumption on the initial data, thus improving the previous \(L^2\)-boundedness results. For \(\alpha>1\), we further prove that the solution converges exponentially to the spatially homogeneous state \((a/b,0)\), with an explicit admissible rate. The proof combines a weighted Gagliardo–Nirenberg type inequality adapted to the degeneracy with the dissipative structure of the signal equation. Therefore logistic damping drives the system toward homogeneity, despite the loss of uniform elliptic regularization.

This paper investigates the large deviation principle for 2D stochastic $g$-Navier–Stokes equations driven by multiplicative noise and defined on an unbounded channel-like domain. We first prove the existence and uniqueness of probabilistically strong solutions for the stochastic equation under small noise perturbations by means of the local monotonicity method. Subsequently, we show the well-posedness of solutions to the corresponding control equation and the convergence in distribution of the solutions for the stochastic equation as the noise intensity tends to zero. Finally, we establish the Freidlin-Wentzell type large deviation principle for the solutions of the stochastic equation by using the weak convergence method. To overcome the lack of compactness of Sobolev embedding on unbounded domains, we employ a scalar stream function together with a truncation technique to derive uniform tail estimates for the solutions of the control equation.

The KP-II equation was introduced by Kadomtsev and Petviashvili to explain the transverse stability of line solitary waves in shallow water. Using Darboux transformations, we prove the spectral stability of Y-shaped line solitons in an exponentially weighted space that captures the amount of waves receding behind the line solitons. We show that the spectrum of the linearized operator consists of the continuous spectrum of the zero background together with three residual spectral domains whose boundaries contain continuous spectra associated with the constituent line solitons. Exploiting the analytic dependence of the pairings with adjoint eigenfunctions on the spectral parameters, we prove that the orthogonality conditions associated with all adjoint eigenfunctions can be reduced to orthogonality conditions against three families of adjoint continuous eigenfunctions. Moreover, we investigate linear stability of Y-solitons in a subspace satisfying orthogonality conditions against adjoint continuous eigenfunctions associated with outgoing modulations along each line soliton constituting the Y-soliton.

We establish global existence, uniqueness and quantitative stability for one-dimensional pressureless flow governed by \[ \partial_t\rho+\partial_x\bigl(\rho f(u)\bigr)=0, \qquad \partial_t(\rho u)+\partial_x\bigl(\rho u f(u)\bigr)=0, \] where $\rho$ is the mass density, possibly measure-valued, $u$ is the physical velocity, and $f\in C^1(\mathbb R)$, with $f'>0$, determines the transport velocity $f(u)$. The initial mass has finite positive mass and finite second moment, allowing atoms and unbounded support, and the initial physical velocity is bounded. The difficulty at a cluster collision is that mass and momentum conservation determine a mass-weighted average of physical velocity, whereas transport is governed by the nonlinear law $f$ applied to that average. We construct global entropy solutions by a Bregman variational time discretization whose cost is chosen to reproduce the transport velocity determined by this mass-weighted average, coupled with conditional-expectation updates of physical velocity. For every entropy solution satisfying the Oleinik condition on the transport velocity and initial weak continuity of the kinetic energy measure, we establish a Lagrangian representation and prove uniqueness within this class by comparing flows with possibly different cluster intervals. For two such solutions with the same velocity law and common mass, we obtain a $W_\infty$ estimate for the normalized mass measures in terms of the initial position and physical-velocity discrepancies in mass coordinates, whenever the initial $W_\infty$ distance is finite. The Lagrangian evolution has the semigroup property.

In this paper we deal with the celebrated Gamow liquid drop model, namely, for $m>0$, $N\geq 2$ and $\alpha\in (0,N)$, we study the problem \[ \min\Big\{P(\Omega)+ V_\alpha(\Omega) : \Omega\subset \mathbb{R}^N,\;|\Omega|=m\Big\}\,, \] and focus mostly on the nonexistence of minimizers for large masses. First we show that any possible minimizer in the asymptotic regime must satisfy an optimal thickness property. Eventually, in the simplified setting of sets made up as a tiling of cubes, we show that nonexistence occurs for large masses for the whole range of parameters $N\geq 2$ and $\alpha\in (0,N)$.

We study weighted and Sobolev estimates for two fractional Schrödinger operators on $\mathbb{R}^n$, where $n\geq3$ and $0<s<1$. For the additive operator $L=(-\Delta)^s+V$, with a nonnegative, nontrivial potential $V$ belonging to a suitable reverse Hölder class, $RH_q$, we establish a critical-radius form inequality and decay estimates for the semigroup mass. These yield a heat-kernel bound combining polynomial off-diagonal decay with a fixed positive exponent of decay relative to the fractional critical radius $\rho_s$. A decomposition at this scale gives weighted strong-type estimates for the semigroup maximal operator and weighted Hardy–Littlewood–Sobolev estimates for $L^{-\alpha/2}$, $0<\alpha s<n$, together with the corresponding endpoint weak-type estimates. The admissible weights satisfy explicit growth restrictions determined by the kernel decay, and the fractional integration order is $\alpha s$. We also consider the spectral operator $Q=(-\Delta+V)^s$. Assuming $V\in RH_{q_1}$ with $n/(1+s)\leq q_1<n$ and classical critical radius $\rho_1(x)\geq1$, we prove that $\nabla Q^{-1/2}$ maps $H^{1-s,p}(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n)$ for $1<p\leq p_0$, where $1/p_0=1/q_1-1/n$. By duality, its adjoint maps $L^{p^{'}}(\mathbb{R}^n)$ into $H^{s-1,p^{'}}(\mathbb{R}^n)$ for $p_0^{'}\leq p^{'}<\infty$.

We study the incompressible Oldroyd-B system on $\mathbb R^d$, $d\ge2$, in the regime in which the velocity is the only dissipated quantity: the stress tensor undergoes neither relaxation nor diffusion. In this regime the operator $\mathbb P\operatorname{div}$ acting on symmetric stresses has an infinite-dimensional kernel consisting of exact stationary states, so no decay estimate can hold for the full stress tensor and the existing sharp-decay theory for dissipative systems does not apply. We identify the active pair $U=(u,\Lambda^{-1}\mathbb P\operatorname{div}\tau)$ as the correct decaying observable and establish, in a critical hybrid Besov framework, a complete characterization of its large-time behavior. For $-d/2<\sigma_1<\sigma\le d/2-1$ we prove that $\|U(t)\|_{\mathcal H^{\sigma}}=O(t^{-(\sigma-\sigma_1)/2})$ holds if and only if the low-frequency part of $U_0$ belongs to $\dot B^{\sigma_1}_{2,\infty}$, and, for $\sigma<d/2-1$, that the two-sided estimate $\|U(t)\|_{\mathcal H^{\sigma}}\simeq t^{-(\sigma-\sigma_1)/2}$ holds if and only if the low-frequency part of $U_0$ belongs to a nondegenerate subclass $\mathfrak B^{\sigma_1}_{2,\infty}$ of $\dot B^{\sigma_1}_{2,\infty}$.

We study two related questions for nonlocal equations of Bellman type: the regularity of solutions when the set of controls depends on the state variable, and the periodic homogenization of stationary nonlocal Hamilton-Jacobi equations driven by uniformly elliptic integro-differential operators of order $\sigma\in(0,2)$, whose kernels depend simultaneously on the slow variable, the fast periodic variable and the control. The effective equation is again a nonlocal Bellman equation, but its supremum runs over a set of viscosity Mather measures that varies with the space variable, in a merely Hölder continuous way. We first develop a $C^{\sigma+\alpha}$ regularity theory for nonlocal Bellman equations with Hölder continuous, state-dependent index sets, extending the Schauder estimates of Caffarelli-Silvestre and Serra, in a general framework independent of homogenization. We then use these estimates as a source of compactness to construct a classical solution of the effective problem, unique among bounded classical solutions. Finally, combining this regularity with discounted approximate correctors, we make the perturbed test function method quantitative, and prove that the whole family of solutions converges to the effective solution, with the explicit rate $\epsilon^{\sigma\alpha/(2+\alpha)}$, where $\alpha$ is the Hölder exponent of the data.

We construct a counterexample to integral Aleksandrov-Bakelman-Pucci estimates for nonlocal inequalities of the form \begin{equation*} L_au(x):=\int_{\mathbb R^n}\delta_yu(x) \frac{a(x,y)}{|y|^{n+\sigma}}\,dy \ge-f(x)\quad\text{in }B_1, \end{equation*} with $u\le0$ in $\mathbb R^n\setminus B_1$, where $\delta_yu(x)=u(x+y)+u(x-y)-2u(x)$, and the measurable coefficient satisfies \begin{equation}\tag{C} \lambda\le a(x,y)\le\Lambda,\qquad a(x,y)=a(x,-y),\qquad x\in\mathbb R^n,\quad y\ne0. \end{equation} More precisely, for every $n\ge1$, $0<\sigma<2$, and $0<\lambda<\Lambda$, there exist a fixed coefficient $a(x,y)$ satisfying (C), a constant $d>0$, and nonnegative functions $u_j,f_j\in C_c^\infty(B_1)$ such that $L_au_j\ge-f_j$ in $B_1$, $0\le f_j\le1$, and $|\operatorname{supp}f_j|\to0$ as $j\to\infty$, while $u_j\ge d$ on $B_{1/4}$ for every $j$. Consequently, this fixed operator admits no $L^p$ maximum estimate for any $1\le p<\infty$.

We identify Gaussian curvature as a local mechanism for cubic cancellation of Laplace eigenfunctions on Riemannian surfaces, without arithmetic or dynamical assumptions. For $L^2$-normalized local solutions at frequency $\lambda$, cubic integrals against smooth compactly supported weights tend to zero whenever the curvature is nonzero almost everywhere where the weight is nonzero. Finite-order nondegeneracy yields sharp polynomial decay rates. In particular, on closed surfaces with nowhere-vanishing curvature, we obtain the rate $O(\lambda^{-1/6})$, attained by spherical harmonics on the round sphere. This proves the cubic case of the Gaussian moment conjecture for all normalized real eigenfunctions on closed negatively curved surfaces. The local proof rests on an exact identity relating the reduced phase at a Fermat point to Gaussian curvature. On complete hyperbolic surfaces with positive systole, we also prove global $O((\log\lambda)^{-3/2})$ decay for $L^2$-normalized eigenfunctions and smooth weights with bounded covariant derivatives through order two, without assuming finite area.

We prove boundary Hölder regularity for fractional porous medium-type equations in bounded smooth domains with zero exterior data. For $0<\sigma<1$, $n>2\sigma$, and $m>1$, we show that nonnegative weak dual solutions $v$ of $\partial_t v+(-\Delta)^\sigma(v^m)=0$ satisfy \[ \frac{v}{d^{\sigma/m}}\in C_{x,t}^{\alpha,\frac{\alpha m}{\sigma(m+1)}} \] up to the boundary and away from the initial time, for some $\alpha\in(0,\sigma)$, where $d$ denotes the distance to the boundary. More generally, we establish boundary Hölder estimates for $u/d^\sigma$ for nonnegative viscosity solutions of $\partial_t u=u^\gamma Lu$, with $0<\gamma<1$ and initial data comparable to $d^\sigma$. Here $L$ is a symmetric stable operator of order $2\sigma$ whose measurable angular density is bounded above and below by positive constants. Our proof combines comparison principles and boundary barriers with a blow-up and compactness argument based on a Liouville theorem for ancient solutions in a half-space.

We use centered differences to construct Chernoff families and derive even asymptotic expansions for their iterated products. Richardson extrapolation is then used to cancel polynomial part and thereby raise the convergence order. As illustrations, we apply the construction to uniformly parabolic, Schrödinger-type second-order operators and to a third-order dispersive operator, obtaining, at extrapolation depth $m$, the rates $ O(n^{-2m}) $ on the Sobolev scales $H^{2m+\delta}$, $H^{4m}$, and $H^{5m}$, respectively, with arbitrary $\delta>0$. In the analytic category, quantitative all-order estimates allow the extrapolation depth to grow with the base resolution and yield the super-fast rates $e^{-cn^2}$ for the two second-order models and $e^{-cn} $ for the third-order model. The constructions are explicit in terms of the coefficients of the underlying differential operators and do not use the target semigroup in their definition. This provides a constructive positive answer to the challenge of super-fast Chernoff approximation formulated by Remizov in Problem 30 of arXiv:2410.00416.

We study the two-dimensional non-rotating primitive equations with positive horizontal and vertical viscosity and without thermal diffusion. Our main results are fourfold. First, for every integer \(s\ge4\), we prove that sufficiently small, compatible, doubly periodic \(H^s\) perturbations of a stably stratified rest state with an affine temperature profile generate unique global strong solutions; the horizontal velocity and temperature remain uniformly small at the initial Sobolev order despite the derivative loss in the full diagnostic vertical velocity, and strict stratification persists. Second, we establish an inverse-map conservation law that determines the terminal temperature solely from the initial temperature: the terminal label is identified as the normalized increasing rearrangement of the initial label modulo the vertical period, and the terminal perturbation admits a quadratic expansion with cubic remainder. Third, the limiting equilibrium uniquely minimizes the quadratic perturbation energy among compatible, strictly stratified states with the same periodic label distribution. Finally, we establish sharp algebraic decay in suitable lower-order Sobolev norms for both velocity components and for the temperature and pressure differences from equilibrium. Exact nonlinear solutions provide matching lower bounds, establishing optimality for uniform estimates over the small-data class. The proofs combine weighted energy methods, spectral analysis, and inverse-map identities.

We generalize Moser's stability theorem for minimal hypersurface foliations to arbitrary codimension. Let $\mathbb{T}^N=\mathbb{R}^N/\mathbb{Z}^N$ be the standard torus, and let $P\subset\mathbb{R}^N$ be an $m$-plane satisfying a Diophantine condition relative to $\mathbb{Z}^N$. We prove that, under every sufficiently small smooth $\mathbb{Z}^N$-periodic perturbation of the Euclidean metric, the affine foliation of $\mathbb{T}^N$ induced by $P$ persists as a smoothly conjugate foliation by minimal $m$-dimensional submanifolds. The normalized conjugating displacement depends smoothly and tamely on the metric and is locally unique. The proof uses Hamilton's formulation of the Nash-Moser implicit function theorem.

We prove several regularity theorems concerning Lipschitz viscosity solutions to the minimal surface system. The main lemma (i.e. “Decoupling Lemma") says that the component functions of such solutions satisfy the homogeneous Pucci inequalities. This allows us to apply the regularity theory for scalar viscosity solutions for elliptic PDE component-wise to a viscosity solution for the system. As a result of the Decoupling Lemma, we deduce that 2D Lipschitz viscosity solutions to the system are analytic in the interior, and obtain a quantitative partial regularity theorem. We also prove a geometric Harnack inequality for Lipschitz solutions. The Harnack inequality yields a strong maximum principle and half-space Liouville theorem. Another significant consequence of the lemma is a boundary regularity theorem for Lipschitz viscosity solutions. Notably, our theorem does not require any density assumption at the graph boundary, and we do not assume stationarity. This is used to prove a classification theorem for the singular set of a 4D stationary solution on a bounded domain. We also prove an $\varepsilon$-regularity theorem extending a result of the author. To conclude, we survey open problems. As an example, we have included a Bernstein-type theorem for 3D homogeneous entire Lipschitz viscosity solutions. Taken together, our results suggest that viscosity methods are a useful tool for studying the regularity of minimal graphs.

We prove sharp residual stability estimates for the critical Poincaré-Sobolev equation near sums of one positive hyperbolic solution and one projected Euclidean bubble in dimensions at least four. The estimates are uniform in the relative positions of the centers. The sharp rate is linear except in dimension six, where it contains a square root of a logarithm. For nonnegative functions, the same estimates hold under an energy constraint near the sum of the two profile energies for \(\lambda\) outside a finite set, and an additional mass constraint removes this restriction. For arbitrary finite numbers of profiles, we identify and realize all nonnegative Palais-Smale levels. We characterize the parameters at which different pairs of profile numbers have the same energy and show that the limiting energy and mass determine both profile numbers.

In this paper we prove that the classical thermoelastic Timoshenko and Bresse systems with thermal coupling acting only on the bending moment are not exponentially stable under full homogeneous Dirichlet boundary conditions if the equal wave speed condition is not satisfied. Combined with their well-known converses, our results provide complete characterizations of exponential stability in terms of the coefficients of each system. To the best of our knowledge, these characterizations were previously known only under mixed boundary conditions. The proof is based on the introduction of a boundary correction term in the standard trigonometric ansatz so that the full homogeneous Dirichlet conditions are satisfied.

We propose k-Hessian equations as auxiliary equations for degenera- tions of Kähler classes to nef classes of numerical dimension k. As proof-of-concept results, we provide a uniform mean-value inequality and uniform à priori estimates for Hessian equa- tions.

Based on a semigroup approach, we prove quantitative uniform-in-time propagation of chaos for two systems of interacting particles with singular interaction kernels $K$ of type $L^k$, $k>d$, and a contractive Lipschitz confinement force field. First, we consider a system with moderate interaction, which means that the particles interact through the mollified kernel $K\ast V^N$, for some approximation of identity $V^N = N^{d\alpha} V(N^\alpha \cdot),\, \alpha>0$. Assuming that $\|K\|_{L^k}$ is sufficiently small, we obtain that for any $\varepsilon>0$ the uniform-in-time rate is of order $N^{-\min\left(\alpha,\frac12-\frac{d\alpha}{2}-\varepsilon\right)}$, plus an exponentially small in time contribution of the initial condition. Second, for the classical, non-mollified particle system we establish uniform-in-time propagation of chaos with rate $N^{-\min\left(\alpha,\frac12-\frac{d\alpha}{2}-\varepsilon\right)} + \|K-K\ast V^N\|_{L^{k}}$, plus an exponentially small in time contribution of the initial condition. The method relies on the regularity properties of the solution of the associated nonlinear Fokker-Planck equation combined with estimates of the Duhamel formulation of the mollified empirical measure, together with partial Girsanov transforms for the non-mollified system of particles.

We establish the first partial regularity results for minimizers of a class of suitably quasiconvex variational integrals defined on anisotropic Sobolev spaces. These variational integrals are related to higher gradient continuum models for pantographic metamaterials.

We develop an $N$-adic modification of the construction of David-Jeznach-Julia for compact subsets of the real line $\mathbb{R}\subset\mathbb{C}$ for which harmonic measure for the complementary domains has no dimension drop. In particular, we show that for every $s\in(0,1/2)$, there exists an $s$-Ahlfors regular compact set $K\subset\mathbb{R}\subset\mathbb{C}$ such that $\mathrm{dim}\omega^\infty=s$, where $\omega^\infty$ denotes harmonic measure for $\mathbb{C}\setminus K$ with pole at infinity. Combined with a result of the third author, this shows that the threshold $1/2$ is sharp.

Frank and Loss proposed critical Sobolev inequalities for the Dirac operator and for vector fields. They conjectured optimality of their spinorial family and wrote that it is conceivable that the higher-dimensional Loss-Yau potential is an optimizer. Wang and Zhang recently disproved both optimizer claims using direct comparison and instability arguments. We compute closed-form quotients for two elementary families of explicit competitors with strictly smaller quotients. For every $d\geq2$, the spinorial competitor is a constant spinor multiplied by a scalar Aubin-Talenti optimizer. For every $d\geq3$, the vector competitor is the planar azimuthal field \[ A_d(x)=\frac{d}{(1+\lvert x\rvert^2)^2} (-x_2,x_1,0,\ldots,0). \] It lies in the nonlinear gauge and its quotient is computed in closed form. For every odd $d\geq3$, it gives an explicit proof of nonoptimality of the higher-dimensional Loss-Yau-Dunne-Min potential. In dimension three, this field is a specific member of the competitor family of Wang and Zhang: it is a linear combination of the Loss-Yau potential and its reflected twin. Its axisymmetry gives a negative answer to their question about minimization within the $\mathrm{SO}(2)\times 1$-invariant class.

We consider singular Schrödinger operators with an attractive $\delta'$ interaction of strength $\beta$ supported by a star graph and prove that the ground-state eigenvalue is uniquely maximized by stars of the maximum symmetry. Two different proofs are presented, one for an even number of edges using a comparison to the $\delta$ interaction star, the other general based on Birman-Schwinger technique.

For every $0<s<1$ we construct a compact $s$-Ahlfors regular set $K\subset\mathbb C$ for which harmonic measure of $\mathbb C\setminus K$ with pole at infinity is comparable to $\mathcal H^s|_K$. In particular, there is no dimension drop for any dimension $s<1$. The construction is $N$-adic and based on configurations close to the $N$-th roots of unity. We adjust the positions of the children by small polynomial deformations so that the logarithmic potential of the natural Bernoulli measure becomes constant in the limit. A first-mode correction gives the range $s<2/3$, while corrections of arbitrary finite order yield the full range $0<s<1$.

We prove orbital stability of minimizing solitary waves at sufficiently small positive speeds for an Airy equation with a quadratic interaction filtered by a Bessel potential of order one. The stability hypotheses are verified: constrained minimization gives one negative direction, the small-speed KdV limit and operator-norm perturbation of the profile resolvent give a one-dimensional translation kernel, and a differentiable rescaled branch with uniformly bounded derivative gives a positive slope of the quadratic invariant. A conserved Lyapunov functional then proves orbital stability in the three-halves-derivative energy space and the perturbation also gives uniqueness of centered minimizers at small speed. The dynamical framework is supplied by a direct Bourgain-space bilinear estimate, local well-posedness in the square-integrable class, unconditional uniqueness from one-quarter derivative onward, and global well-posedness in the half-derivative energy space. For every prescribed positive speed, constrained minimization constructs smooth positive profiles that are symmetric, strictly monotone away from their center, and exponentially localized. At general positive speeds the stability criterion remains conditional on nondegeneracy and the slope condition.

It is known that there can be at most 3 disjoint nontrivial minimal graphs in R^3 with 0 boundary values, and conjectured that there can be at most 2. We explore properties that the graphs would have if there were 3.

We study admissible entire subsolutions and radial boundary blow-up solutions of $S_k^{\frac1k}[\Delta uI-D^2u]=f(u)+g(u)|Du|^q$ for $1\leq k<n$, continuing our study of the determinant case $k=n$ in Part I. Under structural assumptions on the continuous, non-decreasing coefficients $f$ and $g$, we establish necessary and sufficient conditions for the existence of entire subsolutions when $q<2$. Our results cover the non-Lipschitz case $0<q<1$ with a possibly vanishing zero-order term, the gradient-singular case $q<0$, and sign-changing gradient coefficients. We also characterize finite-radius blow-up of radial solutions and derive two-sided estimates for the blow-up radius, as well as pointwise upper bounds for their profiles in terms of inverse Keller–Osserman functions.

We study positive $(n-1)$-convex entire subsolutions of the normalized real $(n-1)$ Monge–Ampère equation $\operatorname{det}^{\frac1n}\bigl(\Delta uI-D^2u\bigr)=f(u)+g(u)|Du|^q$ in $\mathbb R^n$. Under two sets of structural assumptions on the continuous, non-decreasing coefficients $f$ and $g$, we establish necessary and sufficient conditions for the existence of entire subsolutions when $q<2-\frac1n$. Our results cover the pure-gradient degenerate case $f\equiv0$ with $0<q<1$, the gradient-singular case $q<0$, and the case where $g$ changes sign. For $q=2-\frac1n$, we recover the Keller–Osserman condition for the gradient-independent equation, prove nonexistence when $g$ is eventually positive, and obtain an exponentially weighted condition when $g$ has a nonpositive limit at infinity. This is the first of two companion papers and treats the determinant case.