Papers

We investigate a three-dimensional parabolic system that arises as a hyperviscous and penalized approximation of the incompressible Navier–Stokes equations. The model combines three complementary dissipative mechanisms: the classical viscous diffusion, a biharmonic (hyperviscous) regularization, and a divergence penalization. In addition, a Temam-type correction is incorporated into the nonlinear convection term to compensate for the weak compressibility effects generated by the penalization procedure. We prove the global existence of weak solutions for arbitrary initial data belonging to $L^2(\mathbb{R}^3)$. For sufficiently small initial data in $H^2(\mathbb{R}^3)$, we establish the existence and uniqueness of global strong solutions. Furthermore, for initial data in $L^1(\mathbb{R}^3)\cap H^2(\mathbb{R}^3)$, we derive optimal large-time decay estimates, showing that the solutions exhibit the same asymptotic decay rates as those of the classical heat equation. A key feature of our analysis is that all the obtained a priori estimates are uniform with respect to the positive penalization parameter $\varepsilon$. These uniform bounds provide a stable and rigorous analytical foundation for the study of the penalized approximation of incompressible flows.

We study periodic partitions of the plane into two distinct cells minimizing the anisotropic $\ell_1$-perimeter. For a rectangular lattice $G$, we compute explicitly the $(G,\ell_1)$-isoperimetric profile and classify all minimizers. When one cell has small area, the optimal tiling is generated by a square and a chipped rectangle, while in the remaining regime the tiling is generated by two adjacent rectangles. Further minimizing over all possible planar lattices, we show that the $\ell_1$-isoperimetric profile is attained by the Pythagorean double tiling of two axis-aligned squares sharing a vertex. This configuration is unique unless the two cells are assigned the same area. Finally, we prove that the limiting (non periodic) partitions obtained by sending one volume to infinity are locally $\ell_1$-isoperimetric.

We study the nonisotropic fractional analogs of Laplace, Helmholtz and Yukawa equations. We provide a Duffin correspondence for the Yukawa equation and a Feynman–Kac reconstruction for the Helmholtz equation. The foundation of our analysis is the fact that the nonisotropic fractional Laplace equation is related to a symmetric $\alpha$-stable Lévy process with independent identically distributed components. By using this relation we provide a Walk-on-Cubes algorithm that simulates the solutions of Helmholtz and Yukawa equations.

Two-sided threshold crossings arise in discrete monitoring systems whenever intervention is triggered by departure from an operating band. This paper develops a finite-state first-passage framework based on a biased random walk with symmetric absorbing barriers. To remove parity gaps of the raw hitting time, we analyse a parity-corrected lifetime through even- and odd-state Markov skeletons. Adjacent-row likelihood-ratio inequalities verify the monotonicity condition and establish increasing failure rate and new-better-than-used properties. These ageing properties yield log-concave survival probabilities and geometric persistence bounds. Substochastic-matrix formulas are derived for survival, hazard, mean lifetime, state-conditioned remaining useful life, and finite-horizon crossing risk. A renewal-cost criterion then converts the first-passage distribution into an optimal preventive-intervention schedule. Model parameters are obtained by matching the drift and variance of observed increments to a two-point random-walk approximation, with a rolling-window extension for nonstationary regimes. Numerical experiments compare exact calculations with Monte Carlo diagnostics, moment-matched inverse-Gaussian and Weibull benchmarks, serially correlated increments, calibration perturbations, and fixed-interval policies. A synthetic remote-patient-monitoring example illustrates how discrete physiological deviations can be mapped to transparent risk scores and personalized review schedules. The framework provides an auditable link between discrete stochastic dynamics, remaining-lifetime prediction, and condition-based intervention, while separating mathematical validation from clinical validation.

This paper studies the Cauchy problem for the fully parabolic Keller-Segel system. The main results show that there exists a critical threshold $A_{\rm crit}>0$ for steady states $(A,A)$ such that the steady states are nonlinearly stable when $A\le A_{\rm crit}$ and nonlinearly unstable when $A>A_{\rm crit}$. We discuss asymptotic convergence rates as well. In the subcritical case $A<A_{\rm crit}$, the rates correspond to those of the heat equation, and in the critical case $A=A_{\rm crit}$, the rates correspond to half those of the heat equation.

In this paper we investigate the global existence, blow-up and standing waves for the $L^{2}$-supercritical nonlinear Klein-Gordon equations with quadratic asymmetric interaction (LSNKG). First, by introducing a suitable auxiliary functional, using concavity analysis and virial estimates, we obtain a finite time blow-up result for solutions to the Cauchy problem of (LSNKG) when the initial energy is negative. Next, by defining appropriate functionals, manifolds and a constrained variational problem, we employ variational method and the Lagrange multiplier method to derive the existence of ground state solutions for the corresponding nonlinear elliptic (steady-state) system, thereby to establish the existence of standing wave with the ground state for (LSNKG). Then, using the variational characterization of the ground state solutions and constructing invariant sets under the flow generated by the Cauchy problem for (LSNKG), we combine the potential well argument with concavity analysis to establish a sharp threshold between blow-up in finite time and global existence. Finally, by exploiting the variational characterization of the ground state, introducing appropriate scalings, and choosing suitable initial data based on the ground state, we justify the instability of standing wave with the ground state for (LSNKG).

Recent well-posedness results for evolutionary partial differential equations with state-dependent inhomogeneity are extended to a larger problem class incorporating nonautonomous material behaviour. This generalization facilitates the formulation of a framework able to accommodate delayed boundary value problems related to evolutionary partial differential equations. The results permit the ad hoc treatment of Dirichlet- and Neumann-like boundary conditions with state-dependent delay. Complex boundary conditions involving nonautonomous delay in the material law and state-dependent delay in the forcing term can be accommodated by use of extended state spaces. This approach provides the first systematic treatment addressing well-posedness of several classical boundary conditions involving state-dependent delay. The viability and versatility of the theory is showcased by applications to parabolic and hyperbolic partial differential equations with Dirichlet, Neumann, Robin, Wentzell-Robin and Leontovich boundary conditions.

In this paper, we study a viscous conservation law with discontinuous flux. We define a weak solution concept, show its existence via an explicit formula, and prove that the weak solution is unique. In addition, we obtain the large-time behavior of the solution. The asymptotic limit of the solution is a steady-state solution. Here, the large-time asymptotic governed by the convection terms either converges to a constant steady state or to a non-constant steady state, depending on the behavior of the flux functions at a threshold value, which is the meeting point of two fluxes. This phenomenon does not occur in the single-flux case.

In this article, we study the existence and uniqueness of a transport equation with discontinuous coefficients that comes from the solution of a balance law with a time-dependent source term. We also give a simplified proof providing an explicit formula for the solution of the balance law using a variational formulation. This system can be considered as a generalized version of the one-dimensional model for the large-scale structure of the formation of the universe.

We prove global well-posedness of the hard-sphere Boltzmann equation in $\mathbb{R}^d$ for small initial data in the critical Besov space $B_{1,1}^{d-1}L_v^1$ and in the critical Sobolev space $W^{d-1,1}L_v^1$. The proof is based on new bilinear estimates for the nonlinear collision operator, which rely on transversality considerations.

We study the weakly coupled elliptic Anderson model with spatial white noise on the four-dimensional torus, which provides a basic example of a critical SPDE requiring renormalization at arbitrarily high orders. With coupling $\lambda |\log\varepsilon|^{-\frac12}$ where $\lambda>0$ is sufficiently small, we prove that the Green's function of the corresponding random Schrödinger operator, suitably centered and rescaled, converges to a centered Gaussian random field with explicit covariance. The main difficulty is that, for such critical models, one must expand up to order $|\log\varepsilon|$, while the perturbative expansion contains factorially many pairings and a growing number of renormalization terms. To overcome this, we construct a truncated renormalized parametrix and prove sharp high-order bounds for its remainder. A central ingredient is a multiscale analysis based on a new version of Hepp trees, combined with new estimates for summations over permutations. These estimates reveal a precise balance between logarithmic losses from scale summation and factorial gains from the structure of primitive pairings. The methods developed here are intended as a first step toward a general theory for critical SPDEs with weak couplings.

In this paper, we prove the quantitative propagation of chaos results that allow us to derive multi-species cross-diffusion equations from moderately interacting stochastic particle system. The quantitative propagation of chaos result in $L^1$-norm is obtained by the relative entropy method, and the proof is carried out in two steps. In the first step, we quantify the relative entropy between the joint distribution of the particle system and the tensorised solution of the PDE at the intermediate level. In the second step, we establish a rigorous convergence rate to the multi-species cross-diffusion equations by analyzing the $L^2$-distance between the solution of the intermediate-level PDE and that of the limiting PDE. Furthermore, combining the strong $L^1$-convergence for the propagation of chaos with the $L^p$-estimates $(2\le p<\infty)$ for the marginal distribution of multi-species particle system, we derive the corresponding $L^q$-result $(1<q<\infty)$ via interpolation.

It is well-established that single-axis Transverse Ray Transform tomography data from high-energy X-ray diffraction is insufficient for general reconstruction of three-dimensional elastic strain. In this paper we show that, when combined with the constraint of mechanical equilibrium, this problem becomes uniquely solvable for isotropic elastic samples with known elastic constants and non-zero Poisson's ratio. Building on the reconstruction framework of Desai and Lionheart [N.M. Desai, W.R.B. Lionheart, An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data, Inverse Problems 32 (11) (2016) 115009], we formulate this constrained single-axis tomography problem in the Fourier domain and show that the resulting system is invertible everywhere outside a finite collection of measure zero characteristic planes. In the case of bounded samples, this is sufficient to establish uniqueness for reconstructions. We further derive conditional stability estimates that quantify the regularity requirements associated with reconstruction of the various strain components.

We study the sedimentation of stationary random suspensions of rigid particles in Stokes flow. Batchelor's formula predicts the first dilute correction to the infinite-volume mean settling speed due to hydrodynamic interactions between suspended particles. A rigorous derivation has long been obstructed by the long-range nature of the Stokes flow, which gives rise to infrared divergences in the large-volume limit. In dimension $d>2$, for stationary suspensions satisfying quantitative decorrelation assumptions, we construct the infinite-volume mean settling speed and show that it governs the relative settling speed of particles in large containers, independently of the container shape. We then establish a renormalized cluster expansion of this mean settling speed in the dilute regime and compute it up to the two-particle term, thereby justifying Batchelor's formula. The proof is based on the infrared renormalization of hydrodynamic interactions. Infinite-volume observables are decomposed into an explicit singular part, carrying the non-integrable large-scale contribution, and a regular remainder controlled by elliptic estimates. The singular part is renormalized through counterterms that encode the diverging mean backflow generated by the suspension. At the level of the dilute cluster expansion, the renormalization is implemented cluster by cluster and the singular-regular decomposition is achieved through a finitary diagrammatic expansion of hydrodynamic interactions, inspired by the method of reflections, which isolates the leading divergent substructures and exposes the key cancellations.

We study an infinite-horizon optimal stopping problem for a two-dimensional normally reflected diffusion in the quadrant with payoff \(G(x_1,x_2)=x_1\vee \alpha x_2\). The problem combines three features that complicate the usual free-boundary analysis: reflection on the coordinate axes, a genuinely two-dimensional stopping region, and a nonsmooth max-type reward. We formulate the associated reflected obstacle problem, prove a verification theorem under explicit Itô–Krylov–Tanaka admissibility and measure-superharmonicity assumptions, and derive a conditional epigraph structure for the stopping set. The main technical point is that the stopping-gain object \(\Gamma=c+rG-\mathcal LG\) is a signed measure rather than a function. Its diagonal component is $\Gamma^\Delta(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+\alpha^2}}\sigma_\Delta(dx)$, $n=(1,-\alpha)$, which shows that pointwise stopping-gain sign conditions must be interpreted with care. We also prove that the correct potential representation is the killed-resolvent formula $V(x)=G(x)-R_r^{\mathcal C}\Gamma(x)$, rather than the unrestricted reflected resolvent. A constant-coefficient reflected Brownian example illustrates the diagonal singular term explicitly.

We study finite-horizon optimal control problems with scalar isoperimetric constraints from a function-space duality perspective. Controls are treated as elements of weakly compact subsets of \(L^\infty(0,T;\mathbb R^m)\), while the state equation induces a control-to-state map into \(W^{1,\infty}(0,T;\mathbb R^n)\). For linear dynamics, concave payoff, and an affine isoperimetric functional, we prove that the constrained value function has an interval domain, is concave, and admits a Fenchel–Moreau dual representation. This yields a superdifferential formula identifying the negative of the dual multiplier with the sensitivity of the value function with respect to the constraint level. A constraint qualification is then used to identify the dual multiplier with the normal Pontryagin multiplier of the augmented isoperimetric system. We also treat linear-quadratic problems with a single quadratic equality constraint by reducing them to quadratic forms on a Hilbert space. The resulting analysis separates the validity of the envelope formula from the regularity needed for Riccati synthesis, showing that sensitivity may persist even when the modified control-weight operator becomes singular.

We consider the three-dimensional stochastic Navier–Stokes equations with additive stochastic forcing. We study temporal discretizations based on a semi-implicit Euler as well as a Crank-Nicolson scheme. We prove that locally in time the former converges with rate 1 while the latter converges with rate 3/2. These theoretical results are confirmed by extensive numerical simulations. To the best of our knowledge this is the first time that numerical simulations for the three-dimensional stochastic Navier–Stokes equations have been performed.

We investigate Dirichlet boundary value problems for a class of second-order parabolic equations in divergence-form with coefficient matrices that exhibit singular and degenerate behaviors characterized by a Muckenhoupt weight class. This framework serves as the parabolic analogue to the singular-degenerate elliptic equations pioneered by Fabes, Kenig, and Seraponi. Under a smallness assumption on the partially weighted mean oscillation of the coefficients, we establish the existence, uniqueness, and local interior and boundary regularity estimates for weak solutions within appropriately defined weighted Sobolev spaces. The proofs rely on the freezing coefficient technique alongside the level-set method introduced by Caffarelli and Peral. Additionally, we develop the necessary weighted Sobolev space framework and related weighted inequalities. Finally, a compactness argument is utilized to demonstrate that solutions to these equations remain locally close, in the weighted Sobolev norm, to their frozen-coefficient counterparts.

We consider backwards globally self-similar solutions of the 3D incompressible Navier-Stokes equations which are invariant under the joint action of scaling (the natural parabolic scaling) and rotation about a given axis, at a constant angular speed $\alpha$ in self-similar time. For these so-called rotated self-similar solutions (RSS), we prove that if they satisfy a Type~I upper bound, and if the rotation parameter $\alpha$ is either too small, or too large, then they must be trivial. This Liouville-type result extends the classical works of Nečas-Růžička-Šverák ('96) and Tsai ('98), which only consider $\alpha=0$, to the case of similarity profiles which experience nontrivial rotation. Our results partially answer a question posed by Perelman. For backwards globally self-similar solutions which are invariant under the discrete action of scaling and rotation, the so-called rotated discretely self-similar solutions (RDSS), we obtain similar Liouville-type results under a Type~I upper bound, assuming extreme values of the rotation parameter $\alpha$, and if the scaling factor $\lambda$ is sufficiently close to $1$. We also establish a new regularity criterion for 3D Navier-Stokes which is local in nature: if the solution satisfies a Type~I upper bound in a unit parabolic cylinder, and there is a single time-slice at which the solution is locally approximately self-similar, then the top-center of the parabolic cylinder is a regular point of the Navier-Stokes flow. The proof of all these results rests on the introduction of a robust weighted-$L^2$ framework. In particular, our method is quantitative and is not sensitive to whether the Bernoulli head pressure satisfies a maximum principle, which was a key obstruction in previous works.

Let $H_a=-\Delta+a|x|^{-2}$ be the Friedrichs extension on $L^2(\mathbb{R}^d)$, where $d\ge 3$ and $-(d-2)^2/4\le a<0$ lies in the attractive Hardy range. Starting from the known positive two-sided comparison for the kernel of $H_a^{-s/2}$, we determine the complete strong non-endpoint mapping range for two power weights. If $\sigma=(d-2-\sqrt{(d-2)^2+4a})/2$ and $0<s<d-2\sigma$, then [ ||x|^{-\beta}H_a^{-s/2}f|{L^q} \lesssim ||x|^\alpha f|{L^p} ] holds for $1<p,q<\infty$ precisely under the exponent ordering, scaling, sum, and origin conditions stated in the main theorem. At either origin-critical boundary, the strong estimate and the corresponding weighted $L^p\to L^{q,\infty}$ estimate fail, whereas the Lorentz replacement $L^{p,1}\to L^{q,\infty}$ holds. We also derive weighted Sobolev consequences and treat the Hardy-critical Friedrichs case separately. No new heat-kernel, spectral multiplier, Bernstein, or Littlewood–Paley theorem is claimed.

This study investigates the boundary conditions for fluid-dynamic equations at the interface of a gas and its condensed phase during steady evaporation of a polyatomic gas. Evaporation curves illustrating the dependence of the temperature and pressure ratios on the Mach number are considered for a calorically perfect gas whose molecules behave like rigid rotors. At the condensed phase, complete absorption conditions are assumed. Also an extension to cases in which a part of the molecules is diffusely reflected at the condensed phase is also considered. We revisit the half-space evaporation problem for polyatomic gases based on previous results, applied to a slightly modified entropy functional. The structure of the (modified) maximal total entropy production curves is investigated, and simple functions that accurately fit the numerical results are proposed. Functions of the proposed form, with modified coefficients, also fit the evaporation curves previously obtained by different numerical methods surprisingly well. The approximation is performed for different numbers of internal degrees of freedom or ratios of specific heats. Simple functions that depend on the ratio of specific heats are found to fit the obtained evaporation curves for different ratios of specific heats very well.

We consider the inverse problem for the quantitative reconstruction of physical properties in the context of passive imaging, where ambient wavefields are used to infer a medium. The data are modeled as a superposition of waves generated by stochastic sources. In this work, we focus on time-harmonic acoustic wave propagation and assume that the stochastic sources exciting the medium are zero-mean and spatially uncorrelated. Under these assumptions, the expected value of the cross-correlation between signals recorded at two locations can be related to the deterministic Green's function and the covariance of the source terms. We follow a first-order formulation of the wave equation, which enables the treatment of correlations between different types of wavefields. A numerical framework is developed for the resulting nonlinear inverse problem. The quantitative reconstruction is carried out using an iterative minimization scheme, in which the gradient of the misfit functional is computed via the adjoint-state method. Numerical experiments in two and three dimensions are performed using synthetic data, and inversions based on the expected value of cross-correlations are compared with those relying on direct wavefield measurements from active-source acquisitions.

The Einstein-de Sitter universe is the prevailing model used in cosmology to describe the cold dark matter-dominated epoch of the universe. This model is a spatially homogeneous and isotropic spacetime undergoing decelerated expansion, and is linearly unstable under the Einstein-Euler equations with a pressureless fluid equation of state. We show that every initial data set for the Einstein-Euler equations on $\mathbb{T}^3$ with a near-flat metric and positive fluid energy density converges to a flat metric under the Einstein-Euler flow with a polytropic equation of state. This means the metric asymptotes to an Einstein-de Sitter spacetime. In particular, this settles the question of whether the Einstein-de Sitter model can be nonlinearly stable for an appropriate matter model.

We consider the Cauchy problem for the nonlinear Dirac equation on a noncompact $N$-star metric graph $G$, \[ \mathrm{i}\partial_t \psi = D\psi - |\psi|^{p-2}\psi, \qquad \psi(0)=\psi_0, \] where $p\ge3$, $\psi:\mathbb{R}\times G\to\mathbb{C}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. Using Bourgain-type spaces defined through the spectral resolution of $D$, together with elementary $L^\infty$ bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well-posedness for initial data \[ \psi_0\in H_D^s(G)\cap L^\infty(G;\mathbb C^2), \qquad 0\le s<\frac12 . \] The corresponding solution belongs to \[ C([0,T];H_D^s(G))\cap X_T^{s,b}\cap L^\infty([0,T]\times G). \] Moreover, $\|\psi(t)\|_{L^2(G;\mathbb{C}^2)}$ is conserved along the solution on the existence interval. We also establish a blow-up alternative in the combined $H_D^s$ and space-time $L^\infty$ control norm.

The study of vacuum is important in understanding compressible flows. In particular, physical vacuum, which ensures a finite, nontrivial pressure-induced acceleration in the normal direction on the moving boundary, naturally arises in the study of the motion of gaseous stars. In this paper, we analyze the free boundary problem for the three-dimensional compressible Navier–Stokes–Poisson equations with degenerate viscosities for self-gravitating viscous gaseous stars. For the spherically symmetric and barotropic motion, we establish the global existence of the unique classical solution without any restriction on the size of the initial data. Our solutions are classical up to the moving boundary for positive times, and preserve the physical vacuum boundary behavior of the Lane–Emden star configuration.

Seismic surface waves, particularly low-frequency Rayleigh waves, are notoriously destructive and remain difficult to control using conventional engineering methods. Recent advances in elastic metamaterials have opened new avenues for manipulating and guiding seismic waves at subwavelength scales. In this work, we present a rigorous mathematical study of subwavelength resonances and seismic-wave localization in arbitrarily shaped Matryoshka-type elastic metamaterials, which are composed of nested high-contrast resonators made of materials much stiffer than the background medium. By employing the displacement-to-traction map and a variational formulation, we derive necessary and sufficient conditions that characterize subwavelength resonances. Using the Gohberg–Sigal theory and Puiseux series of multivalued algebraic functions, we establish the existence of subwavelength resonances and obtain their asymptotic characterization in terms of the eigensystem of the generalized stiffness tensor, which serves as the elastic analogue of the capacitance matrix in the celebrated Minnaert acoustic-cavitation systems. Furthermore, in the concentric spherical case, the stiffness tensor exhibits a block-diagonal structure separating translational and rotational components, with each block being tridiagonal and possessing positive, simple eigenvalues. Based on this matrix formulation,we rigorously demonstrate that, for structures with a sufficiently large number of layers, ring defects in layered subwavelength resonators can act as effective radially laminated seismic waveguides, supporting both wave localization and guided propagation along the defects. In particular, appropriately placed ring defects can induce eigenmodes that are exponentially localized at multiple defect sites simultaneously,with precisely quantified amplitude ratios.

We prove lossless inhomogeneous Strichartz estimates for solutions to the Schrödinger equation on compact manifolds with nonpositive curvature over frequency-dependent time intervals of length $\log \lambda \cdot \lambda^{-1} $. As applications, we improve upon the Sobolev norm growth bounds for the cubic NLS established by Planchon, Tzvetkov and Visciglia on 3-dimensional compact manifolds when the manifold also has nonpositive curvature and extend the lossless homogeneous Strichartz estimates on logarithmic time intervals established by the first author and Sogge to Schrödinger operators with critically singular potentials.

Reconstructing the location and shape of an unknown impenetrable scatterer from far-field measurements is a fundamental inverse problem in elastic scattering. In this paper, we propose monotonicity-based shape characterization theorems and develop corresponding algorithms for rigid and traction-free impenetrable scatterers. By establishing the factorization of the elastic far-field operator and constructing localized wave functions, we derive a sharp monotonicity-based characterization criterion for determining the shape and position of the impenetrable scatterer. This criterion is based on the spectral properties of the monotonicity operator, defined as a specific linear combination of the far-field and Herglotz probing operators. Building on this theoretical foundation, we first present a counting-based monotonicity sampling method that evaluates the number of negative eigenvalues of the monotonicity operator. To address the inherent sensitivity of eigenvalue-counting to measurement noise, we further develop two novel monotonicity spectral sampling algorithms that exploit the magnitudes, rather than merely the signs, of the negative eigenvalues. The single-frequency monotonicity spectral sampling method provides robust stability against data perturbations, while the multi-frequency monotonicity spectral sampling method extension aggregates frequency information into a multiscale indicator that balances noise robustness with high-resolution geometric fidelity. Numerical experiments across various scatterer geometries and noise levels demonstrate sharp boundary localization and accurate reconstruction of complex concave features, confirming the effectiveness of the single-frequency and multi-frequency monotonicity spectral sampling methods.

We study pathwise well-posedness of the stochastic heat equation (SHE) with a multiplicative noise on the circle. By combining the convolution Young and rough integration theory, introduced by Gubinelli and Tindel (2010), with the random tensor estimate approach to pathwise well-posedness of stochastic dispersive PDEs with multiplicative noises, introduced by Chapouto and the second and third authors (2026), we establish pathwise well-posedness of SHE in both the Young and rough cases, improving the results in Gubinelli and Tindel (2010). In particular, in the rough case (= the white-in-time case), our result covers the case of almost space-time white noise, thus establishing an optimal result within the framework of one-parameter rough paths.

We study scattering and inverse scattering generated by sources and penetrable media with corner singularities. We introduce the notion of split corners, a local model that unifies geometric singularities and coefficient discontinuities arising in source and medium scattering. Within this framework, we establish scattering and inverse scattering results that extend the classical theory to split corners, allowing the source or medium contrast to approach different limiting values in distinct sectors meeting at the corner tip. For source scattering, we prove that every admissible two-split corner necessarily radiates in a general bounded inhomogeneous background, thereby generalizing the classical corner radiation principle to configurations involving both geometric corners and jump discontinuities. For penetrable media, we establish analogous results together with explicit compatibility conditions for incident waves of arbitrary vanishing order. As consequences, we obtain several uniqueness results in inverse source and inverse medium scattering, including recovery of polygonal convex hulls from a single far-field measurement.