Papers

We investigate the low Mach number limit of the three-dimensional full compressible Navier-Stokes-Fourier (NSF) equations on \(\mathbb{R}\times\mathbb{T}^2\) for two different classes of initial data, corresponding respectively to a well-prepared regime and an ill-prepared regime. The density and temperature are allowed to approach different asymptotic states at infinity. For the well-prepared regime, the solutions of compressible NSF equations converge to a planar diffusion wave solution globally in time as the Mach number tends to zero, where the difference between the states at the far fields is small independently of the Mach number and the non-zero modes of the initial perturbations are exponentially small. Moreover, the optimal time-decay rate can be obtained. It can be viewed as the first global-in-time result on the low Mach number limit of three-dimensional NSF equations with large temperature variations. The corresponding local-in-time result is proved while the difference between the states at the far fields can be arbitrarily large for the ill-prepared data.

For the infinite Fermi-Pasta-Ulam (FPU) system, general solutions can be approximated by counter-propagating waves associated with solutions to the Korteweg-de Vries (KdV) equation as the lattice mesh size goes to zero. We show that, by exploiting the conservation of the FPU Hamiltonian, the continuum limit from the FPU system to the KdV equation with $L^2$-level initial data holds in appropriate norms on an arbitrary time interval, thereby answering an open question posed by Hong, Kwak, and Yang (2021). Moreover, for the local-in-time continuum limit of the FPU system to the KdV equation, we lower the required Sobolev regularity to $H^s$ with $s > - \frac 34$. Our continuum limit results also apply to the case of the Toda lattice in Flaschka's form, thereby lowering the regularity requirement in our previous work (2026). To establish this low-regularity continuum limit, we prove key trilinear estimates that are sharp up to the endpoint by combining linear estimates, the transversality of characteristic curves, and multilinear dispersive smoothing properties of the linear FPU flow.

In this paper, we establish a Bourgain–Brezis–Mironescu formula for BV functions on arbitrary open sets for a general class of radial mollifiers. Building on this formula, we identify the \(\Gamma\)-limit of the improved nonlocal energies with respect to \(L^1\)-convergence as a constant multiple of the total variation. As a consequence, we obtain a characterization of BV functions in terms of the finiteness of the corresponding nonlocal energies. We also establish local \(L^1\)-compactness on arbitrary open sets and global \(L^1\)-compactness on bounded open sets under a natural uniform integrability assumption. In particular, except for the global compactness result, all these results hold on arbitrary open sets, with no assumptions on boundedness, connectedness, or boundary regularity. Counterexamples show that for global compactness, both the boundedness of the domain and the uniform integrability assumption are essential. The proofs rely primarily on one-dimensional restrictions of BV functions and a concentration argument for radial mollifiers.

We systematically investigate breather solutions in the modified Korteweg-de Vries (mKdV) equation via numerical simulations. We show that the breather solutions are stable under a variety of perturbations, including amplitude changes, modifications of internal parameters, and perturbations of the nonlinearity of the equation. Our results are not only consistent with the known analytical studies on the stability of breathers in integrable systems, extending them further, but also provide numerical evidence that such breather-type structures can persist and remain stable over the simulated time scales in some non-integrable settings with symmetric potentials.

We study the long-time behavior of Wasserstein gradient flows for interaction energies \[ \mathsf E[\mu] = \frac12\iint_{M\times M}K(x,y)\,\mathrm d\mu(x)\,\mathrm d\mu(y) \] on a closed manifold $M$. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density $u_0\in L^p(M)$, $p>1$, the energy gap is integrable in time and satisfies \[ \mathsf E[\mu_t]-\mathsf E_{\min}=o(t^{-1}). \] If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the $o(t^{-1})$ rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any $\delta>0$, we construct a solution of the linearized flow whose energy is comparable to $t^{-1-\delta}$ along a sequence of times tending to infinity. Moreover, for any $\delta>0$, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to $t^{-1-\delta}$. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.

Nonlocal models have found great success in recent years. In particular, models that replace spatial derivatives with strongly singular integral operators involving finite interaction radii are used in a wide range of applications, including image processing, continuum mechanics, and many other areas. However, from the perspective of mathematical analysis, difficulties arise when considering the appropriate notion of “boundary” conditions. (The nonlocal analogue of the boundary is often called a “collar” and is typically not a lower-dimensional set.) For instance, merely enforcing homogeneous Dirichlet-type constraints on the collar fails to guarantee a number of essential properties used in the analysis and numerical simulation of these models, as we show in the present work. In the local setting, working in a Sobolev space of functions that are zero on the boundary allows one to integrate by parts with no boundary term, approximate by test functions, extend by zero to the full space, and apply the Hardy and Poincaré inequalities. We prove that the nonlocal analog of each of these fails if one only requires the functions to be zero on the collar, and that one must move to a smaller, more restrictive space to enjoy these properties. In addition, we introduce a new lifting operator, correct an error in a previously published result on a nonlocal Green's theorem, and establish how the properties listed above relate to existing notions of fractional Sobolev spaces.

We establish an exact parametrized reduction of the classical Willmore flow of Hopf tori in the round three-sphere to the spherical elastic flow. In Willmore-flow time the reduction is $(\partial_t\gamma)^{\perp}=-4\nabla_{L^2}\mathcal E(\gamma)$; its global reconstruction uses a pullback-circle-bundle normalization and does not require preservation of simplicity along flow lines. Combining the known global subconvergence of the spherical elastic flow with Pozzetta's full convergence theorem then yields full smooth convergence of flow lines, modulo domain diffeomorphisms, for every simply parametrized Hopf torus at arbitrary initial energy. We then obtain quantitative conclusions for the unreparametrized trajectory: finite normal $L^2$-metric length, total-variation convergence of the pullback area measures, bounded-Lipschitz convergence of the image measures, and explicit tail estimates. The density of the full varifold endpoint equals the covering multiplicity of the limiting elastic profile, and Pinkall's marked flat lattices converge. In particular, a simple initial Hopf immersion with energy arbitrarily close to $4\pi^2$ from above can smoothly converge to some parametrization of the Clifford torus with multiplicity two, although its limiting conformal class in moduli space is exactly the square class.

In a recent preprint (arXiv:2511.09556), we exhibited an inverse energy cascade for the 3D Navier-Stokes equations which results in "instantaneous" Type I blow-up and failure of uniqueness in certain sharp regularity classes. Our purpose here is to elucidate this phenomenon further in the setting of the Obukhov dyadic model, and to compare it to the better-known phenomenon of finite-time blow-up. When intermittency is low ($\alpha\leq 2$), we recover our previous result from the Navier-Stokes setting; in the energy-supercritical case where intermittency is high ($\alpha>2$), we show that the same can occur in the class of finite energy solutions. We present two different proofs: a soft approach using a Lyapunov function and an explicit multiscale construction. For the inviscid system we illustrate that a similar phenomenon occurs at lower regularity. Finally, we state a finite-time blow-up theorem for a mixed Desnyansky-Novikov-Obukhov model, in which a forward cascade from finitely supported data and no force produces a singularity that is of Type II only by a small margin, with features comparable to the recent forced Navier-Stokes blow-up of OpenAI.

Let \(\mathbf{E}\to M\) be a finite-rank Riemannian vector bundle over a compact Riemannian manifold with nonempty smooth boundary, equipped with a compatible connection. We develop an intrinsic trace theory for Sobolev sections defined through weak covariant derivatives. For every integer \(m\ge1\) and every \(1\le p<\infty\), we construct continuous trace operators for the covariant derivatives up to order \(m-1\) and characterize the Sobolev space with homogeneous Dirichlet boundary conditions as the kernel of the corresponding trace operator. As an application, we establish the existence of ground state solutions for a class of quasilinear elliptic equations on Riemannian vector bundles with generalized \((p,q)\)-growth. The proposed framework unifies several important geometric operators, including the Bochner \(p\)-Laplacian, the \((p,q)\)-Bochner Laplacian, and prescribed mean curvature-type operators.

Families of explicit analytic solutions of the magnetohydrodynamic equilibrium equations are presented, equivalent to steady incompressible Euler flow. The solutions are non-axisymmetric and possess exact nested toroidal flux surfaces. No expansion is made in inverse aspect ratio or in the deviation from axisymmetry. The magnetic field and flux surfaces are given explicitly in Cartesian coordinates using elementary functions. The field, current density, and scalar pressure are smooth over the toroidal domain. The pressure gradient vanishes only on the magnetic axis. One family of solutions has uniform rotational transform $\iota=2$, while another family has a sheared $\iota$ profile. These counterexamples to Grad's conjecture are valuable for understanding the existence and regularity of 3D equilibria and for testing numerical codes.

We prove the Mumford–Shah conjecture, formulated in 1989, by completing the classification of blow-up limits of minimizers in the plane. The remaining case concerns limits whose discontinuity set is noncompact and has connected complement. In this setting, we rule out bounded connected components using second-variation inequalities obtained from combined inner and outer variations associated with two perpendicular translations of a compact subset separated from the rest of the discontinuity set. Established results then imply that the discontinuity set is connected, and the known classification for connected discontinuity sets completes the proof.

We consider variational image denoising with spatially dependent Orlicz regularization and introduce an Orlicz–Sobolev approximation of the ROF model based on a scaled $L\log L$-type density. For a general class of uniformly superlinear integrands satisfying a $\Delta_2$-condition, we establish existence and uniqueness of minimizers and derive a Fenchel dual problem with dual attainment and pointwise optimality conditions. We then specialize the framework to a logarithmic density whose correction is activated above a prescribed local gradient scale. For this model, we obtain explicit expressions of the Fenchel dual and optimality conditions as well as a pointwise radial formula for the dual proximal map involving the Lambert $W$-function. As the logarithmic parameter tends to zero, we prove equicoercivity and $\Gamma$-convergence to the ROF functional, together with convergence of the corresponding minimizers. Numerical illustrations indicate that the logarithmic correction can reduce staircasing for diffuse transitions, while retaining behavior comparable to ROF for sharp interfaces and a natural image.

This paper studies both stability and maximal regularity of the diffusion equation on an unbounded surface without any assumptions on the curvature. We show that the surface diffusion system admits a unique global-in-time strong solution satisfying $L^2$-asymptotic stability when the slope of the surface is gentle. We also derive maximal $L^2$-regularity of the solution when the exterior force satisfies both surface divergence form and tangential condition. Moreover, we investigate the Hölder continuity with respect to time of the solution. The key idea of showing our stability and maximal regularity results is to apply nice properties of both Laplace and weighted Laplace-Beltrami operators in order to derive the properties of the Laplace-Beltrami operator on the unbounded surface.

We prove the sharp two-disk comparison conjectured by Langford and Laugesen for the second positive Neumann eigenvalue under an upper Gaussian-curvature bound, and strengthen it to a shifted reciprocal inequality. Let $\Omega\subset\mathbb{C}$ be bounded, simply connected and Lipschitz, with $g=\omega|dz|^2$, where $\omega$ is continuous and positive on $\overline{\Omega}$ and $\Delta\log\omega+2K\omega\ge0$ in distributions. Write $A=|\Omega|_g$ and enumerate the spectrum by $0=\lambda_0<\lambda_1\le\lambda_2\le\cdots$. If $KA<4\pi$ when $K>0$, then $\frac{1}{\lambda_2}+\frac{1}{\lambda_3}>\frac{2}{\lambda_1(D_K(A/2))}$ and $\lambda_2<\lambda_1(D_K(A/2))$, where $D_K(A/2)$ is the constant-curvature disk of area $A/2$. The same conclusions hold for disk-type Lipschitz domains in smooth surfaces. In the subcritical regime, both bounds are sharp, as witnessed by an extremizing sequence of smooth connected domains of exact area and curvature whose Neumann eigenvalues converge, at every fixed index, to those of two disjoint equal model disks. At $KA=4\pi$ we obtain the sharp non-strict bounds $\lambda_2\le2K$ and $\lambda_2^{-1}+\lambda_3^{-1}\ge K^{-1}$. The proof combines two-pole Green coordinates, positive-kernel comparison, and simultaneous centering. We also establish a global quadratic deficit for the model area split.

We prove a priori interior $L^2$ bounds for the second fundamental form of area-minimizing currents modulo $2$ in every dimension $n\geq 2$ and arbitrary codimension. The bounds depend only on the dimension, codimension, and an upper bound for the mass. As a corollary, we show that area-minimizing surfaces modulo $2$ with bounded area lie in finitely many diffeomorphism classes. This answers a question of \cite{Liu2020WeakCounterexamples}.

We study a passive scalar on the two-dimensional torus driven by an arbitrary finite set of white-in-time divergence-free Fourier modes. We prove that the exponential $L^2$ dissipation rate of the advection-diffusion equation is uniformly bounded as the diffusivity tends to zero as long as there exist two nonparallel forced modes. If not, then the dissipation rate can be arbitrarily fast depending on the initial data, but the top Lyapunov exponent remains uniformly bounded below. The proof uses a projection onto a finite set of modes which depends on the forcing set. Part of our argument follows the structure of the four-mode case investigated by Chemnitz and Chemnitz, but uses a strictly weaker condition than their matrix inequality between Frobenius and operator norms. Combined with an accessibility argument, this allows us to consider an arbitrary set of forcing modes and initial data.

We introduce the nonlocal zero-mass equation \[ - \Delta u + \Phi_u(|x|)\, |u|^{p-2}\, u = f(u) \quad \text{in } {\mathbb R}^N, \qquad \Phi_u(r) = a \int_r^\infty \rho^{-b}\, h_u^{q-1}(\rho)\, d\rho, \] where $h_u(\rho)$ is the mass of $|u|^p$ in the ball of radius $\rho$. For radial functions, this equation includes the defocusing inverse-power Schrödinger equation, the Chern-Simons-Schrödinger equation, and the Schrödinger-Poisson-Slater equation as special cases. We develop the associated ball-mass Lebesgue and Sobolev spaces, which are uniformly convex, and prove sharp compact radial embeddings above a new critical exponent, a Brézis-Lieb type splitting, and a Pohožaev identity. Exploiting a scaling invariance of the operator, we construct an unbounded sequence of eigenvalues of a scaled eigenvalue problem using the Fadell-Rabinowitz cohomological index, and obtain existence and multiplicity results in the subscaled, superscaled, and critical regimes. Our main result is a Brézis-Nirenberg type theorem, proved by means of a new scaled linking theorem, which gives a nontrivial radial solution for every $\lambda > 0$ that is not an eigenvalue. Specializing to the three models recovers several known results in a unified way and yields new ones, including a Brézis-Nirenberg type result for the Schrödinger-Poisson-Slater equation in dimensions $N \ge 4$.

We study an optimal boundary control problem for a quasilinear hyperbolic system describing isothermal gas flow in pipelines. The control acts at network nodes and models the action of compressor stations. The system admits non-constant steady states, which are strictly monotone in space. We work in the framework of H2-solutions and consider an optimal control problem that steers the system toward a prescribed steady state. We establish the existence of optimal controls and then prove the exponential turnpike property, showing that optimal trajectories remain exponentially close to the corresponding steady state over most of the time horizon. The analysis relies on a cheap control property obtained via a Lyapunov approach and on a constructed control. Numerical simulations illustrate the theoretical result.

We study a social contagion model describing the propagation of opinions across a network. Our model is a generalization of a model introduced by Daley and Kendall, and consists of a system of coupled differential equations set on the nodes of an infinite graph. The modeling is inspired by SIR-type models from epidemiology. The key idea is that individuals can transmit their opinion to their neighbors (as in epidemiological models), until they perceive that the opinion is already known in their neighborhood. After introducing the model, we analyze the long-time behavior of its solutions. We prove that the solutions converge and we study their asymptotic equilibrium. We also establish estimates on the speed of propagation of the solutions through the network. These results give estimates on the rate of adoption of the opinion and on its velocity in the social network. We also provide simulations that indicate that this model can give rise to patterns: when the individuals take into account individuals that are ”far enough” in estimating whether or not the opinion is already known, then we observe the emergence of localized ”bubbles” where the opinion is adopted.

Given three normed spaces $X\_0 , X\_1 , \tilde{X\_1}$, with $\tilde{X\_1}$ a closed subspace of $X\_1$ with same norm, how does $[X\_0 , X\_1 ]\_\theta$ compare to $[X\_0 , X\_1 ]\_\theta\,? This classical problem of interpolation theory has been mostly considered in the case of finite codimensional subspaces. We show here that quotient J norms are a useful tool which allows to tackle several relevant cases.

We study the high-temperature derivation of $\Phi^4_d$ measures from grand-canonical Bose gases on two- and three-dimensional tori. Building on the upper-symbol variational approach of the author's earlier work, we use relative entropy to give a simple, unified proof of relative free-energy convergence and Hilbert–Schmidt convergence of all fixed-order rescaled reduced density matrices to their Hartree counterparts. For interaction ranges $\varepsilon=\lambda^\eta$, where $\lambda$ is the inverse temperature, these results hold for every fixed $0<\eta<1/2$ in two dimensions and $0<\eta<1/26$ in three. In two dimensions, we improve the result of Jougla and Rougerie (2026) and the local $\Phi^4_2$ derivation of Fröhlich, Knowles, Schlein and Sohinger (2025). Combined with classical approximation, the two-dimensional result gives the first local $\Phi^4_2$ derivation valid for every fixed exponent $0<\eta<1/2$, allowing polynomial exponents arbitrarily close to the diluteness threshold from below. In three dimensions, we use the identification of the local measure and the stationary SPDE estimates of Nam, R.~Zhu and X.~Zhu (2025). From these estimates we strengthen their classical correlation convergence to the Hilbert–Schmidt norm. This yields the local limit at every fixed order in the same polynomial range.

We prove the existence of rogue waves and breathers in nonlinear lattice wave equations including the nonlinear Klein–Gordon and the FPUT lattice (with added local forces). The main feature of our lattice wave equation is that the kinetic part $\frac{\mathrm{d}^2}{\mathrm{d}t^2}$ is multiplied by a non-constant function $\frac{1}{d(t)}$ such that gaps in the spectrum of $\frac{1}{d(t)}\frac{\mathrm{d}^2}{\mathrm{d}t^2}$ open wide enough to include the spectrum of the spatial linear operator. We find solutions as critical points of an indefinite functional using a saddle-point method combined with concentration-compactness arguments. In case of temporally $T$-periodic coefficients and under identical assumptions as for rogue waves, the same variational method also provides existence of breather solutions whose temporal period is an arbitrary prescribed integer multiple of $T$.

We establish global existence, decay, and scattering for sufficiently small, smooth, and localized perturbations of a constant non-vacuum equilibrium of a relativistic quantum hydrodynamic system in three space dimensions. In logarithmic-amplitude and phase variables, the equations form a semilinear system of coupled wave equations. The skew-symmetric coupling between the time derivatives cancels in the energy identity, yielding a natural derivative energy. The linearized system has two dispersion branches. At low frequencies, the slow branch exhibits Schrödinger-type dispersion, while the fast branch has a spectral gap; at high frequencies, both branches are wave-like. The main nonlinear difficulty arises from quadratic interactions with nontrivial time and space-time resonances. We show that the symbol of every active quadratic interaction contains the corresponding interaction phase as an exact factor. This structural cancellation removes the resonant denominator and allows us to eliminate the quadratic terms by a nonsingular normal-form transformation. Combining this transformation with dispersive and energy estimates, we obtain uniform-in-time bounds for the derivative energy and $\langle t\rangle^{-3/2}$ decay of the first derivatives in $L^\infty$. We further prove that the nonlinear solution scatters to a solution of the linearized system in the high-order derivative energy norm.

Floquet bundle theory serves as a natural extension of the eigenvalue theory for elliptic and periodic parabolic operators to the spectral theory of non-autonomous, non-periodic parabolic operators, which provides a useful spectral tool for studying threshold dynamics in more general non-autonomous systems. However, current understanding and applications of the normalized principal Floquet bundle remain rather limited compared with the eigenvalue theory for elliptic or periodic parabolic problems. In this work, we first extend several fundamental properties of the principal eigenvalue to the normalized principal Floquet bundle of non-autonomous, non-periodic parabolic operators, and obtain a series of parallel results. Furthermore, we focus on the asymptotic behavior of the principal Floquet exponent with respect to the generalized frequency and diffusion rate under various typical parameter limits (including both independent and coupled parameter regimes) for the operator under homogeneous Neumann boundary conditions. The main tools are a parabolic comparison principle developed in this work for long-time limit processes, which is used to derive bounds for the principal Floquet exponent via suitably constructed sub- and super-solutions, supplemented by Harnack's inequality and long-time averaging techniques. Finally, we employ the normalized principal Floquet bundle to define two critical numbers (as extensions of the basic reproduction number) for a non-autonomous spatially diffusive SIS epidemic model, examine the extinction and weak persistence of the disease, and investigate the limiting behavior of the basic critical numbers with respect to the generalized frequency and diffusion rate.

We establish lower bounds for the critical CR GJMS operator on closed pseudo-Hermitian manifolds. In dimension $2n+1$, assuming embeddability, we obtain a lower bound involving the Heisenberg Sobolev norm of order $n+1$, modulo an $L^{2}$ term, on the orthogonal complement of the kernel. In dimension three, we prove that the CR Paneitz operator is bounded below without assuming embeddability. The proofs combine the Heisenberg calculus with a square-root construction based on the joint functional calculus of the sub-Laplacian and the Reeb vector field on the Heisenberg group. As an application, we show that the infinitely many negative eigenvalues of the CR Paneitz operator on the Rossi sphere have zero as their only accumulation point. Consequently, this operator does not have closed range.

We study the uniform stability of composite waves consisting of a degenerate Oleinik shock and a rarefaction wave for scalar viscous conservation laws. We consider cubic-type fluxes and arbitrarily large $L^2$ initial perturbations of the composite wave within the $H^1$ strong-solution class. We obtain a uniform $L^2$ estimate for such large perturbations. In particular, the estimate implies the contraction of a degenerate Oleinik shock, up to a dynamical shift. As a further consequence, the uniformity with respect to viscosity implies that the associated self-similar solution, composed of the degenerate shock and rarefaction, is stable in the class of inviscid limits. The main analytical difficulty is that the standard weighted Poincaré inequality underlying the $a$-contraction method fails to yield the required coercivity at the linear level. We overcome this obstruction by establishing a new degenerate Hardy–Poincaré inequality adapted to the degenerate structure of the shock profile. Combined with an exact nonlinear decomposition, this linear inequality yields a strict coercivity estimate that controls the quadratic energy together with the higher-order nonlinear terms, without any smallness assumption on the perturbation.

For the semistable $J$-equation, we prove that the numerical $J$-null locus coincides with the ambient $C^2$-singular locus of Murakami's weak solution. Consequently, this singular locus is a proper analytic subset with finitely many irreducible components, and the weak solution is locally smooth outside the $J$-null locus. The proof relies on two analytic ingredients. First, we establish a regularization theorem for singular $J$-subsolutions, showing that Demailly's global regularization preserves quantitative strict cone conditions. Thus, singular strict subsolutions with prescribed analytic poles can be replaced by smooth strict subsolutions away from their pole sets. Second, we derive relative a priori estimates adapted to these subsolutions: a relative $L^\infty$-estimate from determinant control and a weighted second order estimate yielding uniform $C^2$-bounds on compact subsets of the regular locus.

For $n\geq5$, let $(u,p)$ be a smooth solution of the stationary incompressible Navier-Stokes equations in $\R^n\setminus\{x'=0\}$, where $\{x'=0\}$ is the $x_n$-axis. We prove that the scale-invariant transverse bound $|u(x)|\leq C |x'|^{-1}$ forces $u\equiv0$. This result relies on a quantitative local regularity theorem, which shows that the transverse bound condition yields uniform control of $u$ and $\nabla u$ in a smaller ball and removes the possible singularity along the axis. The proof combines weak extension across the axis, approximate Green functions for an adjoint drift operator, a one-sided bound for the total head pressure, a localized Frehse-Růžička weighted estimate, and a finite Stokes-Morrey bootstrap. Applying the local theorem at arbitrarily large scales yields whole-space rigidity and removability of line singularities. This together with the asymptotic expansions on the exterior domains in the case $|u(x)|\leq C |x|^{-1}$ in [1], yields the same expansion in exterior domains under the condition $|u(x)|\leq C |x'|^{-1}$.

Compliant arterial wall mechanics, contributing to the Windkessel effect in the carotid artery (CA), have significant impact on hemodynamic patterns and surface shear indicators for cardiovascular diseases, e.g. atherosclerosis. In this study, we extend the linear elastic Fluid-Structure Interaction (FSI) framework to account for nonlinear strain-dependent wall behavior. The model incorporates a Young's modulus generalized from tensile tests on silicone phantoms to capture the nonlinear stress-strain relation of arterial tissue. The final computational model, using a resistance-type boundary condition and in vitro measured stress-strain relation is validated against both, in vitro assessed silicon CA phantom as well as published clinical data on the flow splitting to the daughter branches in CA bifurcations. To better reflect physiological conditions, our model is subsequently extended to incorporate prestress of patient-specific geometries, and clinically measured stress-strain relations, followed by validation against clinical CA data. The present study demonstrates the feasibility of strain-dependent Young's (elastic) modulus as a means to enhance the capacity of the linear elastic framework to accurately represent the physiologically nonlinear mechanics of arterial walls, striking a balance between implementation effort and physiological fidelity. This approach yields realistic strains, volumetric inflation behavior and nonlinear pressure-volume relationships. Furthermore, the study reveals the importance of proper outlet boundary conditions and the shortcomings of resistance-type boundary condition in patient-specific modeling, leading to non-physiological pressure profiles and non-physiological temporal flow splitting.

We establish small-energy regularity for stationary weak solutions of the Liouville equation $ -\Delta u=e^u $ in every dimension. It solves a problem left open in Da Lio [Commun. Partial Differ. Equations, 2008] and posed explicitly again by Da Lio and Hyder in [Commun. Contemp. Math. 2025]. The proof relies on the analysis of concentration phenomena for almost stationary fields with structure constraints.