75,924 papers in this slice of arXiv.
Kotaro Motegi
We prove the existence of a genuinely time-dependent Brakke flow starting from Γ0⊂Rn+1
Changfeng Gui, Tuoxin Li, Juncheng Wei +1
We prove that for every integer N≥3 and α≥21, Beckner's inequality 2α∫SNu(PNu)dw+(N−1)!∫SNudw−N(N−1)!log∫SNeNudw≥0 holds for every u∈H2N(SN) whose center of mass is at the origin. The proof is mainly based on an integral representation formula and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively for every integer N≥3.
Rui Chen, Daniel Hauer
We study fundamental gaps for the Dirichlet p-Laplacian on bounded convex domains with convex potentials. We prove log-concavity of the positive first eigenfunction by a regularization and two-point maximum principle. For N≥2, we identify a sharp transition at p=2 through collapsing smooth convex domains: the gap vanishes for 1<p<2, remains of order D−2 for p=2, and diverges for p>2. For p≥2 and convex potentials, we first establish a degenerate weighted Poincaré inequality, which yields quantitative stability estimates for the Lp-Poincaré inequality and, in turn, dimension-free bounds for the fundamental gap; for zero potential, we further obtain an enhanced gap estimate involving both the first eigenvalue and the diameter. We also prove existence of diameter-normalized gap minimizers for p>2 and show that they degenerate as p↓2. Finally, for N=1, we prove the sharp inequality λ2,p(ID,V)−λ1,p(ID,V)≥(p−1)(2p−1)(Dπp)p for every p>1 and every convex potential, with equality precisely for constant potentials.
Jan Haskovec
We perform a linearized stability analysis of the Vlasov--Fokker--Planck equation obtained as the mean-field description of a stochastic spontaneous aggregation model with memory. Memory is represented by a chain of K internal variables. We characterize the spatially homogeneous equilibria and derive a scalar dispersion relation for spatially inhomogeneous perturbations. When all relaxation rates are equal, we show that every Fourier mode that is unstable in the memoryless model possesses a unique critical relaxation parameter: sufficiently long memory stabilizes the mode, whereas it remains unstable for short memory. Although memory may destabilize individual modes associated with negative Fourier coefficients of the sensing kernel, we prove that, for radially symmetric distance-decreasing kernels, it cannot destabilize a homogeneous equilibrium that is stable in the memoryless model. Thus, memory cannot create an overall instability of an otherwise stable homogeneous state, although it may change the set of the unstable modes. We present a numerical example for the normalized top-hat kernel, demonstrating that increasing the effective memory length successively stabilizes the Fourier modes, with higher frequencies being stabilized before lower ones. This is consistent with the coarsening effect observed in recent particle simulations. Finally, under suitable nondegeneracy and regularity assumptions, we use the Crandall--Rabinowitz theorem to obtain branches of spatially inhomogeneous stationary solutions bifurcating from the homogeneous state.
Agus L. Soenjaya
We study the stochastic resistive Hall--magnetohydrodynamic (Hall--MHD) system on bounded convex polyhedral domains, subject to nonlinear perfectly conducting boundary conditions. The system is driven by multiplicative Gaussian forcing in the momentum equation together with structured curl-type noise in the induction equation arising from stochastic perturbations of the resistive and Hall parts in the generalised Ohm law. We develop a fully discrete, linearly implicit, structure-preserving mixed finite element method based on a compatible discrete de Rham complex, which preserves the magnetic Gauss law exactly. We prove subsequential convergence of the discrete solutions to a finite-energy weak martingale solution, thereby obtaining a constructive existence result for the stochastic Hall--MHD system with curl-type noise. Numerical experiments illustrate the stochastic dynamics, exact divergence preservation, and convergence of the method.
Harprit Singh
We develop an intrinsic theory of weighted, inhomogeneous Besov spaces on general homogeneous Lie groups without recourse to group-specific arguments. Starting from a definition in terms of localised test functions, we establish an equivalent, wavelet-like, multiscale characterisation. This provides a unified mechanism for deriving Besov embeddings, a Taylor-remainder characterisation, Young-type product estimates, Schauder estimates for convolution semigroups, and a weighted Kolmogorov criterion for random distributions. We apply this framework to parabolic Anderson-type equations associated with positive Rockland operators and with singular initial data. For space-time noise, our results cover the Young regime. For purely spatial noise, we also establish well-posedness in the first singular regime using a variant of the Cole-Hopf transform for Rockland operators. Both applications rely on a mild sewing lemma that accommodates time-dependent Banach spaces and increments with a singularity at the initial time.
Nam Q. Le, Qi Sun, Hung V. Tran
For each nonnegative integer m, we construct smooth symmetric 3×3 coefficient matrices Am satisfying the fixed ellipticity bound I≤Am≤281I for which the smooth solutions of uniformly elliptic equations in nondivergence form tr(Am(x)D2um)=Am(x):D2um=0in B2⊂R3 have common Dirichlet data, satisfy ∥um∥L∞(B2)≤1, but m→∞lim∥Dum∥L1(B1)=∞. Thus, there is no interior W1,1 estimate depending only on ellipticity in dimension three, and consequently no such W1,p estimate for any p≥1. This resolves in the negative an open question raised by Nadirashvili, Tkachev, and Vlăduţ. The construction also gives a uniformly convergent limit u∈/BVloc(B1) for a measurable uniformly elliptic coefficient matrix obtained as an L1 limit of the Am.
Zhida Chang, Mingying Zhong
We study the Vlasov--Poisson--Boltzmann system on the three-dimensional torus under uniform shear flow. For Maxwell molecules with Grad's angular cutoff and sufficiently small shear rate, we consider perturbations around the spatially homogeneous self-similar profile of the sheared Boltzmann equation. In self-similar variables, we prove the global stability of this profile, including global existence and uniqueness, the exponential decay of nonzero-spatial derivatives in weighted L∞ spaces, and the uniform boundedness of the renormalized perturbation. The analysis combines a Caflisch's decomposition, Guo's L∞--L2 estimates, macro--micro analysis, and a spectral study of the zero-frequency mode. We further derive a closed zero-frequency system for a renormalized total energy and suitable second-order moments, which yields a precise large-time description of the shear-induced energy growth.
Nilasis Chaudhuri, Milan Pokorný, Ewelina Zatorska
We study a viscous one-velocity Baer--Nunziato system for two barotropic compressible fluids in a bounded three-dimensional domain. In contrast with models in which the volume fraction is merely transported or determined by an algebraic equilibrium constraint, it satisfies a differential closure driven by the pressure gap between the phases. For arbitrary finite-energy initial data and adiabatic exponents γ±>1, we construct global-in-time dissipative measure-valued solutions and prove dissipative measure-valued--strong uniqueness relative to any sufficiently regular solution with the same initial data. The principal difficulties are the singular and non-continuous behavior of the pressure-relaxation source at vanishing volume fractions, the associated concentration defects, and the lack of direct coercivity of the standard thermodynamic relative energy with respect to the volume fraction. These are resolved by an endpoint cutoff argument and an augmented relative energy coupled to the renormalized volume-fraction equation.
Antonio Agresti, Michael Kniely, Bao Quoc Tang
The existence of global classical solutions for reaction-diffusion systems arising from chemical reaction networks remains a major open problem in the deterministic setting, especially for reactions with high polynomial growth. We prove that a suitably chosen, physically motivated transport noise yields unique strong solutions for complex balanced chemical reaction networks that are global in time with arbitrarily high probability. These solutions possess paths in CtθCx∞ for all θ<1/2 and are, in particular, classical in space. Furthermore, we show that a suitable transport noise can enhance the dissipation of spatial fluctuations at an arbitrarily prescribed exponential rate. Our proofs rely on a combination of scaling-limit arguments, maximal Lp(Lq)-regularity, and entropy-entropy dissipation estimates.
Shilpa Dutta, Anja Schlömerkemper
We establish the existence of an energy minimizer for a variational model of compressible magnetoelastic solids. The analysis is carried out in a new admissible class of deformations consisting of mappings of finite distortion, which extends previously available existence frameworks. A key ingredient is a compactness result under the critical integrability assumption on the outer distortion coefficient, which significantly weakens the regularity requirements imposed in earlier works. To obtain this result, we prove a diameter estimate for finite-distortion mappings satisfying the Ciarlet-Nečas condition and derive an open mapping theorem under the optimal integrability assumption, that is, the outer distortion is in Ln−1. This provides a partial positive result in the direction of the Iwaniec-Šverák conjecture and implies that admissible deformations are homeomorphisms. These topological and compactness properties allow us to apply the direct method of the calculus of variations and establish the existence of minimizers for compressible magnetoelastic solids within the admissible class of deformations consisting of mappings of finite distortions.
Floris Roodenburg
In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. These operators are studied on Sobolev spaces with power weights measuring the distance to the boundary. We prove optimal estimates for the resolvent operators and the corresponding heat semigroups. In addition, it is proved that the Dirichlet and Neumann Laplacians admit a bounded H∞-functional calculus on Sobolev spaces with certain compatibility conditions at the boundary. We show that these compatibility conditions cannot be omitted in general. The results in this paper are a direct extension of those obtained by Lindemulder, Lorist, the author, and Veraar in [J. Funct. Anal., 289(8):110985, 2025].
Mu Gao, Huicheng Yin
We are concerned with the global classical solution problem of 3D compressible isentropic Euler equations of Chaplygin gases ⎩⎨⎧∂tρ+div(ρv)=0,∂t(ρv)+div(ρv⊗v)+∇p=0,ρ(0,x)=ρˉ+ερ0(x), v(0,x)=εv0(x). where ρˉ>0 is a constant, ε>0 is small, the state equation is p=p(ρ)=P0−ρD with P0 and D being some positive constants. For the 3D compressible Euler equations of Chaplygin gases, which are a prototype of multidimensional nonlinear symmetric hyperbolic systems with totally linearly degenerate eigenvalues, there is a basic conjecture imposed by A. Majda: it typically has a global classical solution (ρ,v) with (ρ−ρˉ,v)∈C([0,∞),Hs(R3))∩C1([0,∞),Hs−1(R3)) when (ρ0,v0)∈Hs(R3) with s>25 unless (ρ,v) itself blows up in finite time. In this paper, under the assumptions that for any fixed constant μ with 0<μ<1/2, integer N≥15, rotv0(x)≡0 and ∥(ρ0,v0)∥HN(R3)+∣a∣≤13∑∥⟨x⟩1+μ∇a(ρ0,v0)∥L2(R3)≤1, we show that the classical solution (ρ,v) exists globally. Our main ingredients include: establishing a series of new decay estimates of energy bounds, weighted pointwise space-time L∞-L2 estimates and weighted Strichartz-type estimates for the 3D potential flow equation of Chaplygin gases.
Zhibing Zhang
In this paper, we establish two improved Liouville-type theorems for the three-dimensional stationary tropical climate model without imposing any temperature assumptions. Owing to the special structure inherent to the equations, we fully exploit the L2-estimate for the oscillation of the temperature, refined interpolation techniques and an iteration argument to establish a Liouville-type theorem under more flexible and relaxed power-law growth conditions. As an interesting consequence, we obtain the triviality of solutions under the integrability conditions imposed only on (u,∇v) or (∇u,∇v). Furthermore, by developing a systematic framework to handle the energy function associated with the non-trivial solutions and with the aid of delicate ODE analysis and a contradiction argument, we refine our Liouville-type theorem by introducing logarithmic corrections to the growth rates.
De Huang, Jiajun Tong, Xiaopeng Zheng
For the 2-D incompressible Euler equation, we prove the existence of m-fold symmetric uniformly rotating vortex patches with 90-degree corners for all m≥12. Properties of these patches and the induced stationary flows in the co-rotating frame are characterized. Our construction is based on a novel fixed-point method.
Lauri Oksanen, Rakesh, Mikko Salo
We prove uniqueness in the Lorentzian Calderón problem in a semiglobal setting, comparing a potentially large perturbation of the Minkowski metric against a small one. Our proof uses a reduced amount of data, solving a formally determined version of the problem. In contrast to traditional approaches, it employs neither control-theoretic arguments nor a reduction to a geometric inverse problem via microlocal methods or high-frequency solutions. Instead, it relies on distorted plane waves and weighted L2-estimates.
Suman Pal, Prasanta chatterjee
This paper derives the second-order positive Burgers' equation from the standard Burgers' hierarchy to explore its complete integrability and exact analytical solutions. We construct this higher-order nonlinear evolution equation by systematically applying the recursion operator to the classical Burgers' equation. Expanding on this structural framework, we then derive the explicit third-order equation and use Complete Bell Polynomials to generalize the n-th order hierarchy. Through the Cole-Hopf transformation, the second-order nonlinear equation rigorously maps to the linear third-order dispersion equation. We establish complete integrability by explicitly formulating the scalar Lax pair (zero-curvature representation), which allows us to directly derive the associated differential and algebraic Darboux transformations. To systematically classify explicit stationary, time-dependent traveling wave, and self-similar profiles, we apply several techniques to the linear domain: separation of variables, the Hirota perturbation method, traveling wave reduction via the Complete Discrimination System for Polynomial Method (CDSPM), and Lie similarity reduction. Crucially, our analysis demonstrates the exact finite truncation of the Hirota perturbation series. We conclude by outlining how this equation impacts the theoretical understanding of fluid dynamics, nonlinear transport phenomena, and higher-order wave propagation.
Fan Zheng
We audit the exact-arithmetic certificate in arXiv:2310.19781v2, its 2025 Communications in Mathematical Physics version, and their shared MATLAB archive. CMP corrects none of the audited items; we therefore treat both versions together and use the openly accessible arXiv source for exact provenance. We found 11 proof-affecting defects (alongside typographical slips) and give reproducible exact counterexamples to several implemented bounds and reconstruction steps. To the best of our knowledge, this is the first independent, version-pinned, source-level audit of a published computer-assisted proof (CAP) in analysis to identify multiple proof-affecting defects in its released computational certificate. The findings do not refute the intended theorem or its analytic reduction, but they show that the published certificate does not prove the claimed conclusion. Future work will give a corrected, structurally simpler proof with substantially less machine assistance.
Michael Eden, Hari Shankar Mahato
We study the multiscale behavior of a coupled visco-poroelastic system arising in the modelling of fibre-reinforced hydrogels (FIHs) used in tissue engineering scaffolds. The composite material consists of a periodic fibre scaffold, which is governed by quasi-static linear elasticity, and a hydrogel phase saturating the interstitial space, which is modelled as a Biot linear poroelastic medium enhanced with Kelvin--Voigt structural damping. The two phases are coupled through continuity of displacement and traction across their shared interface. Both the fibre scaffold and the hydrogel phase are connected, so that mechanical forces can be transmitted through the composite and interstitial fluid can flow directly through the hydrogel network. Starting from a microscopic (ε-scale) model, we derive uniform a-priori estimates and establish well-posedness via a Rothe time-discretisation argument for both the case of standard Biot fluid content η=0 and the case of viscous fluid content η=αδ>0. We then perform a rigorous two-scale homogenization in the limit ε→0 using periodic unfolding. In addition to the usual effective elasticity, storage, coupling, and permeability coefficients, the homogenized constitutive laws contain nonlocal-in-time memory terms generated by the microscopic viscoelastic relaxation. All effective coefficients and memory kernels are explicitly characterized in terms of the microscale geometry and material parameters.
Hiroyuki Hirayama, Masahiro Ikeda
In the present paper, we consider the Cauchy problem of a system of three nonlinear Schrödinger equations with quadratic nonlinearity. We first prove the existence of ground states in the form of solitary wave solutions with two frequency parameters. We then show that the conditions on the frequency parameters for the existence of ground states depend on the resonance structure of the system. Next, we give the two results for global solutions. The first is an improvement of global well-posedness for initial data below the ground state threshold. The second is an improvement of global well-posedness for oscillating initial data. We also prove the orbital stability of the ground state sets with small speed parameter.