Analysis & PDEs

Partial differential equations, functional analysis, and harmonic analysis

Includes:Analysis of PDEs(math.AP)Functional Analysis(math.FA)Classical Analysis and ODEs(math.CA)

Analysis & PDEs

Partial differential equations, functional analysis, and harmonic analysis

Includes:Analysis of PDEs(math.AP)Functional Analysis(math.FA)Classical Analysis and ODEs(math.CA)

Trending in Analysis & PDEs

2512.01966

Semigroups For Initial-Boundary Value Problems

We review the theory of one-sided coupled operator matrices with a focus on evolution equations with inhomogeneous boundary conditions. (The original article had no abstract.)

2512.0196617

Dec 2025Analysis of PDEs

Absence of Exponential Stability and Polynomial Stabilization in a Class of Beam Models with Tip Rotary Inertia

We investigate the impact of dissipative dynamic boundary conditions applied at one end of a beam, analyzing their influence on model stability within the Euler-Bernoulli framework. Our primary finding is that hybrid dissipation does not alter the decay characteristics of the original model. We examine two scenarios: first, when hybrid dissipation is the sole dissipative mechanism, and second, when it complements other dissipative mechanisms. In the first case, we demonstrate that hybrid dissipation fails to induce exponential decay, instead producing a slow decay rate of $t^{-1/2}$ for large $t$. In the second case, when acting as a complementary mechanism, hybrid dissipation neither enhances nor diminishes the decay behavior of the original model.

2512.01964

Dec 2025Analysis of PDEs

15,452 papers

Uniformly Bounded Cochain Extensions and Uniform Poincaré Inequalities

In this paper, we construct a novel global bounded cochain extension operator for differential forms on Lipschitz domains. Building upon the classical universal extension of Hiptmair, Li, and Zou, our construction restores global commutativity with the exterior derivative in the natural $HΛ^k(Ω)$ setting. The construction applies to domains and ambient extension sets of arbitrary topology, with strict commutation holding on the orthogonal complement of harmonic forms, as dictated by the underlying topological obstruction. This provides a missing analytical tool for the rigorous foundation of Cut Finite Element Methods (CutFEM). We also obtain continuous uniform Poincaré inequalities and lower bounds for the first Neumann eigenvalue on non-convex domains.

2604.04927Apr 2026

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The entropy production is not always monotone in the space-homogeneous Boltzmann equation

We show an example of a function and a collision kernel for which the entropy production increases in time when we flow it by the space-homogeneous Boltzmann equation. The collision kernel is not any of the physically motivated kernels that are commonly used in the literature. In this particular setting, our result disproves a conjecture of McKean from 1966.

2604.04861Apr 2026

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2604.04819

Boundary estimates for parabolic non-divergence equations in $C^1$ domains

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence form parabolic equations in parabolic $C^1$ domains, providing explicit moduli of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with $C^{1,\mathrm{Dini}}$ boundaries, while also recovering the known $C^{1-\varepsilon}$ regularity for parabolic Lipschitz domains, unifying both regimes with a single proof.

2604.04819Apr 2026

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2604.04799

On a $_2F_1\big(\frac{1}{4}\big)$-identity due to Gosper

It is only in exceptional cases that a $_2F_1(z)$-series with rational parameters and a rational argument, apart from the cases for $z \in \{ \pm 1, \frac{1}{2} \}$ associated with classical hypergeometric identities, admits an evaluation given by a combination of $Γ$-values with rational arguments. In this paper, we present a new and integration-based approach toward the construction of special values for $_2F_1$-series of the desired form. We apply this approach using a $_2F_1\big(\frac{1}{4}\big)$-identity originally due to Gosper and later considered by Vidunas, Ebisu, and Zudilin, to evaluate a ${}_{2}F_{1}$-series of convergence rate $\big(\frac{172872}{185039}\big)^2$. With regard to extant research on so-called ``strange'' ${}_{2}F_{1}$-evaluations, as in the work of Ebisu and Zeilberger, our new series seems to have the largest numerator/denominator in its argument.

2604.04799Apr 2026

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2604.04776

Optimal $C^{1,α}$ regularity up to the boundary for fully nonlinear elliptic equations with double phase degeneracy

In this paper we establish optimal $C^{1,α}$ regularity up to the boundary for viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy law and oblique boundary conditions. The approach developed here relies on first deriving uniform boundary Hölder estimates for perturbed models with oblique boundary data in ``almost $C^{1}$-flat'' domains. Building upon these estimates, the desired regularity is obtained through a compactness and stability framework for viscosity solutions. As a byproduct of our analysis, we determine the optimal Hölder exponent for solutions when the governing operator is quasiconvex or quasiconcave. In addition, we establish an improved regularity result along vanishing points of the source term.

2604.04776Apr 2026

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2604.04676

A Trudinger-Moser inequality under a refined constraint in fractional dimensions and extremal functions

We establish a Trudinger-Moser type inequality with a Tintarev-type constraint in fractional-dimensional spaces and prove the existence of maximizers in the critical regime. Our results provide a refinement of those in (Calc. Var. 52 (2015), 125-163) in the setting of fractional-dimensional spaces, as well as of those in (Ann. Global Anal. Geom. 54 (2018), 237-256) for classical Sobolev spaces.

2604.04676Apr 2026

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2604.04626

A seminorm-only characterization of analytic Besov spaces on the disc

We introduce the space $\mathcal{W}^{s,p}(\mathbb{D})$ of analytic functions $u$ on the unit disc such that the radial restrictions $u_{r}(ξ):=u(rξ)$ satisfy the Gagliardo seminorm-only bound \[ \sup_{0<r<1}[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}<\infty, \] with no $\emph{a priori}$ control of $\sup_{r}\|u_{r}\|_{L^{p}(\mathbb{S}^{1})}$. Our main result shows that this assumption already forces $u\in H^{p}(\mathbb{D})$ and that the radial boundary trace $u^{*}$ belongs to $W^{s,p}(\mathbb{S}^{1})$, with $u_{r}\to u^{*}$ in $W^{s,p}(\mathbb{S}^{1})$ as $r\to1^{-}$. The key mechanism combines the mean-value property (which pins the constant mode at $u(0)$) with a fractional Poincar$é$ inequality on $\mathbb{S}^{1}$, recovering $L^{p}$ control from oscillation alone. As a consequence, the trace map $u\mapsto u^{*}$ is a surjective isomorphism $\mathcal{W}^{s,p}(\mathbb{D})\xrightarrow{\sim}B^{s}_{p,p,+}(\mathbb{S}^{1})$ with explicit norm equivalence.

2604.04626Apr 2026

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2604.04609

Nonlinear Schrödinger equations with critical Hardy potential and Choquard nonlinearity

We study the Cauchy problem for the nonlinear Schrödinger equation characterized by contrasting effects between the concentration at the origin of a critical Hardy potential and the intrinsic nonlocality of a Choquard nonlinearity. We prove the existence of a ground state solution through optimizers of an interpolation Hardy-Gagliardo-Nirenberg inequality and derive a non-existence result via Poho\v zaev identities. Using these results, we provide various criteria for the global existence and finite-time blow-up for the problem in the energy-subcritical regime. Finally, we establish a key compactness result, which enables us to obtain a characterization of finite-time blow-up solutions with minimal mass.

2604.04609Apr 2026

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2604.04592

$W^{2,1}$ approximation of planar Sobolev homeomorphisms by smooth diffeomorphisms

The approximation of Sobolev homeomorphisms by smooth diffeomorphisms is well understood in first-order spaces $W^{1,p}$, but remains largely open in the second-order space $W^{2,1}$ due to a fundamental tension between curvature control and injectivity. In this paper we isolate and resolve the local analytical component of this problem. We construct explicit local regularisations both across flat interfaces and near multi-cell vertices, and prove convergence in $W^{2,1}$ together with quantitative preservation of the Jacobian. The resulting maps are $C^{1}$ on the whole domain and smooth inside each cell of the partition; in particular they are $C^{2}$ away from the interfaces. These local constructions are combined into a global smoothing theorem: any piecewise quadratic $C^{1}$-compatible planar homeomorphism satisfying a quantitative bi-Lipschitz condition can be approximated in $W^{2,1}$ by maps that are $C^{1}$, injective, and have positive Jacobian. As a consequence, we show that the general $W^{2,1}$ approximation problem reduces to a purely geometric question: the construction of piecewise quadratic approximations with quantitative injectivity and nondegeneracy.

2604.04592Apr 2026

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2604.04526

Low-regularity global well-posedness for the Boltzmann equation near vacuum

We study the Boltzmann equation near vacuum in anisotropic low-regularity Besov spaces. We establish the global existence and uniqueness of strong solutions with the critical regularity index $2/p$ for $p\in[1,\infty)$ in $\mathbb{R}^3$. The proof relies on a new bilinear estimate for the nonlinear collision operator. Combined with a div-curl type lemma we develop, this allows us to close the a priori estimates and thereby obtain global well-posedness.

2604.04526Apr 2026

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From hyperbolic to complex Euler integrals

Hyperbolic hypergeometric integrals are defined as Barnes-type integrals of products of hyperbolic gamma functions. Their reduction to ordinary hypergeometric functions is well known. We study in detail their degeneration to complex hypergeometric functions. Namely, using uniform bounds on the integrands, we prove that the univariate hyperbolic beta integral and the conical function degenerate to two-dimensional integrals over the complex plane.

2604.04471Apr 2026

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2604.04416

Rigidity for a semilinear Neumann problem with exponential nonlinearity in the large diffusion limit

We consider a semilinear Neumann problem with exponential nonlinearity in a smooth bounded domain $Ω\subset \mathbb{R}^2$. We prove that there exists a threshold $\bar{\varepsilon}>0$ such that for all $\varepsilon>\bar{\varepsilon}$, any classical solution must be constant. This result provides a positive answer to a conjecture recently posed by Calanchi, Ciraolo, and Messina (2026). Our proof relies on a combination of $L^1$-estimates, a Jensen-type argument via the Neumann Green's function to obtain uniform exponential integrability, and elliptic regularity.

2604.04416Apr 2026

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2604.04391

On the Viscosity Solutions of Parabolic p-Laplacian Equations with Capillary-Type Boundary Conditions

In this paper, we establish the well-posedness and large-time asymptotic behavior of viscosity solutions to singular/degenerate parabolic $p$-Laplacian equations with general capillary-type boundary conditions, including Neumann and prescribed contact angle cases, on strictly convex domains. By establishing a gradient estimate independent of the $C^0$ norm of the solution via the maximum principle, and by analyzing the problem through an approximation procedure together with associated elliptic eigenvalue problems, we prove the existence, uniqueness, and asymptotic behavior of solutions. For the elliptic problem with Neumann boundary conditions, we first focus on flat domains with the zero Neumann condition. By reflecting $u$ across the flat boundary $T_1$ and then using inf- and sup-convolution arguments in the reflected domain, we obtain the $C^{1,α}$ result. For the general elliptic case, we obtain sharp global $C^{1,α}$ regularity by flattening the boundary and employing compactness arguments together with an ``improvement of flatness'' iteration. With an extra condition in the iteration, we can also deal with the singular case $1<p<2$. In the parabolic setting, the spatial Hölder regularity of $Du$ follows from elliptic estimates combined with the Lipschitz continuity of $u$ in time, which in turn yields joint Hölder continuity in $(x,t)$. Extensions to non-convex domains are also discussed by incorporating a suitable forcing term.

2604.04391Apr 2026

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2604.04320

A Combinatorial Formula for Recursive Operator Sequences and Applications

We study sequences of bounded operators $(T_n)_{n \ge 0}$ on a complex separable Hilbert space $\mathcal{H}$ that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients $A_0, A_1, \dots, A_{r-1}$ are pairwise commuting bounded operators on $\mathcal{H}$. \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for $T_n$. As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

2604.04320Apr 2026

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2604.04268

Bernstein inequality on parabolic domains

Several families of sharp Bernstein inequalities are established on the weighted $L^2$ space over parabolic domains, which include bounded or unbounded rotational paraboloids and parabolic surfaces. The main tool is a second-order differential operator satisfied by a specific basis of orthogonal polynomials in weighted $L^2$ space.

2604.04268Apr 2026

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2604.04257

Bernoulli cylinder frame operators: filtration, Haar structure, and self-similarity

We study the finite-rank frame operators generated by cylinder indicator functions for the Bernoulli Cantor measure $μ_{p}$. In the symmetric case $p=\frac{1}{2}$, the natural Haar differences diagonalize these operators. For general $0<p<1$, we show that the weighted Haar basis still yields a sparse tree-banded matrix form, although diagonalization is lost. We also prove a filtration representation in terms of conditional expectations and level-wise mass operators. This leads to a norm convergent limit operator $K_{\infty}$, which is compact, positive, and self-adjoint. Finally, we show that $K_{\infty}$ is characterized by a self-similar operator identity induced by the first-level Cantor decomposition, and we derive corresponding block and scalar resolvent renormalization formulas.

2604.04257Apr 2026

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2604.04256

Modified scattering for the Vlasov-Riesz system with long-range interactions

We study the long-time asymptotic behavior of small-data solutions to the three-dimensional Vlasov--Riesz system with the inverse power-law potential $λ|x|^{-α}$ in the strictly long-range regime ($0 < α< 1$). By introducing finite- and infinite-time modified wave operators for the characteristic flows, we describe the asymptotic dynamics via convergence to an effective profile along a suitably modified reference flow, and establish modified scattering of solutions. Our proof relies mainly on ODE techniques for the characteristic flows, while also using PDE methods for weighted $W^{1,\infty}$-bounds. Compared with the earlier result (of Huang and Kwon), our Lagrangian approach extends modified scattering to the broader regime $\frac{1}{2}<α<1$ and provides a distinct and more robust argument.

2604.04256Apr 2026

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2604.04169

An Aronson-Bénilan / Li-Yau estimate in the JKO scheme in small dimension

We derive an Aronson-Bénilan / Li-Yau estimate in the JKO scheme associated to the porous-medium, heat, and fast-diffusion equations, in dimensions $1$ and $2$, and on simple domains (cubes, quarter-space, half-spaces, whole space, and the torus). Our method is based on a maximum principle for the determinant of the Hessian of Brenier potentials, iterated as a one-step improvement along the scheme. As a consequence, we obtain local $L^\infty$ bounds on the density, uniform in the time step, consistent with the continuous-time result. As a byproduct, we rigorously derive the optimality conditions in the fast-diffusion case, filling a gap in the literature.

2604.04169Apr 2026

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Nonlocal Hyperdissipative Perturbations of the Three Dimensional Navier-Stokes System

We study the three-dimensional incompressible Navier-Stokes system on $\mathbb{R}^3$ with an additional dissipative nonlocal term \[ \partial_t u + (u\cdot\nabla)u + \nabla p = νΔu + Lu, \qquad {\rm div}\, u = 0, \] where $L$ is a self-adjoint Fourier multiplier whose symbol is comparable to $-|ξ|^{2α}$ for some $α>1$. We first identify a sharp Fourier-symbol criterion distinguishing lower-order convolution perturbations from genuinely regularizing nonlocal corrections. In the resulting hyperdissipative class we prove the exact $L^2$ energy identity, global weak solvability for every $α>1$, and local strong well-posedness in $H^s(\mathbb{R}^3)$ for $s>\frac52$. We then show that the Lions exponent $α=\frac54$ remains the critical energy-growth threshold in this nonlocal setting: if $α\ge \frac54$, every $H^s$ solution is global, while for every $α>1$ one has global strong solvability for sufficiently small $H^s$ data. Finally, for the vanishing-hyperdissipation approximation of the classical three-dimensional Navier-Stokes equations, we prove a near-singular divergence principle: if the classical flow blows up at a first singular time $T_*$ in a continuation norm $X$, then the corresponding regularized family cannot remain uniformly bounded in $X$ on any interval approaching $T_*$. This identifies the precise point at which the fixed-parameter global theory degenerates in the Navier-Stokes limit.

2604.04167Apr 2026

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2604.04148

Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System

We study a logarithmic fractional Schrödinger--Poisson system in $\R^{3}$: \begin{equation*} \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+φu=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\ \varepsilon^{2α}(-Δ)^αφ=u^{2}, & \text{in }\R^{3}. \end{cases} \end{equation*} Here $α\in\bigl(\frac34,1\bigr)$, $4<p<2_α^{*}=\frac{6}{3-2α}$, and $V$ satisfies a global potential condition. Using a suitable Orlicz-type Banach space, we establish a $C^{1}$ variational framework for the problem and combine the Nehari manifold method with Lusternik--Schnirelmann category theory. We then prove that, for every fixed $δ>0$ and all sufficiently small $\varepsilon>0$, the system admits at least $\operatorname{cat}_{M_δ}(M)$ distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of $V$ as $\varepsilon\to0$.

2604.04148Apr 2026

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Page 1 of 773

Analysis & PDEs Papers - ScienceStack

15,452 papers

Uniformly Bounded Cochain Extensions and Uniform Poincaré Inequalities

2604.04927Apr 2026

View

The entropy production is not always monotone in the space-homogeneous Boltzmann equation

2604.04861Apr 2026

View

2604.04819

Boundary estimates for parabolic non-divergence equations in $C^1$ domains

2604.04819Apr 2026

View

2604.04799

On a $_2F_1\big(\frac{1}{4}\big)$-identity due to Gosper

2604.04799Apr 2026

View

2604.04776

Optimal $C^{1,α}$ regularity up to the boundary for fully nonlinear elliptic equations with double phase degeneracy

2604.04776Apr 2026

View

2604.04676

A Trudinger-Moser inequality under a refined constraint in fractional dimensions and extremal functions

2604.04676Apr 2026

View

2604.04626

A seminorm-only characterization of analytic Besov spaces on the disc

2604.04626Apr 2026

View

2604.04609

Nonlinear Schrödinger equations with critical Hardy potential and Choquard nonlinearity

2604.04609Apr 2026

View

2604.04592

$W^{2,1}$ approximation of planar Sobolev homeomorphisms by smooth diffeomorphisms

2604.04592Apr 2026

View

2604.04526

Low-regularity global well-posedness for the Boltzmann equation near vacuum

2604.04526Apr 2026

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From hyperbolic to complex Euler integrals

2604.04471Apr 2026

View

2604.04416

Rigidity for a semilinear Neumann problem with exponential nonlinearity in the large diffusion limit

2604.04416Apr 2026

View

2604.04391

On the Viscosity Solutions of Parabolic p-Laplacian Equations with Capillary-Type Boundary Conditions

2604.04391Apr 2026

View

2604.04320

A Combinatorial Formula for Recursive Operator Sequences and Applications

2604.04320Apr 2026

View

2604.04268

Bernstein inequality on parabolic domains

2604.04268Apr 2026

View

2604.04257

Bernoulli cylinder frame operators: filtration, Haar structure, and self-similarity

2604.04257Apr 2026

View

2604.04256

Modified scattering for the Vlasov-Riesz system with long-range interactions

2604.04256Apr 2026

View

2604.04169

An Aronson-Bénilan / Li-Yau estimate in the JKO scheme in small dimension

2604.04169Apr 2026

View

Nonlocal Hyperdissipative Perturbations of the Three Dimensional Navier-Stokes System

2604.04167Apr 2026

View

2604.04148

Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System

2604.04148Apr 2026

View

Page 1 of 773