Analysis of PDEs

arXiv:math.AP

Existence and uniqueness, boundary conditions, linear and nonlinear operators, stability, soliton theory, integrable PDEs.

Looking for a broader view? This category is part of:

Analysis & PDEs

Analysis of PDEs

arXiv:math.AP

Existence and uniqueness, boundary conditions, linear and nonlinear operators, stability, soliton theory, integrable PDEs.

Looking for a broader view? This category is part of:

Analysis & PDEs

Trending in Analysis of 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

Related categories:

Commutative Algebra Algebraic Geometry Algebraic Topology Classical Analysis and ODEs Combinatorics

10,428 papers

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

View

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

View

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

View

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

View

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

View

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

View

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

View

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

View

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

View

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

View

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

View

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

View

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

View

2604.04000

Koch-Tataru theorem for 3D incompressible active nematic liquid crystals

We investigate the incompressible hydrodynamic system of the active nematic liquid crystals in the Beris-Edwards framework. Although we focus on constant activity in this paper, the simplified system derived from it exhibits the potential to perform computations and transmit information in active soft materials \cite{defect-active}. More precisely, by employing Kato's strategy for constructing mild solutions combined with the Banach contraction principle, we show the existence and uniqueness of the Koch-Tataru type solution in $\mathbb{R}^3$ for small initial data $(Q_0,u_0)\in L^\infty\times {\rm BMO}^{-1}$. This is the first well-posedness result for the system with initial data in critical space.

2604.04000Apr 2026

View

2604.03931

Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators

In this work, we address a parabolic problem featuring a potentially doubly nonlinear term, governed by a combination of local and nonlocal operators (see Problem P1 below). We first establish the local existence of weak energy solutions via a semidiscretization in time applied to an auxiliary evolution problem. The uniqueness of these solutions is subsequently obtained through a novel generalization of the classical inequality of Diaz and Saa, suitably adapted to the mixed local nonlocal setting. This generalization provides a new comparison principle and establishes the T-accretivity of a corresponding operator in L2. By employing this comparison principle, we construct suitable barrier functions that allow the global in time extension of solutions. Furthermore, we demonstrate the convergence of weak solutions to a nontrivial stationary state. Our approach relies on methods from the theory of contraction semigroups. It is noteworthy that these results are underpinned by a detailed analysis of the stationary problems associated with Problem P1, which also reveals several qualitative properties of the solutions.

2604.03931Apr 2026

View

Inhomogeneous Scaling Function and Heat Kernel Estimates on Fractals Satisfying Some Resistance Conditions

In this paper, we focus on strongly local regular Dirichlet forms, especially those satisfying Morrey-type inequalities. We prove the equivalence between resistance estimates and heat kernel estimates in this case. Self-similar forms on fractals serve as a major application, where we construct a spatially inhomogeneous scaling function and characterize all the doubling self-similar measures. Further, on some special examples, the resistance conditions are reduced to some geometric conditions, on which a complete theory on self-similar Dirichlet spaces is established therein. In particular, we construct a concrete example on rotated triangle fractals, where the optimal heat kernel estimate is not related at all to the lower scaling exponent.

2604.03915Apr 2026

View

2604.03876

Local null controllability of the complete N-dimensional Ladyzhenskaya-Boussinesq model

This work investigates both local null controllability and large time null controllability for a class of complete Ladyzhenskaya Boussinesq systems, where the controls are distributed and supported on small subsets of the domain. The proof of local null controllability relies on classical techniques, including Carleman estimates and Liusternik Inverse Mapping Theorem. Nevertheless, the presence of nonlinearities in both the velocity and temperature equations necessitates careful treatment.

2604.03876Apr 2026

View

2604.03856

Asymptotic behavior of the wave equation subject to a Kelvin-Voigt nonlocal damping

In this article, we examine the well-posedness and asymptotic behavior of the energy associated with the wave equation that incorporates a Kelvin-Voigt nonlocal damping structure given by $-||\nabla u_t(t)||_2^2 Δu_t$. Utilizing the robust framework of nonlinear semigroups, we successfully demonstrate the existence of both strong and weak solutions. Our findings reveal that the decay rate for these solutions is optimally characterized by $1/t$, highlighting the effectiveness of this dissipative structure. This work not only enhances our understanding of the wave equation under nonlocal damping but also emphasizes the crucial balance between mathematical rigor and physical relevance.

2604.03856Apr 2026

View

2604.03848

Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping

In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t^2 u - \partial_x^2 u + \fracμ{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable ($\mathcal{C}^1$). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ($μ=0$).

2604.03848Apr 2026

View

2604.03811

From Linear to Nonlinear: A Resolvente criterion for Polynomial Stability of Semigroups Generated by Monotone Operators

The Borichev--Tomilov theorem \cite{BT2010} provides a sharp characterization of polynomial decay for linear $C_0$-semigroups in terms of resolvent growth along the imaginary axis. In the nonlinear setting, the absence of a spectral theory renders the imaginary-axis approach inapplicable. In this paper, we develop a new framework for nonlinear maximal monotone operators in Hilbert spaces by replacing spectral analysis on $i\mathbb{R}$ with the asymptotic analysis of the \textit{real resolvent equation} \[ λx_λ+ \mathcal{A}(x_λ) \ni y, \quad λ\to 0^+. \] We show that, for homogeneous operators (and suitable perturbations), the blow-up rate of $\|x_λ\|$ at the origin reveals the effective nonlinear scaling of the operator and determines the corresponding polynomial decay rate of the associated semigroup through a coercive dissipation mechanism. This provides a nonlinear Tauberian-type principle for a broad class of degenerate dissipative systems. The approach recovers, in particular, the optimal $1/t$ decay for the wave equation with nonlocal Kelvin--Voigt damping recently obtained by Cavalcanti et al.\ (2025), and allows one to justify decay estimates for weak solutions in situations where classical multiplier methods require higher regularity. It also clarifies the structural limitations of the method, identifying regimes where additional geometric or time-domain arguments are necessary.

2604.03811Apr 2026

View

Page 1 of 522

Analysis of PDEs Papers (math.AP) - ScienceStack

10,428 papers

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

2604.04776

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

2604.04526

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

2604.04526Apr 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

2604.04256

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

2604.04000

Koch-Tataru theorem for 3D incompressible active nematic liquid crystals

2604.04000Apr 2026

View

2604.03931

Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators

2604.03931Apr 2026

View

Inhomogeneous Scaling Function and Heat Kernel Estimates on Fractals Satisfying Some Resistance Conditions

2604.03915Apr 2026

View

2604.03876

Local null controllability of the complete N-dimensional Ladyzhenskaya-Boussinesq model

2604.03876Apr 2026

View

2604.03856

Asymptotic behavior of the wave equation subject to a Kelvin-Voigt nonlocal damping

2604.03856Apr 2026

View

2604.03848

Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping

2604.03848Apr 2026

View

2604.03811

From Linear to Nonlinear: A Resolvente criterion for Polynomial Stability of Semigroups Generated by Monotone Operators

2604.03811Apr 2026

View

Page 1 of 522