Recent Research Directions in Analysis of PDEs 2021–2026

The arXiv section of Analysis of Partial Differential Equations is one of the most prolific areas of pure mathematics, producing over 400 preprints per month as of early 2026. The period 2021–2026 has witnessed landmark breakthroughs — including a computer-assisted proof of finite-time singularity in the 3D Euler equations, the resolution of Hilbert’s Sixth Problem via kinetic theory, and the emergence of probabilistic and nonlocal operator methods as dominant paradigms. This survey identifies, categorises, and profiles the key research directions and landmark papers in math.AP during this era.

Overview #

The landscape of math.AP in 2021–2026 organises into several major research directions:

Background #

The math.AP Landscape #

Analysis of PDEs is the mathematical study of equations involving unknown functions and their partial derivatives, arising in physics, geometry, probability, and engineering. The arXiv math.AP category encompasses everything from regularity theory for elliptic and parabolic equations to global well-posedness for dispersive equations, from geometric flows to inverse problems, and from kinetic theory to stochastic PDEs. With roughly 300–400 papers per month (408 in February 2026 alone), it is one of the most active and interconnected areas of pure mathematics.

The period 2021–2026 is characterised by three broad trends. First, grand-challenge resolutions: several longstanding open problems — including Hilbert’s Sixth Problem and the existence of finite-time singularities for 3D Euler equations with smooth data — were settled using novel combinations of rigorous analysis, Feynman-diagram combinatorics, and computer-assisted numerics. Second, new paradigm emergence: mixed local-nonlocal operators, double phase functionals, and normalised solutions have matured from isolated curiosities into systematic research programmes with their own regularity theories. Third, interdisciplinary expansion: MFG systems, optimal transport, SPDEs, and AI-assisted methods have become structural parts of the math.AP ecosystem.

Recent Developments #

1. Mathematical Fluid Dynamics: Singularity, Non-Uniqueness, and Stability #

Finite-Time Blowup of the 3D Euler Equations #

The question of whether the 3D incompressible Euler equations

$$\partial_t u + (u \cdot \nabla) u + \nabla p = 0, \qquad \operatorname{div} u = 0,$$

can develop a singularity from smooth initial data — open since Euler introduced the equations in 1757 — saw a decisive resolution in a bounded-domain setting through a landmark two-part series by Jiajie Chen and Thomas Y. Hou (arXiv:2210.07191, arXiv:2305.05660, PNAS 2025). Their work proves finite-time, nearly self-similar blowup of both the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data and finite energy in the presence of a solid boundary. The proof employs weighted $L^\infty$ and $C^{1/2}$ norms, sharp functional inequalities inspired by optimal transport, and computer-assisted rigorous numerics to verify nonlinear stability constants. The result was praised as one of the most significant advances in mathematical fluid mechanics in decades.

Prior to Chen–Hou, Tarek Elgindi (2021) showed finite-time singularity for the 3D axisymmetric Euler equations without swirl from $C^{1,\alpha}$ initial vorticity. The Chen–Hou 2021 paper on the Hou-Luo model proved asymptotically self-similar blowup from smooth data for the HL model. Concurrently, Hou and collaborators presented numerical evidence for singularity in 3D Navier-Stokes achieving a $10^7$-fold increase in maximum vorticity, and DeepMind (2025) used AI-assisted methods to discover families of unstable singularities in the Incompressible Porous Media and Boussinesq equations.

Non-Uniqueness of Leray–Hopf Solutions for Navier-Stokes #

A 2021 breakthrough by Dallas Albritton, Elia Brué, and Maria Colombo proved non-uniqueness of Leray–Hopf solutions to the forced 3D Navier-Stokes equations: they exhibited two distinct Leray solutions with zero initial velocity and identical body force, exploiting the extreme instability of a self-similar background solution. Recognised as the most influential 2021 math.AP paper on arXiv by Paper Digest, the result was subsequently extended to bounded domains via gluing methods (arXiv:2209.03530) and to stochastic settings (Electronic Journal of Probability, 2024).

Stability of Shear Flows and Kinetic Theory #

Parallel to the singularity programme, sharp asymptotic stability results for 2D monotone shear flows with no-slip boundary conditions, and extensive work on inviscid damping and enhanced dissipation near shear flows, have appeared throughout 2025–2026.

Arguably the most monumental result in kinetic PDE theory during this period: Yu Deng, Zaher Hani, and Xiao Ma provided a rigorous long-time derivation of the Boltzmann equation from hard-sphere dynamics (arXiv:2408.07818, 2024), extending Lanford’s 1975 short-time theorem to all times within the lifespan of the Boltzmann solution. In a companion paper (arXiv:2503.01800, 2025), they completed the derivation of the compressible Euler and incompressible Navier-Stokes-Fourier equations from Newton’s laws — effectively resolving Hilbert’s Sixth Problem for rarefied hard-sphere gases. The proof uses cumulant ansätze, Feynman-diagram combinatorics, and a molecule-reduction algorithm. This followed the same team’s 2021 derivation of the wave kinetic equation from the cubic NLS.

2. Nonlocal and Fractional PDEs: Mixed Local-Nonlocal Operators #

One of the dominant new paradigms of the 2020s is the study of operators of the form

$$\mathcal{L} u = -\Delta u + (-\Delta)^s u, \quad s \in (0,1),$$

which superpose a classical Laplacian with a fractional (nonlocal) Laplacian. These arise naturally in models combining Brownian and Lévy diffusion processes. The foundational paper by Biagi, Dipierro, Valdinoci, and Vecchi (2020/2021) initiated a systematic theory of regularity and maximum principles for such operators.

Between 2021 and 2026 an explosion of activity produced: gradient regularity for mixed local-nonlocal problems via De Filippis and Mingione (2022, minimisers of mixed functionals are locally $C^{1,\beta}$-regular); Hölder regularity for mixed local-nonlocal degenerate elliptic equations (Garain & Lindgren, 2022); the Wiener criterion for nonlocal Dirichlet problems (Kim, Lee & Lee, 2022); and a Faber-Krahn inequality for mixed operators (Biagi, Dipierro, Valdinoci & Vecchi, 2021). Serena Dipierro and Enrico Valdinoci were among the most prolific contributors, publishing on nonlocal logistic equations with Neumann conditions, ecological niches for mixed dispersal, and Sobolev inequalities for mixed operators.

Giovanni Leoni’s 2023 treatise A First Course in Fractional Sobolev Spaces provided a self-contained reference covering definitions, embeddings, Hardy inequalities, and interpolation inequalities, and ranked among the most-cited arXiv math.AP papers of 2023. Concurrently, a 2025 paper established well-posedness and regularity theory for time-fractional stochastic PDEs involving Caputo derivatives and general nonlocal operators driven by Gaussian and Lévy noise (arXiv:2512.03754).

3. Double Phase Operators and Nonstandard Growth #

The double phase functional

$$\mathcal{H}(u) := \int_\Omega \bigl(|Du|^p + a(x)|Du|^q\bigr),dx, \quad q > p > 1,\ a(x) \geq 0,$$

introduced by Colombo and Mingione, generated a remarkable surge of activity throughout 2021–2026.

The work of Cristiana De Filippis and Giuseppe Mingione is particularly prominent throughout, providing a comprehensive regularity theory for double phase and nonuniformly elliptic functionals (arXiv:2308.10222).

4. Normalized Solutions and Variational Methods for Schrödinger Equations #

The problem of finding solutions $u \in H^1(\mathbb{R}^N)$ with prescribed $L^2$-norm — the mass constraint

$$\int_{\mathbb{R}^N} |u|^2,dx = c$$

— has become a central theme in the study of nonlinear Schrödinger equations. The influential papers by Louis Jeanjean and Thanh Trung Le on multiple normalized solutions for Sobolev critical equations (2020–2021) and by Juncheng Wei and Yuanze Wu on normalized solutions with critical Sobolev exponent and mixed nonlinearities (2021) launched a wave of activity. Key directions include: normalized ground states for NLS with potential (Bartsch, Molle, Rizzi & Verzini); normalized solutions for Schrödinger-Poisson-Slater equations; and standing waves and stability for Choquard equations. The March 2026 arXiv listings confirm that sharp exponents, existence and asymptotics for Choquard equations, and boosted ground states for pseudo-relativistic Schrödinger equations remain highly active.

Parallel work on eigenvalue problems addresses Steklov eigenvalues (monotonicity for regular $N$-gons, sharp geometric bounds), eigenvalues of Pucci’s extremal operator in 3D, and biharmonic Steklov problems on thin sets.

5. Mean Field Games and Aggregation-Diffusion PDEs #

Mean field game theory generated a prolific suite of PDE questions between 2021 and 2026. Highlights include: Imanuvilov, Liu, and Yamamoto (2023) proving Lipschitz stability for determining states and inverse sources in MFG equations using Carleman estimates; Klibanov, Li, and Liu (2023) on Hölder stability via Carleman estimates; the inverse boundary problem for first-order master equations (Liu & Zhang, 2022); and Bresch, Jabin, and Soler (2022) introducing a novel probabilistic derivation of the mean-field limit applicable to Vlasov-Poisson-Fokker-Planck in 2D. By 2025–2026, nonlocal MFG models with spatial interactions and new work on Wasserstein gradient flows of kernel mean discrepancies with connections to machine learning appeared on arXiv (arXiv:2506.01200).

Optimal transport has deeply influenced aggregation-diffusion equations and gradient flows. The March 2026 arXiv listings include a major 73-page paper by Carrillo, Gwiazda, and Skrzeczkowski presenting a new formula for the Wasserstein distance between solutions to nonlinear continuity equations.

6. Chemotaxis and Reaction-Diffusion Systems #

Chemotaxis systems — in particular Keller-Segel models with signal-dependent motility (density-suppressed diffusion) — generated intense activity. Key papers include logistic damping effects and global classical solutions for reaction-diffusion systems with density-suppressed motility (Lyu & Wang, 2021), refined regularity analysis for Keller-Segel-consumption systems (Li & Winkler, 2022), and global existence with uniform boundedness under signal-dependent motility (Jiang & Laurençot, 2021). In 2024, a construction of smooth finite-time blowup solutions for the 3D Keller-Segel-Navier-Stokes (chemotaxis-fluid) system with buoyancy appeared, using a quantitative method that directly constructs the singular solution (arXiv:2404.17228).

In parallel, free boundary reaction-diffusion models for species spreading and SIS epidemic models — including 2026 work on asymmetric kernels in advective periodic environments — continue to produce threshold and long-time dynamics results.

7. Free Boundary Problems #

The Stefan problem (modelling solidification and melting) remained highly active throughout 2021–2026. Key results include $C^{1,\alpha}$ regularity of flat free boundaries for the inhomogeneous one-phase Stefan problem (Ferrari, Forcillo, Giovagnoli & Jesus, 2024; arXiv:2404.07535); regularity of the free boundary for the supercooled Stefan problem in arbitrary dimensions (2025; arXiv:2512.10136), where the free boundary decomposes into regular, singular, and jump parts with the singular part having controlled parabolic dimension; and well-posedness and regularity of physical solutions for the supercooled Stefan problem assuming only integrable initial temperature, with explicit classification of free boundary points (2025; arXiv:2506.18741). These results use obstacle problem techniques, non-degeneracy estimates, and sharp free boundary classification arguments.

Shape optimisation for principal eigenvalues of Pucci operators and $\Gamma$-convergence of convolution-type functionals for free discontinuity problems are active related directions in 2026.

8. Stochastic PDEs and Regularity Structures #

Martin Hairer’s theory of regularity structures generated deep ongoing activity. The period 2021–2026 saw Bailleul and Bruned (2021) extending the algebraic renormalisation framework of regularity structures to a broader class of singular SPDEs (arXiv:2101.11949); the publication of “A tourist’s guide to regularity structures” by Bailleul and Hoshino (2025/2026) in EMS Surveys as an essentially self-contained treatment; applications to stochastic quantisation ($\Phi^4_3$), the KPZ equation, and stochastic geometric flows (Hairer, 2021); and variance renormalisation in regularity structures for the 2D generalised Parabolic Anderson Model (Gerencsér & Hsu, 2026).

On the fluid side, global unique solvability for stochastic Navier-Stokes-Korteweg equations and stochastic Allen-Cahn-Navier-Stokes systems with ergodic invariant measures appeared in 2025, and non-uniqueness of Leray-Hopf solutions was extended to the stochastic forced setting.

9. Dispersive PDEs: Wave Turbulence, Well-Posedness, and Blowup #

The full derivation of the wave kinetic equation from the cubic NLS by Deng and Hani (arXiv:1912.09518, 2021) was the most impactful dispersive result of the era. Their analysis relies on absolutely convergent Feynman-diagram (paired-tree) expansions and identifies favourable scaling laws $\alpha \sim L^{-\varepsilon}$ for the kinetic limit.

Ongoing work includes polynomial growth of Sobolev norms for the fractional NLS on $\mathbb{T}^d$ (Wang, 2026); low-regularity global well-posedness for generalised Zakharov-Kuznetsov equations (Nowicki-Koth, 2026); modulated dispersive equations (modulated KdV with normal form reduction; Gubinelli, Li, Li & Oh, 2025; arXiv:2505.24270); and probabilistic well-posedness of dispersive PDEs beyond variance blowup (2025; arXiv:2509.02344). Scattering results for the quintic generalised Benjamin-Bona-Mahony equation and the 3D Zakharov-Kuznetsov equation, and long-time asymptotics via Riemann-Hilbert and inverse scattering methods for integrable equations, appear in the March 2026 listings.

10. Geometric PDEs #

Ricci flow uniqueness in the non-compact setting (Lee, 2025; arXiv:2503.20292) and a new non-Kähler expanding Ricci soliton construction with Kähler tangent cone at infinity (Bamler, Chen & Conlon, 2026) reflect the continued health of geometric flows. The volume-preserving mean curvature flow regularity in dimensions 2 and 3 appeared in March 2026 (Arya, Jeon & Julin).

Regularity theory for Monge-Ampère equations received major contributions via a geometric approach: Brendle, Léger, McCann, and Rankin (2023; arXiv:2311.10208) derived the Pogorelov second-derivative bound using Kim-McCann-Warren’s pseudo-Riemannian geometry, providing a new approach to $C^1$ estimates for optimal transport maps. Liouville theorems and sharp solvability for the parabolic Monge-Ampère equation with periodic data appeared in March 2026.

11. Inverse Problems for PDEs #

The Calderón problem — recovering a coefficient from boundary Dirichlet-to-Neumann data — attracted major advances: the quasilinear setting (Cârstea, Feizmohammadi, Kian, Krupchyk & Uhlmann, 2021), inverse problems for fractional semilinear elliptic equations (Lai & Lin, 2020), the Calderón problem via Vekua theory (Clifford analysis framework, 2026; arXiv:2601.17313), and the convex lifting approach (Alberti, Petit & Sanna, 2025; arXiv:2507.00645). The anisotropic Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds (Krupchyk, 2025) was an important further advance.

Inverse moving source problems for parabolic equations (Zhao, 2023), reconstruction of scalar parameters in subdiffusion, and inverse problems for multi-term time-fractional diffusion with Caputo derivatives are active in 2025–2026.

12. Semi-Classical Analysis, Spectral Theory, and Nonlinear Elliptic Theory #

A 2024 arXiv survey on semi-classical analysis introducing three representative topics ranked as the top 2024 math.AP paper by Paper Digest, and a 2026 paper celebrating the 100th anniversary of the WKB papers (Vũ Ngọc) indicate that semi-classical methods remain foundational.

In nonlinear elliptic and parabolic theory, major contributions include: Regularity Theory for Elliptic PDEs by Fernández-Real and Ros-Oton (2023), a comprehensive self-contained reference; Fujita-type results for degenerate parabolic equations on Heisenberg groups (Fino, Ruzhansky & Torebek, 2023), ranked the highest-impact 2023 math.AP paper; and singularity formation for nonlinear heat equations on infinite graphs (Punko & Zucchero, 2026).

Emerging and Cross-Cutting Themes (2025–2026) #

Computer-assisted proofs and rigorous numerics. The Chen–Hou Euler blowup proof and related work on the CLM model (Hou-Wang, 2026) demonstrate that computer-assisted methods with rigorous error control are becoming standard for complex nonlinear stability analyses. These methods combine spectral Galerkin approximations with interval arithmetic and weighted norm frameworks to certify nonlinear stability constants — a methodology likely to expand further.

AI and machine learning for PDEs. The 2026 workshop MLPDES26 and the NSF/AMS report on AI for the mathematical sciences signal growing interplay between pure math.AP and deep learning. Neural PDE networks for equation discovery (arXiv:2502.18377), geometric operator learning via optimal transport (arXiv:2507.20065), and AI-assisted singularity discovery (DeepMind, 2025) represent this interdisciplinary frontier.

PDE methods in geometry and probability. The intersection of math.AP with differential geometry, probability (SPDEs), and mathematical physics remains extremely active. The March 2026 listings span general relativity (tensorial wave equations), Kähler geometry (Ricci solitons), and stochastic PDEs — confirming that math.AP functions as a hub connecting multiple mathematical disciplines.

Open Problems #

Smooth-data Euler regularity beyond bounded domains. The Chen–Hou result proves blowup in a bounded domain. Whether finite-time singularity occurs for the 3D Euler equations in all of $\mathbb{R}^3$ from smooth, rapidly decaying initial data — the original Euler problem — remains open.

Navier-Stokes uniqueness from smooth initial data. The Albritton-Brué-Colombo result proves non-uniqueness for forced NS from zero initial velocity. Non-uniqueness (or uniqueness) of Leray–Hopf solutions for the unforced equations from smooth $H^1$ initial data is unresolved (see the companion survey on self-similar solutions).

Optimal regularity theory for double phase problems. Despite the comprehensive work of De Filippis and Mingione, optimal Schauder estimates for parabolic double phase systems at the boundary and under critical growth conditions are not fully established.

Complete derivation programme for Hilbert’s Sixth Problem. Deng-Hani-Ma resolved the case of hard-sphere gases in the Boltzmann regime. The derivation of hydrodynamic equations from particle dynamics in other regimes — dense gases, quantum systems, plasma — remains largely open.

Global well-posedness for energy-critical NLS in high dimensions. Despite progress on wave kinetic theory and probabilistic well-posedness, the deterministic global well-posedness theory for energy-critical and supercritical dispersive equations in dimensions $d \geq 5$ has significant gaps.

Quantum and numerical computation in pure math.AP. The growing use of computer-assisted proofs raises methodological questions about standards of verification, reproducibility, and the scope of problems accessible to these techniques.

References #

Albritton, D., Brué, E., & Colombo, M. (2021). Non-uniqueness of Leray solutions of the forced Navier-Stokes equations. https://cvgmt.sns.it/media/doc/paper/5405/main.pdf

Bailleul, I., & Bruned, Y. (2021). Renormalised singular stochastic PDEs. arXiv:2101.11949. https://www.pure.ed.ac.uk/ws/portalfiles/portal/194767736/2101.11949.pdf

Bailleul, I., & Hoshino, M. (2025). A tourist’s guide to regularity structures and singular stochastic PDEs. EMS Surveys in Mathematical Sciences. https://ems.press/journals/emss/articles/14298505

Brendle, S., Léger, F., McCann, R. J., & Rankin, C. (2023). A geometric approach to a priori estimates for optimal transport maps. arXiv:2311.10208. https://arxiv.org/abs/2311.10208

Chen, J., & Hou, T. Y. (2022). Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis. arXiv:2210.07191. https://arxiv.org/abs/2210.07191

Chen, J., & Hou, T. Y. (2023). Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data II: Rigorous numerics. arXiv:2305.05660. https://arxiv.org/abs/2305.05660

Chen, J., & Hou, T. Y. (2025). Singularity formation in 3D Euler equations with smooth initial data. PNAS, 122(28). https://www.pnas.org/doi/10.1073/pnas.2500940122

De Filippis, C., & Mingione, G. (2023). Regularity for double phase problems at nearly linear growth. arXiv:2308.10222. https://arxiv.org/abs/2308.10222

DeepMind. (2025). Discovering new solutions to century-old problems in fluid dynamics. https://deepmind.google/blog/discovering-new-solutions-to-century-old-problems-in-fluid-dynamics/

Deng, Y., & Hani, Z. (2021). On the derivation of the wave kinetic equation for NLS. arXiv:1912.09518. http://arxiv.org/pdf/1912.09518.pdf

Deng, Y., Hani, Z., & Ma, X. (2024). Long time derivation of the Boltzmann equation from hard sphere dynamics. arXiv:2408.07818. https://www.semanticscholar.org/paper/91b67412a6058c1ace054a32fbf36fa2d2998d3d

Deng, Y., Hani, Z., & Ma, X. (2025). Hilbert’s sixth problem: Derivation of fluid equations via Boltzmann’s kinetic theory. arXiv:2503.01800. https://www.semanticscholar.org/paper/01d8f11b5d31f7037fb4914797e938db11d76ec5

Ferrari, F., Forcillo, N., Giovagnoli, D., & Jesus, B. (2024). Free boundary regularity for the inhomogeneous one-phase Stefan problem. arXiv:2404.07535. https://arxiv.org/abs/2404.07535

Gubinelli, M., Li, J., Li, T., & Oh, T. (2025). Nonlinear PDEs with modulated dispersion IV: Normal form reduction for modulated KdV. arXiv:2505.24270. https://arxiv.org/pdf/2505.24270.pdf

Hou, T. Y. (2021). The potentially singular behavior of the 3D Navier-Stokes equations. arXiv:2107.06509. https://arxiv.org/abs/2107.06509

Hu, J., Jin, S., Liu, N., & Zhang, L. (2024). Quantum circuits for partial differential equations via Schrödingerisation. Quantum, 8, 1563.

Imanuvilov, O. Y., Liu, Y., & Yamamoto, M. (2023). Lipschitz stability for determining states and inverse sources in MFG equations. [Journal of Mathematical Analysis].

Ok, J., Scilla, G., & Stroffolini, B. (2025). Partial regularity for parabolic systems of double phase type. arXiv:2510.03849. https://arxiv.org/pdf/2510.03849.pdf

Paper Digest. (2025, March). Most influential arXiv (Analysis of PDEs) papers — 2025-03 version. https://www.paperdigest.org/2025/03/most-influential-arxiv-analysis-of-pdes-papers-2025-03-version/

Segata, J., & Chen, M. (2026). Scattering for the 3D Zakharov-Kuznetsov equation [arXiv preprint]. arXiv math.AP March 2026.

arXiv math.AP listings. (2026, February–March). https://arxiv.org/list/math.AP/2026-03