Advances in PDEs, Infinite-Dimensional Dynamical Systems and Numerical Dynamics

Dear Colleagues,

Partial differential equations (PDEs) are fundamental to modeling complex, continuous phenomena in various fields of science and engineering. The dynamical systems they generate often evolve in infinite-dimensional spaces, exhibiting rich behaviors such as the emergence of patterns, transition to chaos, and long-term evolution on attractors. In particular, stochastic partial differential equations (SPDEs) of evolutionary type play a crucial role in describing dynamics with random influences in natural and man-made complex systems, with their solutions residing in infinite-dimensional state spaces such as Hilbert or Banach spaces.

Numerical dynamics focuses on how well a numerical scheme applied to a differential equation replicates the dynamical behavior of the underlying system, in particular its long-term or asymptotic behavior. This involves comparing the continuous-time dynamical system with the discrete-time system defined by the numerical method. Two classes of systems are of particular interest: dissipative systems, which possess attractors, and non-dissipative systems such as Hamiltonian systems, which preserve structural features or quantities.

This Special Issue aims to gather cutting-edge research and review articles that advance the understanding of PDEs, the infinite-dimensional dynamical systems they generate, and the numerical dynamics of the schemes used to simulate them. We seek contributions that explore the interplay between rigorous mathematical analysis and sophisticated numerical computation, particularly those addressing whether numerical methods correctly replicate key dynamical features.

Topics of interest for this Special Issue include, but are not limited to, the following:

• Advanced Theoretical Analysis: Well-posedness, asymptotic behavior, and stability theory for PDEs and SPDEs.
• Stochastic PDEs: Strong, mild, and weak solutions; stochastic integration in Hilbert spaces; Gelfand triples and the variational approach; Markov property and invariant measures.
• Infinite-Dimensional Dynamical Systems: Existence, regularity, continuity and dimension of pullback (random) attractors, exponential dichotomies, hyperbolic solutions and their stable/unstable manifolds, bifurcations and chaotic dynamics in nonlinear PDEs. 
• Numerical Dynamics and Discretization: Structure-preserving methods, convergence of numerical attractors and saddle points, and stability analysis in dissipative systems.
• Interdisciplinary Applications: Novel applications in fluid mechanics, quantum physics, mathematical biology, and beyond.

We look forward to receiving your original contributions and insightful reviews that will push the boundaries of this vibrant and interconnected area of research.

Dr. Shuang Yang
Guest Editor

Dr. Rodiak Figueroa-López
Guest Editor Assistant

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