arXiv subjects

Hedong Hou

Publications and source records attributed to Hedong Hou.

10 recordsLinked to original sources

arXiv2026

On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations

We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We also establish uniqueness of stationary weak solutions in an endpoint critical space. Similar results are proved for the fractional Navier-Stokes equations with arbitrarily large power of the Laplacian in both Lebesgue and Besov spaces.

math.AP↗

arXiv2025

On well-posedness for non-autonomous parabolic Cauchy problems with rough initial data

We establish a complete picture for existence, uniqueness, and representation of weak solutions to non-autonomous parabolic Cauchy problems of divergence type. The coefficients are only assumed to be uniformly elliptic, bounded, measurable, and complex-valued, without any additional regularity or symmetry conditions. The initial data are tempered distributions taken in homogeneous Hardy--Sobolev spaces $\dot{H}^{s,p}$, and source terms belong to certain scales of weighted tent spaces. Weak solutions are constructed with their gradients in weighted tent spaces $T^{p}_{s/2}$. Analogous results are also exhibited for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$.

math.AP↗

arXiv2024

On regularity of solutions to the Navier--Stokes equation with initial data in $\mathrm{BMO}^{-1}$

We prove that any mild solution in the Koch--Tataru space to the incompressible Navier--Stokes equation with initial data in $\mathrm{BMO}^{-1}$ is weak*-continuous in time, valued in $\mathrm{BMO}^{-1}$. We also show that the global mild solution vanishes in $\mathrm{BMO}^{-1}$ at infinity in time.

math.AP↗

arXiv2024

On well-posedness for parabolic Cauchy problems of Lions type with rough initial data

We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of $p$ for which tempered distributions in homogeneous Hardy--Sobolev spaces $\dot{H}^{s,p}$ with regularity index $s \in (-1,1)$ are initial data. Source terms of Lions' type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$.

math.AP↗

arXiv2023

On well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces

We prove well-posedness in weighted tent spaces of weak solutions to the Cauchy problem $\partial_t u - \mathrm{div} A \nabla u = f, u(0)=0$, where the source $f$ also lies in (different) weighted tent spaces, provided the complex coefficient matrix $A$ is bounded, measurable, time-independent, and uniformly elliptic. To achieve this, we extend the theory of singular integral operators on tent spaces via off-diagonal estimates introduced by [arXiv:1112.4292] to obtain estimates on solutions $u$, and also $\nabla u$, $\partial_t u$, and $\mathrm{div} A \nabla u$ in weighted tent spaces, showing at the same time maximal regularity. Uniqueness follows from a different strategy using interior representation for weak solutions and boundary behavior.

math.AP↗

arXiv2023

On representation of solutions to the heat equation

We propose a simple method to obtain semigroup representation of solutions to the heat equation using a local $L^2$ condition with prescribed growth and a boundedness condition within tempered distributions. This applies to many functional settings and, as an example, we consider the Koch and Tataru space related to $BMO^{-1}$ initial data.

math.AP↗

arXiv2023

Network comparison via encoding, decoding, and causality

Quantifying the relations (e.g., similarity) between complex networks paves the way for studying the latent information shared across networks. However, fundamental relation metrics are not well-defined between networks. As a compromise, prevalent techniques measure network relations in data-driven manners, which are inapplicable to analytic derivations in physics. To resolve this issue, we present a theory for obtaining an optimal characterization of network topological properties. We show that a network can be fully represented by a Gaussian variable defined by a function of the Laplacian, which simultaneously satisfies network-topology-dependent smoothness and maximum entropy properties. Based on it, we can analytically measure diverse relations between complex networks. As illustrations, we define encoding (e.g., information divergence and mutual information), decoding (e.g., Fisher information), and causality (e.g., Granger causality and conditional mutual information) between networks. We validate our framework on representative networks (e.g., random networks, protein structures, and chemical compounds) to demonstrate that a series of science and engineering challenges (e.g., network evolution, embedding, and query) can be tackled from a new perspective. An implementation of our theory is released as a multi-platform toolbox.

cs.SI↗

arXiv2023

Theoretical foundations of studying criticality in the brain

Criticality is hypothesized as a physical mechanism underlying efficient transitions between cortical states and remarkable information processing capacities in the brain. While considerable evidence generally supports this hypothesis, non-negligible controversies persist regarding the ubiquity of criticality in neural dynamics and its role in information processing. Validity issues frequently arise during identifying potential brain criticality from empirical data. Moreover, the functional benefits implied by brain criticality are frequently misconceived or unduly generalized. These problems stem from the non-triviality and immaturity of the physical theories that analytically derive brain criticality and the statistic techniques that estimate brain criticality from empirical data. To help solve these problems, we present a systematic review and reformulate the foundations of studying brain criticality, i.e., ordinary criticality (OC), quasi-criticality (qC), self-organized criticality (SOC), and self-organized quasi-criticality (SOqC), using the terminology of neuroscience. We offer accessible explanations of the physical theories and statistic techniques of brain criticality, providing step-by-step derivations to characterize neural dynamics as a physical system with avalanches. We summarize error-prone details and existing limitations in brain criticality analysis and suggest possible solutions. Moreover, we present a forward-looking perspective on how optimizing the foundations of studying brain criticality can deepen our understanding of various neuroscience questions.

q-bio.NC↗

arXiv2022

A unified theory of information transfer and causal relation

Information transfer between coupled stochastic dynamics, measured by transfer entropy and information flow, is suggested as a physical process underlying the causal relation of systems. While information transfer analysis has booming applications in both science and engineering fields, critical mysteries about its foundations remain unsolved. Fundamental yet difficult questions concern how information transfer and causal relation originate, what they depend on, how they differ from each other, and if they are created by a unified and general quantity. These questions essentially determine the validity of causal relation measurement via information transfer. Here we pursue to lay a complete theoretical basis of information transfer and causal relation. Beyond the well-known relations between these concepts that conditionally hold, we demonstrate that information transfer and causal relation universally originate from specific information synergy and redundancy phenomena characterized by high-order mutual information. More importantly, our theory analytically explains the mechanisms for information transfer and causal relation to originate, vanish, and differ from each other. Moreover, our theory naturally defines the effect sizes of information transfer and causal relation based on high-dimensional coupling events. These results may provide a unified view of information, synergy, and causal relation to bridge Pearl's causal inference theory in computer science and information transfer analysis in physics.

cond-mat.stat-mech↗

arXiv2022

Self-organized critical dynamics of RNA virus evolution

RNA virus (e.g., SARS-CoV-2) evolves in a complex manner. Studying RNA virus evolution is vital for understanding molecular evolution and medicine development. Scientists lack, however, general frameworks to characterize the dynamics of RNA virus evolution directly from empirical data and identify potential physical laws. To fill this gap, we present a theory to characterize the RNA virus evolution as a physical system with absorbing states and avalanche behaviors. This approach maps accessible biological data (e.g., phylogenetic tree and infection) to a general stochastic process of RNA virus infection and evolution, enabling researchers to verify potential self-organized criticality underlying RNA virus evolution. We apply our framework to SARS-CoV-2, the virus accounting for the global epidemic of COVID-19. We find that SARS-CoV-2 exhibits scale-invariant avalanches as mean-field theory predictions. The observed scaling relation, universal collapse, and slowly decaying auto-correlation suggest a self-organized critical dynamics of SARS-CoV-2 evolution. Interestingly, the lineages that emerge from critical evolution processes coincidentally match with threatening lineages of SARS-CoV-2 (e.g., the Delta virus). We anticipate our approach to be a general formalism to portray RNA virus evolution and help identify potential virus lineages to be concerned.

q-bio.PE↗