arXiv subjects

Yong Zhen Yang

Publications and source records attributed to Yong Zhen Yang.

5 recordsLinked to original sources

arXiv2026

Local/global well-posedness analysis of time-space fractional Schrödinger equation on $\mathbb{R}^{d}$

We investigate a class of nonlinear time-space fractional Schrödinger equations with nonlocal effects in both time and space. The time derivative is of Achar type, and the space operator is a $ϕ(-Δ)$-type operator defined via a Bernstein function $ϕ$. This nonlocality invalidates classical Strichartz estimates. By combining asymptotic analysis of Mittag-Leffler functions, the Hörmander multiplier theorem, and harmonic analysis techniques, we establish a Gagliardo-Nirenberg inequality in $ϕ$-Triebel-Lizorkin spaces and derive key Sobolev estimates for the solution operator. These analyses yield the local and global well-posedness of the equations in appropriate Banach spaces. Our work demonstrates the effectiveness of the $ϕ(-Δ)$-framework for handling fractional dispersive equations with nonlocality.

math.AP↗

arXiv2026

The time fractional stochastic partial differential equations with non-local operator on $\mathbb{R}^{d}$

This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on $\mathbb{R}^d$ driven by mixed Wiener--Lévy noises. The equations feature a Caputo time derivative $\partial_t^α$ ($0<α<1$) and a spatial nonlocal operator $ϕ(Δ)$ generated by a subordinate Brownian motion, leading to a doubly nonlocal structure. For the case $p \ge 2$, we prove the existence, uniqueness, and sharp Sobolev regularity of weak solutions in the scale of $ϕ$-Sobolev spaces $\mathcal{H}_p^{ϕ,γ+2}(T)$. Our approach combines harmonic analysis techniques (Fefferman--Stein theorem, Littlewood--Paley theory) with stochastic analysis to handle the combined Wiener and Lévy noise terms. In the special case of cylindrical Wiener noise, a dimensional constraint $d < 2κ_0\bigl(2 - (2σ_2 - 2/p)_+/α\bigr)$ is obtained.~For the low-regularity case $1 \le p \le 2$, where maximal function estimates fail, we construct unique local mild solutions in $L_p(\mathbb{R}^d)$ for equations driven by pure-jump Lévy space-time white noise, using stochastic truncation and fixed-point arguments. The results unify and extend previous theories by simultaneously incorporating time-space nonlocality and jump-type randomness.

math.AP↗

arXiv2025

Muckenhoupt-weighted $L_q(L_p)$ boundedness for time-space fractional nonlocal operators

We develop a weighted mixed-norm $L_q(L_p)$-estimates for solutions to fractional evolution equations of the form \[ \partial_t^αw(t,x) = ϕ(Δ) w(t,x) + h(t,x), \quad w(0,\cdot) = w_0, \quad t > 0, \; x \in \mathbb{R}^d, \] where $\partial_t^α$ denotes the Caputo derivative of $α\in (0,1)$ and $ϕ(Δ)$ is a nonlocal operator associated with a Bernstein function $ϕ$. For all $p, q \in (1, \infty)$ and $γ\in \mathbb{R}$, we prove the estimate \begin{align*} &\left\| \partial_t^αw \right\|_{L_q(0,T,μ_2dt; H^{ϕ,γ}_p(μ_1))} + \left\| ϕ(Δ) w \right\|_{L_q(0,T,μ_2dt; H^{ϕ,γ}_p(μ_1))} \\ &\qquad\leq C \left( \left\| h \right\|_{L_q(0,T,μ_2dt; H^{ϕ,γ}_p(μ_1))} + \left\| w_0 \right\|_{N_{α,p,ϕ}} \right), \end{align*} where $μ_1\in A_p(\mathbb{R}^d)$ and $μ_2\in A_q(\mathbb{R})$ are Muckenhoupt weights, and $N_{α,p,ϕ}$ is a Banach space characterizing admissible initial data. In particular, when $μ_2\equiv 1$ and $αq>1$, $N_{α,p,ϕ}$ coincides with the weighted Besov space $B^{ϕ,γ+2-\frac{2}{αq}}_{p,q}(μ_1)$. The analysis employs tools from harmonic analysis, including the Fefferman--Stein inequality, Hardy-Littlewood maximal estimates in weighted mixed-norm spaces, and sharp function methods for bounding solution operators. These results extend and unify previous work by K.~H.~Kim et al, providing a general analytic framework for weighted $L_q(L_p)$-theory of time-space nonlocal evolution equations.

math.AP↗

arXiv2025

Cauchy problems for time-space fractional coupled chemotaxis-fluid equations in Besov-Morrey spaces

In this paper, we consider the Cauchy problems for the time-space fractional coupled chemotaxis-fluid equations, which is a generalized form of the coupled chemotaxis-fluid equations studied in \cite{M.H. Yang}. In contrast to \cite{M.H. Yang}, the solution operator of the system does not satisfy the semigroup effect, which makes the approach of \cite{M.H. Yang} inapplicable. Based on the theory of harmonic analysis, using techniques such as real interpolation, embedding in Besov-Morrey spaces, the multiplier theorem, and the Hardy-Littelwood inequality in Morrey spaces, we establish global existence. As an application, we analysis the asymptotic behavior of the solutions.

math.AP↗

arXiv2025

On the well-posedness of time-space fractional Schrödinger equation on $\mathbb{R}^{d}$

This paper considers the well-posedness of a class of time-space fractional Schrödinger equations introduced by Naber. In contrast to the classical Schrödinger equation, the solution operator here exhibits derivative loss and lacks the structure of a semigroup, which makes the classical Strichartz estimates inapplicable. By using harmonic analysis tools -- including the smoothing effect theory of Kenig and Ponce for Korteweg-de Vries equations \cite[\emph{Commun.~Pure Appl.~Math.}]{Kenig}, real interpolation techniques, and the Van der Corput lemma -- we establish novel dispersive estimates for the solution operator. These estimates generalize Ponce's regularity results \cite[\emph{J.~Funct.~Anal.}]{Ponce} for oscillatory integrals and enable us to address the derivative loss in the Schrödinger kernel. For the cases $β<2$~(in one space dimension) and $β>2$~(in higher dimensions), we prove local and global well-posedness in Sobolev and Lorentz-type spaces, respectively. Additionally, we analyze the asymptotic behavior of solutions and demonstrate the existence of self-similar solutions under homogeneous initial data. The results highlight the interplay between fractional derivatives, dispersive properties, and nonlinear dynamics, extending the understanding of nonlocal evolution equations in quantum mechanics and related fields.

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