arXiv subjects

Xieling Fan

Publications and source records attributed to Xieling Fan.

3 recordsLinked to original sources

arXiv2026

Formation of quasi-singularities of shock and implosion type for compressible Euler flows

This paper investigates a novel quasi-singularity formation phenomenon in the isentropic compressible Euler equations in three dimensions. For any prescribed finite set of points $\{z_j\}_{j=1}^N \subset \mathbb{R}^3$ and an arbitrarily large parameter $M \gg 1$, we explicitly construct real-analytic initial data generated by Herglotz-type wave fields. We prove that the corresponding classical solution $$ (p,v)\in C^1([0,T]\times\mathbb{R}^3;\mathbb{R}\times\mathbb{R}^3), $$ with $$ T = C_0 \, M^{-\frac{3γ+1}{γ-1}}, $$ where $γ> 1$ is the adiabatic index and $C_0 > 0$ is a constant independent of $M$. Throughout the time interval $[0, T]$, the velocity field, its spatial gradient, and the pressure gradient exhibit simultaneous localized amplification at each prescribed point $z_j$. More precisely, each component of the velocity field is bounded below by $M$, each component of the pressure gradient is of size at least $M^{\frac{2γ}{γ-1}}$, and the velocity gradient exhibits a prescribed anisotropic sign pattern with entries of size at least $M$ at every $z_j$. All implicit constants in these estimates are independent of $M$ and depend solely on the fixed physical parameters and the locations $\{z_j\}_{j=1}^N$. This mechanism generates highly localized, shock-like gradient concentrations and implosion-like spatial profiles within the smooth regime. The underlying mechanism relies on the construction of the Herglotz kernel via Helmholtz far-field patterns, together with delicate quantitative energy estimates for the quasilinear hyperbolic system.

math.AP↗

arXiv2026

Construction of Solutions with Extraordinary Gradient Amplification and Localization for Schrödinger Equations

This paper constructs solutions to linear and nonlinear Schrödinger-type equations in two and three spatial dimensions that exhibit prescribed, extraordinary gradient amplification and localization. For any finite time interval $[0,T]$, any prescribed collection of $n\in\mathbb{N}$ distinct points on $\partial D$, where $D$ is the compact support of the anisotropic coefficients, lower-order terms, or nonlinearities, and any amplitude threshold $\mathcal{M}>0$, we show that one can design smooth initial and/or boundary data such that the spatial gradients of the resulting solutions exceed $\mathcal{M}$ in neighborhoods of these points outside $D$ for almost every $t\in[0,T]$. Moreover, the ratio between the local $C^{1,\frac12}$-norm of the solution near each prescribed point outside $D$ and the $C^{1,\frac12}$-norm inside $D$ is bounded from below by $\mathcal{M}/2$ for almost every $t\in[0,T]$. We further prove that the spatial measure of the regions where the gradient magnitude exceeds $\mathcal{M}$ tends to zero as $\mathcal{M}\to\infty$, demonstrating that the amplification phenomenon is highly localized. This effect arises from the structure of the Schrödinger-type equation combined with carefully designed input profiles. From a physical perspective, the results provide a deterministic analogue of localization phenomena observed in quantum scattering and Anderson localization. In addition, the observed trade-off between extreme spatial localization and large gradient amplification is fully consistent with the spirit of the Heisenberg uncertainty principle: while the latter is traditionally formulated in a global $L^2$ space--frequency framework, our results offer a complementary deterministic manifestation at the level of localized spatial gradients in Schrödinger dynamics.

math.AP↗

arXiv2025

A New Quasi-Singularity Formation Mechanism for Second-order Hyperbolic Equations

This paper investigates a novel mechanism for quasi-singularity formation in both linear and nonlinear hyperbolic wave equations in two and three dimensions. We prove that over any finite time interval, there exist inputs such that the Hölder norm of the resulting wave field exceeds any prescribed bound. Conversely, the set of such almost-blowup points has vanishing measure when the aforementioned bound goes to infinity. This phenomenon thus defines a quasi-singular state, intermediate between classical singularity and regularity. Crucially, both the equation coefficients and the inputs can be arbitrarily smooth; the quasi-singularity arises intrinsically from the structure of the hyperbolic wave equation combined with specific input characteristics.

math.AP↗