Displaying items by tag: patch solution

Changhui Tan, Liutang Xue, and Zhilong Xue

Abstract

In this paper, we revisit the patch solutions for a class of inviscid whole-space active scalar equations that interpolate between the 2D Euler equation and the \(\alpha\)-SQG equation. Compared with the 2D Euler equation in vorticity form, there is an additional Fourier multiplier \(m(\Lambda)\) (\(\Lambda = (-\Delta)^{1/2}\)) in the Biot–Savart law. If the symbol \(m\) satisfies the Osgood-type condition \[\int_2^{+\infty} \frac{1}{r (\log r) m(r)} = +\infty\] and certain mild assumptions, the system is referred to as the 2D Loglog-Euler type equation.

First, we prove a Yudovich-type theorem establishing the existence and uniqueness of a global weak solution for the Loglog-Euler type equation associated with bounded and integrable initial data. This result directly applies to patch solutions, which are weak solutions corresponding to patch initial data given by characteristic functions of disjoint, regular, bounded domains.

Next, we revisit the seminal result by Elgindi [ARMA 211 (2014) 965-990] and provide a different proof under explicit assumptions on \(m\), showing that for the 2D Loglog-Euler type equation with \(C^{1,\mu}\) (\(0<\mu<1\)) single-patch initial data, the evolved patch boundary globally preserves the \(C^{1,\mu-\varepsilon}\) regularity for any \(\varepsilon \in (0,\mu)\). In contrast to the frequency-space argument in [ARMA 211 (2014) 965-990], we develop an entirely physical-space-based approach that avoids the Littlewood–Paley theory and offers advantages for potential extensions to the half-plane or bounded smooth domains.

Furthermore, we investigate the global propagation of higher-order \(C^{n,\mu}\) boundary regularity for patch solutions with any \(n \in \mathbb{N}^\star\), and analyze the evolution of multiple patches.

This work is supported by NSF grants DMS #2238219

Qianyun Miao, Changhui Tan, Liutang Xue, and Zhilong Xue

Abstract

In this paper, we investigate a class of inviscid generalized surface quasi-geostrophic (SQG) equations on the half-plane with a rigid boundary. Compared to the Biot-Savart law in the vorticity form of the 2D Euler equation, the velocity formula here includes an additional Fourier multiplier operator \(m(\Lambda)\). When \(m(\Lambda) = \Lambda^\alpha\), where \(\Lambda = (-\Delta)^{1/2}\) and \(\alpha\in (0,2)\), the equation reduces to the well-known \(\alpha\)-SQG equation. Finite-time singularity formation for patch solutions to the \(\alpha\)-SQG equation was famously discovered by Kiselev, Ryzhik, Yao, and Zlato\v{s} [Ann. Math., 184 (2016), pp. 909-948].

We establish finite-time singularity formation for patch solutions to the generalized SQG equations under the Osgood condition \[ \int_2^\infty \frac{1}{r (\log r) m(r)} dr < \infty \] along with some additional mild conditions. Under these assumptions, we demonstrate that there exist patch-like initial data for which the associated patch solutions on the half-plane are locally well-posed and develop a finite-time singularity. Our result goes beyond the previously known cases. Notably, our result fills the gap between the globally well-posed 2D Euler equation (\(\alpha = 0\)) and the \(\alpha\)-SQG equation (\(\alpha > 0\)). Furthermore, in line with Elgindi's global regularity results for 2D Loglog-Euler type equations [Arch. Rat. Mech. Anal., 211 (2014), pp. 965-990], our findings suggest that the Osgood condition serves as a sharp threshold that distinguishes global regularity and finite-time singularity in these models.

In addition, we generalize the local regularity and finite-time singularity results for patch solutions to the \(\alpha\)-SQG equation, as established by Gancedo and Patel [Ann. PDE, 7 (2021), no. 1, Art. no. 4], extending them to cases where \(m(r)\) behaves like \(r^\alpha\) near infinity but does not have an explicit formulation.

This work is supported by NSF grants DMS #2108264 and DMS #2238219