Self-similar blow-up profile for the one-dimensional reduction of generalized SQG with infinite energy

Abstract

We study the singularity formation mechanisms of the inviscid generalized Surface Quasi-Geostrophic (gSQG) equation on the whole space $\mathbb{R}^2$ and on the upper half-plane $\mathbb{R}^2_+$, allowing infinite energy. In each case, we derive a one-dimensional reduction that captures the leading-order singular behavior of the original 2D system, and use a fixed-point argument to show the existence of finite-time self-similar blow-up solutions for the 1D systems. We also perform numerical simulations for verification and visualization.

Figures (9)

• Figure 1: Visualization for a self-similar profile for \ref{['Equation: fixedPointEqn']}, the 1D reduction of gSQG on $\mathbb{R}^2$. Configuration: $\alpha=0.3, N=5\times10^4$, domain truncation $[0,5]$, tolerance $\varepsilon=10^{-7}$, and the non-uniform mesh is $x_i=5(i/N)^2$. In the left subplot, we plot the iterations and the approximate profile for $f_*$, together with the lowerbound $\max(0,1-x^2)$. In the right subplot, we plot the iterations and the approximate profile for $\Omega_*=-xf_*$.
• Figure 2: Visualization for self-similar profiles for \ref{['Equation: fixedPointEqn']}, the 1D reduction of gSQG on $\mathbb{R}^2$, under different values of $\alpha\in(0,1)$. Configuration: $\alpha=0.1,0.3,0.5,0.7,0.9, N=1\times10^4$, domain truncation $[0,5]$, tolerance $\varepsilon=10^{-7}$, and the non-uniform mesh is $x_i=5(i/N)^2$. In the left subplot, we plot the approximate profiles $f_*$, together with the lowerbound $\max(0,1-x^2)$, for different $\alpha$. In the right subplot, we plot the iterations and the approximate profile $\Omega_*=-xf_*$ for different $\alpha$.
• Figure 3: Visualization for the limiting behavior of the self-similar profile of \ref{['Equation: fixedPointEqn']} as $\alpha\to0$. Configuration: $\alpha=0.01$, $N=1\times10^4$, domain truncation $[0,5]$, tolerance $\varepsilon=10^{-7}$, and the non-uniform mesh is $x_i=5(i/N)^2$. In the left subplot, we plot the computed profile $f_*$ together with the predicted limiting profile $\sin(\sqrt{6}\,x)/(\sqrt{6}\,x)$. In the right subplot, we plot the difference $f_*(x)-\sin(\sqrt{6}\,x)/(\sqrt{6}\,x)$, showing that the numerical profile is already very close to the predicted limit when $\alpha$ is small.
• Figure 4: Visualization for a self-similar profile for \ref{['Equation: fixedPointEqnHP']}, the 1D reduction of gSQG on $\mathbb{R}^2_+$. Configuration: $\alpha=0.1, N=5\times10^4$, domain truncation $[0,2.43\times 10^{8}]$, tolerance $\varepsilon=10^{-7}$. We have $\Delta x\approx 4\times10^{-4}$ near origin and this $\Delta x$ increases as $x$ increases. Then we compute and plot the profiles after scaling by the factor $\lambda$. In the left subplot, we plot the iterations and the approximate profile for $f_*$. In the right subplot, we plot the iterations and the approximate profile for $\Theta_*=xf_*$.
• Figure 5: Visualization for self-similar profiles for \ref{['Equation: fixedPointEqnHP']}, the 1D reduction of gSQG on $\mathbb{R}^2_+$, under different values of $\alpha\in(0,1/2)$. Configuration: $\alpha=0.05,0.15,0.25,0.35,0.45$, $N=1\times10^4$, domain truncation $[0,1.63\times 10^{6}]$, tolerance $\varepsilon=10^{-7}$. We have $\Delta x\approx 1.5\times10^{-3}$ near the origin and this $\Delta x$ increases as $x$ increases. In the left subplot, we plot the rescaled approximate profiles $f_*(\sqrt{2}x/f_*"(0))$, together with the lower bound $\max(0,1-x^2)$ and the Burgers limiting profile determined by $f_{\textbf{Burgers}}+x^2f_{\textbf{Burgers}}^3-1=0$, for different $\alpha$. In the right subplot, we plot the approximate profile $\Theta_*(\sqrt{2}x/f_*"(0))=\sqrt{2}x/f_*"(0)\cdot f_*(\sqrt{2}x/f_*"(0))$ .

Theorems & Definitions (58)

• Theorem 1.1: gSQG on $\mathbb{R}^2$
• Theorem 1.2: 1D reduction of gSQG on $\mathbb{R}^2_+$
• Remark 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5