We consider ( \(-\alpha \) )-homogeneous solutions (stationary self-similar solutions of degree \(-\alpha \) ) to the two-dimensional inviscid Boussinesq equations in a half-plane. We show their non-existence and existence with both regular and singular profile functions. More specifically, we demonstrate the following: Non-existence of rotational ( \(-\alpha \) )-homogeneous solutions with regular profiles \((u,p,\rho )\in C^{1}(\overline{\mathbb {R}^{2}_{+}}\backslash \{0\})\) for \(0\leqq \alpha \leqq 1\) and \((u,p,\rho )\in C^{2}(\overline{\mathbb {R}^{2}_{+}}\backslash \{0\})\) for \(-1/2\leqq \alpha <0\) ;
Existence of rotational ( \(-\alpha \) )-homogeneous solutions with regular profiles \((u,p,\rho )\in C^{2}(\overline{\mathbb {R}^{2}_{+}}\backslash \{0\})\) for \(\alpha >1\) and \((u,p,\rho )\in C^{1}(\overline{\mathbb {R}^{2}_{+}})\) for \(\alpha <-2\) ;
Existence of rotational ( \(-\alpha \) )-homogeneous solutions with \(x_1\) -symmetric singular profiles \((u,p,\rho ) \in C^{\infty }(\overline{\mathbb {R}^{2}_{+}}\backslash \{x_1=0\}\cup \{x_2=0\})\cap C(\overline{\mathbb {R}^{2}_{+}})\) for \(-1<\alpha < -1/2\) and \((u,p,\rho )\in C^{\infty }(\overline{\mathbb {R}^{2}_{+}}\backslash \{x_1=0\}\cup \{x_2=0\})\) for \(-1/2\leqq \alpha <1\) .
The ( \(-\alpha \) )-homogeneous solutions with continuous profiles \((u,p,\rho )\in C^{\infty }(\overline{\mathbb {R}^{2}_{+}} \backslash \{x_1=0\}\cup \{x_2=0\})\cap C(\overline{\mathbb {R}^{2}_{+}})\) for \(-1<\alpha <-1/2\) provide examples of self-similar weak solutions in \(\mathbb {R}^{2}_{+}\) for the scaling exponent \(\alpha \approx -0.657\) , at which Wang et al. (Phys Rev Lett 130(24):244002, 2023) numerically discovered the existence of backward self-similar solutions with smooth profile functions.
