Forward Discretely Self-similar Solutions of the MHD Equations and the Viscoelastic Navier–Stokes Equations with Damping

Abstract

In this paper, we prove the existence of forward discretely self-similar solutions to the MHD equations and the viscoelastic Navier–Stokes equations with damping with large weak $L^3$ initial data. The same proving techniques are also applied to construct self-similar solutions to the MHD equations and the viscoelastic Navier–Stokes equations with damping with large weak $L^3$ initial data. This approach is based on Bradshaw and Tsai (Ann Henri Poincaré 18(3):1095–1119, 2017).

Appendix

In this appendix, we prove the three inclusions $L^3_w\subset M^{2,1}\subset L^2_{-3/2}\subset L^2_{loc }$. To begin with, the first inclusion can be shown by the inequality

$$\begin{aligned} r^{-1}\int _{B_r(x_0)}|f(x)|^2dx&=~r^{-1}\int _{B_r(x_0)}\int _0^{|f(x)|}2\alpha \,d\alpha dx =r^{-1}\int _{B_r(x_0)}\int _0^\infty 2\alpha 1_{|f|>\alpha }(x)\,d\alpha dx\\&=~r^{-1}\int _0^\infty 2\alpha |\{|f|>\alpha \}\cap B_r(x_0)|d\alpha \\&=~r^{-1}\int _0^{r^{-1}}2\alpha |\{|f|>\alpha \}\cap B_r(x_0)|d\alpha \\&\quad +r^{-1}\int _{r^{-1}}^\infty 2\alpha |\{|f|>\alpha \}\cap B_r(x_0)|d\alpha \\&\le ~r^{-1}\int _0^{r^{-1}}2\alpha |B_r(x_0)|d\alpha +r^{-1}\int _{r^{-1}}^\infty 2\alpha |\{|f|>\alpha \}|d\alpha \\&\le ~r^{-1}|B_r(x_0)|r^{-2}+r^{-1}\int _{r^{-1}}^\infty 2\alpha \Vert f\Vert _{L^3_w}^3\alpha ^{-3}d\alpha \\&\lesssim ~1+\Vert f\Vert _{L^3_w}^3. \end{aligned}$$

Next, the second inclusion is valid as

$$\begin{aligned} \int _{\mathbb {R}^3}\frac{|f(x)|^2}{(1+|x|)^3}\,dx= & {} \int _{|x|<1}\frac{|f(x)|^2}{(1+|x|)^3}\,dx+\sum _{k=0}^\infty \int _{2^k\le |x|<2^{k+1}}\frac{|f(x)|^2}{(1+|x|)^3}\,dx\\\le & {} \int _{B_1(0)}|f(x)|^2dx+\sum _{k=0}^\infty \frac{1}{(1+2^k)^3}\int _{2^k\le |x|<2^{k+1}}|f(x)|^2dx\\\le & {} \Vert f\Vert _{M^{2,1}}^2+\sum _{k=0}^\infty \frac{1}{(1+2^k)^3}\,2^{k+1}\Vert f\Vert _{M^{2,1}}^2\\\lesssim & {} \Vert f\Vert _{M^{2,1}}^2. \end{aligned}$$

Finally, the third inclusion holds since

$$\begin{aligned} \int _{|x|\le M}|f(x)|^2dx\le (1+M)^3\int _{\mathbb {R}^3}\frac{|f(x)|^2}{(1+|x|)^3}\,dx=(1+M)^3\Vert f\Vert _{L^2_{-3/2}}^2. \end{aligned}$$