Evolution of a multiscale singularity of the solution of the Burgers equation in the 4-dimensional space-time

The solution of the Cauchy problem for the vector Burgers equation with a small parameter of dissipation $\varepsilon$ in the $4$-dimensional space-time is studied:
$$\mathbf{u}_t + (\mathbf{u}\nabla) \mathbf{u} = \varepsilon \triangle \mathbf{u}, \quad u_{\nu} (\mathbf{x}, -1, \varepsilon) = - x_{\nu} + 4^{-\nu}(\nu + 1) x_{\nu}^{2\nu + 1},$$
With the help of the Cole–Hopf transform $\mathbf{u} = - 2 \varepsilon \nabla \ln H$, the exact solution and its leading asymptotic approximation, depending on six space-time scales, near a singular point are found. A formula for the growth of partial derivatives of the components of the vector field $\mathbf{u}$ on the time interval from the initial moment to the singular point, called the formula of the gradient catastrophe, is established:
$$\frac{\partial u_{\nu} (0, t, \varepsilon)}{\partial x_{\nu}} = \frac{1}{t} \left[ 1 + O \left( \varepsilon |t|^{- 1 - 1/\nu} \right) \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to -\infty, \quad t \to -0.$$
The asymptotics of the solution far from the singular point, involving a multistep reconstruction of the space-time scales, is also obtained:
$$u_{\nu} (\mathbf{x}, t, \varepsilon) \approx - 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \tanh \left[ \frac{x_{\nu}}{\varepsilon} \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty.$$