*(English)*Zbl 1126.35053

The authors study the equation ${u}_{t}-{u}_{txx}+4u{u}_{x}=3{u}_{x}{u}_{xx}+u{u}_{xxx}$ on the real line: one of several shallow water wave equations. Even though the equation is integrable (e.g., in the sense of Lax pairs), its conserved quantities do not even control the energy norm.

The authors apply the operator $1-{\partial}_{x}^{2}$, which reduces the equation to a Burgers type evolution equation with a convolution term, and they use the conserved quantity $\int (1-{\partial}_{x}^{2})u\xb7{(4-{\partial}_{x}^{2})}^{-1}u\phantom{\rule{0.166667em}{0ex}}dx$, which controls the ${L}^{2}$ norm and eventually leads to an a apriori estimate for the supremum norm.

They improve estimates of the third author to the effect that, if finite time $T$ blowup occurs for initial data ${u}_{0}\in {H}^{s}(s>\frac{3}{2}$, for which local well-posedness holds), then ${inf}_{x}{u}_{x}\sim -1/(T-t)$ as $t\to T$, whereas $u$ remains uniformly bounded. If ${u}_{0}\ne 0$ is odd and $(1-{\partial}_{x}^{2}){u}_{0}\ge 0$ for $x<0$, then the finite blow-up occurs only at $x=0$. Conversely, for ${u}_{0}\in {H}^{1}$ where $(1-{\partial}_{x}^{2}){u}_{0}$ is a Radon measure with bounded variation that is $\le 0$ for $x<{x}_{0}$ and $\ge 0$ for $x>{x}_{0}$ (as defined in terms of support), they show the existence of global weak solutions.

They also show, despite the weak regularity hypothesis on the initial data, the uniqueness of weak solutions in this class.