Global solutions to the shallow-water system
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The classical system of shallow-water (Saint--Venant) equations describes
long surface waves in an inviscid incompressible fluid of a variable depth.
Although shock waves are expected in this quasilinear hyperbolic system for a
wide class of initial data, we find a sufficient condition on the initial data
that guarantees existence of a global classical solution continued from a local
solution. The sufficient conditions can be easily satisfied for the fluid flow
propagating in one direction with two characteristic velocities of the same
sign and two monotonically increasing Riemann invariants. We prove that these
properties persist in the time evolution of the classical solutions to the
shallow-water equations and provide no shock wave singularities formed in a
finite time over a half-line or an infinite line. On a technical side, we
develop a novel method of an additional argument, which allows to obtain local
and global solutions to the quasilinear hyperbolic systems in physical rather
than characteristic variables.