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S.N. Alekseenko, M.V. Dontsova, and D.E. Pelinovsky

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Global solutions to the shallow water system with a method of an additional argument

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Applicable Analysis 96, 1444-1465 (2017)

**Abstract:**

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 quasi-linear hyperbolic system
for a wide class of initial data, we find a sufficient condition on the initial
data that guarantee 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 quasi-linear hyperbolic systems in
physical rather than characteristic variables.

**Keywords**:

Shallow-water system; method of an additional argument; global existence; boundary-initial value problems;