141
In the same way, with the integration of the mass equation over depth, we obtain :
h, + div (uh) = °
Now, to simplify expressions, we are going to denote u by u.
Boundary and initial conditions must be added to those two equations. The initial
conditions are the following conditions :
u(t = O,x) = uo(x)
h(t = O,x) = ho(x)
into D
into n
We present an existence theorem for a Shallow water problem with a depth-mean
velocity formulation and non-homogeneous boundary conditions expressing water entering.
If we have to fix the velocity on the boundary, we have also to fix the water elevation on
the part of the boundary where the flow enters. In this case, we get an a priori estimates
that shows the problem is well-posed. Particularly, about the water elevation, we obtain h
and hlog h bounded into Ll (Q) and into Ll (~+). To verify the boundary condition, we have
shown a trace theorem on the space of the integrable functions whose their divergence is
integrable. The existence theorem is true at least when we consider two cases of boundary
conditions .
• u. n fixed and curl u = 0.
• u fixed on the boundary.
In the first case of boundary conditions, if we want to obtain a solution for each T in
[0,00[, we must take account of the dissipation term at the bottom Du lulR2 if the domain
is not simply connected. Sometimes the modelisators neglect the shear effect at the bottom
and then the theorem remains true if the domain is simply connected. Otherwise we always
obtain a solution for T small.
In the last case, the theorem is true for each T in [0,00[, whatever the domain D.
After giving the theorem, we explain in a second paragraph, how we obtain the a priori
estimates of the problem. And then we present in four lemma how we can pass to the limit
with the approximated solutions whose construction is developped in a fourth part.
We would like to thank the professor P.L. Lions for his kind help in our search for
proving the existence of a weak solution of the problem. His work has been very useful to
bring to this point this study.
In the same way, with the integration of the mass equation over depth, we obtain :
h, + div (uh) = °
Now, to simplify expressions, we are going to denote u by u.
Boundary and initial conditions must be added to those two equations. The initial
conditions are the following conditions :
u(t = O,x) = uo(x)
h(t = O,x) = ho(x)
into D
into n
We present an existence theorem for a Shallow water problem with a depth-mean
velocity formulation and non-homogeneous boundary conditions expressing water entering.
If we have to fix the velocity on the boundary, we have also to fix the water elevation on
the part of the boundary where the flow enters. In this case, we get an a priori estimates
that shows the problem is well-posed. Particularly, about the water elevation, we obtain h
and hlog h bounded into Ll (Q) and into Ll (~+). To verify the boundary condition, we have
shown a trace theorem on the space of the integrable functions whose their divergence is
integrable. The existence theorem is true at least when we consider two cases of boundary
conditions .
• u. n fixed and curl u = 0.
• u fixed on the boundary.
In the first case of boundary conditions, if we want to obtain a solution for each T in
[0,00[, we must take account of the dissipation term at the bottom Du lulR2 if the domain
is not simply connected. Sometimes the modelisators neglect the shear effect at the bottom
and then the theorem remains true if the domain is simply connected. Otherwise we always
obtain a solution for T small.
In the last case, the theorem is true for each T in [0,00[, whatever the domain D.
After giving the theorem, we explain in a second paragraph, how we obtain the a priori
estimates of the problem. And then we present in four lemma how we can pass to the limit
with the approximated solutions whose construction is developped in a fourth part.
We would like to thank the professor P.L. Lions for his kind help in our search for
proving the existence of a weak solution of the problem. His work has been very useful to
bring to this point this study.