Long time behavior for one-dimensional motion of a general barotropic viscous fluid
We study one-dimensional motion of a barotropic compressible viscous fluid in a bounded region with impermeable boundary, for a general pressure term. Without loss of generality we assume that the bounded region is the interval (0,1). In material Lagrangian coordinates, and after a normalization, the equations of motion become \[ (1)\quad v_ t- u_ q=0,\quad u_ t=\mu (v^{-1}u_ q)_ q-p(v)_ q+f(t,\int^{q}_{0}v(t,\xi)d\xi), \] where \(q\in (0,1)\), and \(t\geq 0\). Here the material Lagrangian coordinate q of a particle of fluid is the mass of the portion of fluid that occupies the region between the origin and the given particle. After the normalization, the total mass of the fluid is equal to 1. Hence the coordinate q runs over the interval (0,1). The external force f(t,x) is given as a known function of the Eulerian coordinates (t,x), \(x\in (0,1)\), where \(x=\int^{q}_{0}v(t,\xi)d\xi.\) In the above problem, the boundary conditions are \[ (2)\quad u(t,0)=u(t,1)=0\quad \forall t\geq 0, \] and the initial conditions are \[ (3)\quad u(0,q)=u_ 0(q),\quad v(0,q)=v_ 0(q),\quad q\in (0,1). \] Without loss of generality, we suppose that \[ (4)\quad \int^{1}_{0}v_ 0(q)dq=1,\quad m^{-1}\leq v_ 0(q)\leq m\quad \forall q\in (0,1). \] We assume that a real function \(p\in C^ 1(0,\infty)\) is given, such that \(p(1)=1\) and that \[ (5)\quad p'(s)<0\quad \forall s\in (0,+\infty). \] We investigate, in particular, the existence of global solutions satisfying the estimate (6) \(N_ 0^{-1}\leq v(t,q)\leq N_ 0\forall (t,q)\in Q_{\infty}\) \((N_ 0\geq 1)\) in the presence of external forces which do not become small for large values of t. I proved that there is a positive threshold \(r_ 0\), and real positive decreasing functions \(\rho\) (r), R(r), defined on \((0,r_ 0)\), satisfying \(\lim_{r\to 0}\rho (r)=\lim_{r\to 0}R(r)=\infty\), and such that the following condition holds: If \(| f(t,x)| \leq r\leq r_ 0\), and if \(\| u_ 0\| <\rho (r)\), \(\| (\log v_ 0)_ q\| <\rho (r)\), then there is a global solution (u(t),v(t)) of the problem (1)-(3) which satisfies (6). Moreover \(\| u(t)\|\) and \(\| (\log v(t))_ q\|\) are uniformly bounded by R(r). Furthermore, there is a positive function \(R_ 1(r)\) \((\lim_{r\to 0}R_ 1(r)=0)\) such that after a finite time \(T^*=T^*(\| u_ 0\|,\| (\log v_ 0)_ q\|,r)\) the solution (u(t),log v(t)\({}_ q)\) lies in the sphere of radius \(R_ 1(r)\) in the space \(L^ 2\times L^ 2(*)\). Hence, the ``small'' sphere of radius \(R_ 1(r)\) is an absorbing set with respect to solutions which initial data that lie inside the ``large'' sphere of radius \(\rho\) (r). Specifically, stationary and periodic orbits that intersect the large sphere must lie entirely in the small one. It is worth noting that sets consisting of positive functions subjected to the condition \(\int^{1}_{0}v(q)dq=1\) are bounded with respect to the norm \(\| (\log v)_ q\|\) if and only if they are bounded with respect to the norms \(\| v\|_ 1\), \(| v|_{\infty}\), and \(| v^{-1}|_{\infty}.\) We use the quantity \(\psi^ 2[u,v]\equiv (4/\mu)\| u\|^ 2- 2(u,(\log v)_ q)+\mu \| (\log v)_ q\|^ 2\) instead of \(\| u\|^ 2+\| (\log v)_ q\|^ 2\). These two quantities are equivalent. We emphasize that the conclusions above are not results for small data. In fact, to any arbitrarily large real number \(\rho\) corresponds a positive number \(r=r(\rho)\) such that any solution (u,v) of problem (1)-(3) is uniformly bounded and satisfies (6) provided its initial data \((u_ 0,(\log v_ 0)_ q)\) belongs to the sphere of radius \(\rho\) (in \(L^ 2\times L^ 2)\) and f belongs to the sphere of radius \(r=r(\rho)\) (in \(L^{\infty}(Q_{\infty}))\). In particular, if \(r=0\), then \(\rho =\infty.\) I prove the above results under weak regularity assumptions on the initial data, namely \((u_ 0,v_ 0)\in L^ 2\times H^ 1\). In this case, in fact, the solutions describe continuous trajectories in the phase space \(L^ 2\times H^ 1\). I further prove strong continuous dependence of solutions on the initial data. Note that the results proved in this paper can be extended to stronger norms.
scientific article; zbMATH DE number 4156085