The global attractor and finite determining nodes for the Navier-Stokes equations of compressible flow with singular initial data. (English) Zbl 0977.35105

The authors investigate the compressible Navier-Stokes equations in one dimension: \[ \rho_t+(\rho u)_x=0,\quad (\rho u)_t+(\rho u^2)_x+ P(\rho)_x =\varepsilon u_{xx}+pf,\quad 0<x<1,\tag{1} \] with boundary and initial conditions \[ u(0,t) = u(1,t) = 0,\quad \rho(\cdot,0) =\rho_0,\quad u(\cdot,0) =u_0. \] Here \(\rho\) and \(u\) are density and velocity, \(P(\rho) = c^2\rho\) is isothermal pressure, \(\varepsilon\) is viscosity, and \(f(x,t)\) an exterior force. Based on existence and uniqueness of weak global solutions to (1) in a suitable function space, the authors prove the existence of a global attractor and characterize its elements in terms of certain nodal properties. The paper starts with a definition of weak solutions along standard lines to (1). Next a crucial expression is introduced: \[ H(\rho,n) =\int^1_0 \left(\tfrac 16 \rho u^2 + AG(\rho)\right)dx +\tfrac 1{32} (\text{Var} (\log\rho))^2, \] “Var”= total variation, \(G(\rho)=\rho\log\rho-\rho+1\).
Theorem 1 asserts the global existence and uniquenes of weak solutions to (1) under various assumptions, notably \[ \rho_0,\rho^{-1}_0\in L^\infty(0,1), \quad \rho_0>0\text{ and }H(\rho_0,u_0)\geq c_0. \] In order to proceed further, the authors then introduce three function spaces \((X_0)^2\), \(X^1,X^2\) for pairs \((\rho,u)\), all related to the Lebesgue decomposition of \(\rho_0\) considered as a measure. Theorem 2 then asserts the existence of an attractor \(A\subseteq (X_0)^2\subseteq X^2\), compact in the topology of \(X^2\), and endoved with various properties. Theorem 3 states that if the exterior force is of a particular type (an admissible force) then the solutions to (1), provided by Theorem 1, exhibit certain nodal line properties. The major part of the paper is now devoted to the proof of Theorems 1-3, which is based on a series of rather delicate lemmas and propositions. In the final part of the paper, Theorem 2 is improved and further results on the attractor are obtained.


35Q30 Navier-Stokes equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B41 Attractors
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