Semiflow Selection to Models of General Compressible Viscous Fluids
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Journal of Mathematical Fluid Mechanics
Semiflow Selection to Models of General Compressible Viscous Fluids Danica Basari´c Communicated by F. Flandoli
Abstract. We prove the existence of a semiflow selection with range the space of c` agl` ad, i.e. left-continuous and having right-hand limits functions defined on [0, ∞) and taking values in a Hilbert space. Afterwards, we apply this abstract result to the system arising from a compressible viscous fluid with a barotropic pressure of the type aγ , γ ≥ 1, with a viscous stress tensor being a nonlinear function of the symmetric velocity gradient.
1. Introduction First developed by Krylov [21] and later adapted by Flandoli and Romito [18], Breit et al. [8] in the context of the Navier–Stokes system, the semiflow selection is an important stochastic tool when studying systems that lack uniqueness: it allows to identify a solution satisfying at least the semigroup property. Inspired by the deterministic adaptation of Cardona and Kapitanski [11], in the first part of this work we will prove the existence of a semiflow selection in an abstract setting. More precisely, denoting with H a Hilbert space and with T = D([0, ∞); H) the Skorokhod space of c` agl` ad functions defined on [0, ∞) and taking values in H, we will show the existence of a Borel measurable map u : D ⊆ H → T such that for any x ∈ D and any t1 , t2 ≥ 0 u(x)(t1 + t2 ) = u [u(x)(t1 )] (t2 ). One could think that a more natural choice for T would be the space C([0, ∞); H) of continuous functions, as in [11]. However, this option can be too strong: if we want to apply this abstract setting to the typical systems arising from fluid dynamics, where [0, ∞) is the set of times and T represents the trajectory space, then it is difficult to ensure the energy of the system to be continuous, since it is at most a non-increasing quantity with possible jumps. For the aforementioned reason, in the context of the compressible Euler system, Breit et al. [7,9] considered the energy in the L1 -space. But the choice T = L1 ([0, ∞); H) is still not optimal as it is better to work with a space whose elements are well-defined at any point. In the second part of this work, we will apply the abstract machinery previously achieved to a general model of compressible viscous fluids, described by the following pair of equations: ∂t + divx (u) = 0, ∂t (u) + divx (u ⊗ u) + ∇x p = divx S,
(1.1)
where the unknown variables are the density and the velocity u. In this context, the viscous stress tensor S is connected to the symmetric velocity gradient Du through the relation S : Du = F (Du) + F ∗ (S), where F is a proper convex l.s.c. (lower semi-continuous) function and F ∗ denotes its conjugate; moreover, the barotropic pressure will be of the type p() = aγ , γ ≥ 1. We will deal with the concept of dissipative solutions, i.e. solutions that satisfy our problem in the weak sense but with an extra defect term in the 0123456789().: V,-vol
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D. Basari´ c
JMFM
balance of momentum, arising from possible concentrations and/
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