Radon measures as solutions of the Cauchy problem for evolution equations
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Radon measures as solutions of the Cauchy problem for evolution equations Mathilde Colombeau Abstract. We consider a standard Navier–Stokes system on the n-D torus Tn = Rn /Zn , valid when the flow is not very compressible and the temperature does not vary too much. We construct a sequence of approximate solutions that tend to satisfy the equations in a weak sense for arbitrary physical initial conditions. By weak compactness, we obtain Radon measure solutions in density and momentum when the velocity of the flow is finite, as numerically observed in all tests, and the absence of void regions in the flow. We notice that the method also applies without viscosity and complements results on other systems of evolution equations such as the systems of isothermal and isentropic gas flows considered in Colombeau (Z Angew Math Phys 66(5):2575–2599, 2015). Mathematics Subject Classification. 35D30, 35Q35. Keywords. Partial differential equations, Evolution equations, Asymptotic solutions.
1. Introduction We consider a standard Navier–Stokes system [24, p. 26] with constant value of the dynamic viscosity, valid when the flow is not very compressible and the temperature does not vary too much n ∂ρ ∂(ρui ) (x, t) + (x, t) = 0, ∂t ∂xi i=1
(1)
for j = 1, . . . , n n ∂(ρuj ) ∂(ρuj ui ) ∂p (x, t) + (x, t) + (x, t) = μΔuj (x, t), μ ≥ 0, ∂t ∂x ∂x i j i=1
p(x, t) = Kρ(x, t)γ , K ≥ 0, 1 ≤ γ ≤ 2,
(2) (3)
where ρ is the density, (u1 , . . . , un ) is the velocity vector, p is the pressure, and μ is the constant value of the dynamic viscosity. In the absence of viscosity, i.e., when μ = 0, the equations reduce to the Euler equations for which approximate solutions have been constructed in [10]. When both μ = 0 and K = 0, the equations reduce to the system of pressureless fluids for which approximate solutions have been constructed in [9]. The general Navier–Stokes system [16, p. 1680] involves an energy equation with a complicated viscosity term. System (1–3) is a simplification which is commonly used in the study of fluid dynamics [24]. A survey of theoretical results on solutions of the compressible Navier–Stokes equations can be found in the introduction of [16]. For any physical initial condition, we construct a family of functions ρ(x, t, ), (ρuj )(x, t, ), uj (x, t, ) with 1 ≤ j ≤ n and Φ(x, t, ) defined for x in the n-D torus Tn = Rn /Zn , for t ∈ [0, + ∞[ and where > 0 is a small parameter whose aim is to tend to 0. We denote indifferently w (x, t) and w(x, t, ) 0123456789().: V,-vol
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for w = ρ, u, p, . . . . These functions tend to satisfy the equations when → 0 in the sense that ∀ψ ∈ Cc∞ (Rn ), ∀t ∈ [0, + ∞[ n ∂ψ ∂ρ (x, t, )ψ(x) − (ρui )(x, t, ) (x) dx → 0, (4) ∂t ∂xi i=1 Rn
for j = 1, . . . , n
n ∂ψ ∂(ρuj ) (x, t, )ψ(x) − (ρuj ui )(x, t, ) (x) ∂t ∂x i i=1 Rn ∂Φ + ρ(x, t, ) (x, t, )ψ(x) − μuj (x, t, )Δψ(x) dx → 0, ∂xj if γ = 1: {Φ(x, t, ) − K ln[ρ(x, t, )]}ψ(x)dx → 0, Rn
if 1 < γ ≤ 2:
Kγ
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