2025/10/24 by Leander Claes, Michael Winkler, Claes, Leander +1
Engineering · Mathematics · #35B40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2510.21660
openalex publication_date 2025/10/24 · openalex created_date 2025/10/28 · openalex updated_date 2026/07/28
In bounded n-dimensonal domains with n≥ 1, this manuscript examines an initial-boundary value problem for the system \ utt = ∇ ⋅ (γ(Θ) ∇ ut) + a ∇ ⋅ (γ(Θ) ∇ u) + ∇⋅ f(Θ), Θt = DΔΘ+ Γ(Θ) |∇ ut|2 + F(Θ)⋅ ∇ ut, . which in the case n=1 and with γ≡ Γ as well as f≡ F reduces to the classical model for the evolution of strains and temperatures in thermoviscoelasticity. Unlike in previous related studies, the focus here is on situations in which besides f and F, also the core ingredients γ and Γ may depend on the temperature variable Θ. Firstly, a statement on local existence of classical solutions is derived for arbitrary a>0, D>0 as well as 0<γ∈ C2([0,∞)) and 0≤Γ∈ C1([0,∞)), for functions f∈ C2([0,∞);ℝn) and F∈ C1([0,∞);ℝn) with F(0)=0, and for suitably regular initial data of arbitrary size. Secondly, it is seen that for each p≥ 2 such that p>n there exists δ(p)>0 with the property that whenever in addition to the above we have (a)/(γ(0)) ≤ δ(p) and (|f'(Θ_⋆)| ⋅ |F(Θ_⋆)|)/(D ⋅ γ(Θ_⋆)) ≤ δ(p), for initial data suitably close to the constant level given by u=0 and Θ=Θ_⋆, with any fixed Θ_⋆≥ 0, these solutions are actually global in time and have the property that ∇ ut, ∇ u and ∇Θ decay exponentially fast in Lp. This is achieved by detecting suitable dissipative properties of functionals involving norms of these gradients in Lp spaces.