2021/11/26 by Wenke Tan, Fan Wu, Tan, Wenke +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2111.13547
openalex publication_date 2021/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the problem of energy conservation for the solutions to the incompressible viscoelastic flows. First, we consider Leray-Hopf weak solutions in the bounded Lipschitz domain Ω in ℝd (d≥ 2). We prove that under the Shinbrot type conditions u ∈ Lqloc(0, T ; Lp(Ω)) \text for any (1)/(q)+(1)/(p) ≤ (1)/(2), \text with p ≥ 4, and \bf F ∈ Lrloc(0, T ; Ls(Ω)) \text for any (1)/(r)+(1)/(s) ≤ (1)/(2), \text with s ≥ 4 , the boundary conditions u|∂Ω=0, \bf F⋅ n|∂Ω=0 can inhibit the boundary effect and guarantee the validity of energy equality. Next, we apply this idea to deal with the case Ω= ℝd (d=2, 3, 4), and showed that the energy is conserved for u∈ Llocq(0,T;Llocp(ℝd)) with (2)/(q)+(2)/(p)≤1, p≥ 4 and \bf F∈ Llocr(0,T;Llocs(ℝd))∩ L(4d+8)/(d+4)(0,T;L(4d+8)/(d+4)(ℝd)) with (2)/(r)+(2)/(s)≤1, s≥ 4 . This result shows that the behavior of solutions in the finite regions and the behavior at infinite play different roles in the energy conservation. Finally, we consider the problem of energy conservation for distributional solutions and show energy equality for the distributional solutions belonging to the so-called Lions class L4L4.