Global Well-posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems

Global Existence and Uniqueness of Nonlinear Evolutionary Fluid Equations

“…This chapter is a continuation of Chapter 1 in Qin and Huang [102]. Here we establish the exponential stability of spherically symmetric solutions to an initialboundary value problem for nonlinear non-autonomous compressible Navier-Stokes equations with an external force and a heat source in bounded annular domains G n = {x ∈ R n : 0 < a ≤ |x| ≤ b} in R n (1 ≤ n ≤ 3), based on the uniform estimates obtained in Chapter 1 of Qin and Huang [102].…”

“…This full non-Newtonian fluid model (p > 2) is more complicated than the models considered in Chapters 3-5 in Qin and Huang [102], which are Newtonian fluid models (p = 2). Therefore, we need to more delicate estimates to deal with this non-Newtonian model.…”

“…Here we establish the exponential stability of spherically symmetric solutions to an initialboundary value problem for nonlinear non-autonomous compressible Navier-Stokes equations with an external force and a heat source in bounded annular domains G n = {x ∈ R n : 0 < a ≤ |x| ≤ b} in R n (1 ≤ n ≤ 3), based on the uniform estimates obtained in Chapter 1 of Qin and Huang [102]. In Eulerian coordinates, the equations under consideration read If we denote (ρ,ṽ,θ) still by (ρ, v, θ) and (ζ, t) by (x, t).…”

Cauchy problem for non-autonomous Navier-Stokes equations in an unbounded domain

“…We use the similar method as those in [1][2][3][4][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Given the non-autonomous terms f, g, we need more complex estimates than that in Jiang [2].…”

“…When f = 0, g = 0, Qin and Huang [7], Qin and Wen [8] proved the global existence and the exponential stability of solutions in H i (i = 1, 2, 4) in a bounded domain, but the Cauchy problem to (1.9)-(1.13) is still open. We use the similar method as those in [1][2][3][4][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”