Consider the Navier-Stokes equations in IRn×(0, T), for n≥3. Let 1 < a≤min{2, n/(n-2)} and define β by (2/a)+ (n/β) = 2. Set α′= α/(α-1). It is proved that Dv belongs to C(0, T; Lα′) ∩ Lα′ (0, T;...Consider the Navier-Stokes equations in IRn×(0, T), for n≥3. Let 1 < a≤min{2, n/(n-2)} and define β by (2/a)+ (n/β) = 2. Set α′= α/(α-1). It is proved that Dv belongs to C(0, T; Lα′) ∩ Lα′ (0, T; L2β/(n-2)) whenever Dv ∈ Lα(0, T; Lβ). In pwticular, v is a regular solution. This results is the natural extensinn to α ∈ (1, 2] of the classical sufficient condition that establishes that Lα(0, T; Lγ) is a regularity class if (2/α)+(n/γ) = 1. Even the borderline case α = 2 is significat. In fact, this result states that L2(0, T; W1,n) is a regularity class if n≤ 4. Since W1,n→L∞ is false, this result does not follow from the classical one that states that L2(0, T; L∞) is a regularity class.展开更多
文摘Consider the Navier-Stokes equations in IRn×(0, T), for n≥3. Let 1 < a≤min{2, n/(n-2)} and define β by (2/a)+ (n/β) = 2. Set α′= α/(α-1). It is proved that Dv belongs to C(0, T; Lα′) ∩ Lα′ (0, T; L2β/(n-2)) whenever Dv ∈ Lα(0, T; Lβ). In pwticular, v is a regular solution. This results is the natural extensinn to α ∈ (1, 2] of the classical sufficient condition that establishes that Lα(0, T; Lγ) is a regularity class if (2/α)+(n/γ) = 1. Even the borderline case α = 2 is significat. In fact, this result states that L2(0, T; W1,n) is a regularity class if n≤ 4. Since W1,n→L∞ is false, this result does not follow from the classical one that states that L2(0, T; L∞) is a regularity class.
