On the Global Well-posedness and Stability of the Navier—Stokes and the Related Equations

We study the problem of global well-posedness and stability in the scale invariant Besov spaces for the modified 3D Navier-Stokes equations with the dissipation term, −Δu replaced by % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaGGOaGaeyOeI0cccaGae8hLdqKaaiykamaaCaaaleqabaGaeqyS % degaaOGaamyDaiaacYcacaaIWaGaeyizImQaeqySdeMaeyipaWZaaS % aaa8aabaWdbiaaiwdaa8aabaWdbiaaisdaaaaaaa!4421! $$ {( - \Delta )^\alpha }u,0 \leqslant \alpha < \frac{5}{4} $$. We prove the unique existence of a global-in-time solution in % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGcbWaa0baaSqaaiaaikdacaGGSaGaaGymaaqaamaalaaabaGa % aGynaaqaaiaaikdaaaGaeyOeI0IaaGOmaiabeg7aHbaaaaa!3E02! $$ B_{2,1}^{\frac{5}{2} - 2\alpha } $$ for initial data having small % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGcbWaa0baaSqaaiaaikdacaGGSaGaaGymaaqaamaalaaabaGa % aGynaaqaaiaaikdaaaGaeyOeI0IaaGOmaiabeg7aHbaaaaa!3E02! $$ B_{2,1}^{\frac{5}{2} - 2\alpha } $$ norm for all % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey % icI48aaKGeaeaacaaIWaGaaiilamaalaaabaGaaGynaaqaaiaaisda % aaaacaGLBbGaayzkaaaaaa!3DE1! $$ \alpha \in \left[ {0,\frac{5}{4}} \right) $$. We also obtain the global stability of the solutions % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGcbWaa0baaSqaaiaaikdacaGGSaGaaGymaaqaamaalaaabaGa % aGynaaqaaiaaikdaaaGaeyOeI0IaaGOmaiabeg7aHbaaaaa!3E02! $$ B_{2,1}^{\frac{5}{2} - 2\alpha } $$ for % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey % icI48aaKGeaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiaacYcadaWc % aaqaaiaaiwdaaeaacaaI0aaaaaGaay5waiaawMcaaaaa!3EAE! $$ \alpha \in \left[ {\frac{1}{2},\frac{5}{4}} \right) $$. In the case % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaaGOmaaaaiiaacqWF8aapcqaHXoqycqWF8aapdaWcaaqa % aiaaiwdaaeaacaaI0aaaaaaa!3CAB! $$ \frac{1}{2} < \alpha < \frac{5}{4} $$, we prove the unique existence of a global-in-time solution in % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaDa % aaleaacaWGWbGaaiilaiabg6HiLcqaamaalaaabaGaaG4maaqaaiaa % dchaaaGaey4kaSIaaGymaiabgkHiTiaaikdacqaHXoqyaaaaaa!40A5! $$ B_{p,\infty }^{\frac{3}{p} + 1 - 2\alpha } $$ for small initial data, extending the previous results for the case α = 1.