Well-posedness of the Cauchy problem for the fractional power dissipative equations
By Changxing Miao, Baoquan Yuan, Bo Zhang. 2006
254 citations recorded by the source corpus.
Abstract This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation u t + ( − △ ) α u = F ( u ) for initial data in the Lebesgue space L r ( R n ) with either r ≥ r d ≜ n b / ( 2 α − d ) or the homogeneous Besov space B p , ∞ − σ ( R n ) with σ = ( 2 α − d ) / b − n / p and 1 ≤ p ≤ ∞ , where α > 0 , F ( u ) = f ( u ) or Q ( D ) f ( u ) with Q ( D ) being a homogeneous pseudo-differential operator of order d ∈ [ 0 , 2 α ) and f ( u ) is a function of u which behaves like | u | b u with b > 0 .
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9f0065dba4622f2f4088f5dd0206605e0fad7004
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Citations in the corpus
- Well‐posedness for fractional Navier–Stokes equations in the largest critical spaces Ḃ∞,∞−(2β−1)(Rn) (2012, background): “[42], Runst and Sickel [39], and Zhai [44] for the proof of the following lemma.”
- On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces (2006, unknown): “.43) it follows that (u,b) ∈ Le∞(I;B˙n/p−1 p,r (R n)) → L∞(I;B˙n/p−1 p,r (R n)). (3.50) (2) If p > n and 1 ≤ r ≤ ∞, using the equivalent characterization of negative index homogeneous Besov space [11], we have sup t>0 t 1 2 − n 2p +α 2 k∇αS(t)(u0,b0)kLp ≤ Ck∇α(u0,b0)k B˙n/p−1−α p,∞ ≤ Ck(u0,b0)”
- Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms (2012, background): “They play a crucial role in the study of many important PDEs [1–11], among them the quasi-geostrophic equation in geophysical fluid dynamics [12], which is crucial to the understanding of atmosphere and ocean, as well as being an enlightening model of 3D Euler equations and Navier–Stokes equations.”