N. Ben Abdallah, and J. Dolbeault, Relative entropies for kinetic equations in bounded domains (irreversibility, stationary solutions, uniqueness), in preparation.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... x; v) resembles the steady state f s 2) Can one nd a rate of convergence for some L p norm In particular, does f go to f s exponentially fast The answer to the rst question is armative, there is convergence at least weakly in of f(t; towards f s based on by now standard arguments [9, 1, 2]. However, strong convergence in L and moreover, rates of convergence in L are unknown even for such simple inhomogeneous kinetic equations. The measurement of decay rates for nonlinear di usion equations has been recently addressed using entropy dissipation methods based on the Bakry Emery ....

Ben Abdallah, N; Dolbeault, J. Relative entropies for kinetic equations in bounded domains (irreversibility, stationary solutions, uniqueness). Preprint.

....and get stability results in L with q 6= p: for instance q = 2 and p = 1 in Theorem 1.2. Similar ideas have been used previously in various contexts: for gravitational systems (without confinement) in [42, 44, 30, 31, 32] using the Casimirenergy method, and for systems in bounded domains in [6, 7], using entropy fluxes involving Darroz es Guiraud type estimates. For confinement, we shall refer to [26] and also to [11, 24, 10] in case of models with a Fokker Planck term. Entropy methods have recently been adapted to nonlinear diffusions: see for instance [2] in the linear case and [13, ....

....jxj 1 OE e (x) 1 and x 7 G oe (OE e (x) 4 R 1 s fl(s OE e (x) ds is bounded in L ) 7 The conditions on the growth of OE e and on the decay of fl will be refered as confinement conditions. We are going to adapt the proofs given in [26] for the case fl(s) e and in [6, 7] for the bounded domain case to prove the existence of a stationary solution f1;oe . The existence of ff = ff(M) will be a consequence of the proof. Let M 0 and consider on L ) ff 2 L ) f 0 a:e: kfk L 1 = Mg the functional oe(f) 1 j rOE[f ] j Lemma 2.6. Under ....

[Article contains additional citation context not shown here]

N. Ben Abdallah, and J. Dolbeault, Relative entropies for kinetic equations in bounded domains (irreversibility, stationary solutions, uniqueness), in preparation.

....the maxwellian stationary solutions, which is not included in Theorem 1. 1 (see Section 4) Similar ideas have been used previously in various contexts: for gravitational systems (without confinement) in [42, 44, 30, 31, 32] using the Casimir energy method, and for systems in bounded domains in [6, 7], using entropy fluxes involving Darroz es Guiraud type estimates. For confinement, we shall refer to [26] and also to [11, 24, 10] in case of models with a Fokker Planck term. Entropy methods have recently been adapted to nonlinear diffusions: see for instance [2] in the linear case and [13, ....

....jxj 1 OE e (x) 1 and x 7 G oe (OE e (x) 4 R 1 s fl(s OE e (x) ds belongs to L ) The conditions on the growth of OE e and on the decay of fl will be refered as confinement conditions. We are going to adapt the proofs given in [26] for the case fl(s) e and in [6, 7] for the bounded domain case to prove the existence of a stationary solution f1;oe . The existence of ff = ff(M) will be a consequence of the proof. Let M 0 and consider on L ) ff 2 L ) f 0 a:e: kfk L 1 = Mg the functional IR oe(f) 1 IR j rOE[f ] j Definition ....

[Article contains additional citation context not shown here]

N. Ben Abdallah, and J. Dolbeault, Relative entropies for kinetic equations in bounded domains (irreversibility, stationary solutions, uniqueness), Preprint Ceremade no. 0133.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC