Secondly there is a recent interest on fractional diffusions in biology because the molecules undergo specific interactions with the overall medium. Tus we ave 1. We describe now te matematical situation. The proof of theorem 6. Explicit in time numerical schemes for the equations of the inverse distribution function are proposed keeping the contraction of the Wasserstein distance at the discrete level. As soon as these two have been stated, and some sort of homogeneity holds for the functional, we can prove uniqueness of the functional s minimizer which is unsurprisingly the stationary state of item. However unicity is not clear.
Boundary conditions are zero-flux. Te efficiency of te mass redistribution metod depends on te position of te nodes were te mass is redistributed. To weaken the convexity assumption on the interaction kernel and to find under which conditions stationary states continue to be global attractors of the dynamics are issues of great interest for applications in mathematical biology. We combine the proof of lemma. It is worth noticing that self-organization is driven by the constrain of mass conservation. The radial case has already been treated by several authors, see for instance . The last interaction contribution is more tricky to handle with, and involves variables doubling, as it is known from granular media, see .
As soon as equi-integrability is gained, the remaining work is usually to propagate L p regularity for the cell density. Dd libres amortis des mouvements oscillatoires dont l amplitude diminue au cours du temps jusqu. Te second step consists to rewrite 2. We again pay attention to make the equation for the equilibrium state appear.
Intuitively, constant densities are stationary states when the extension is done by meanvalue.
Biographie de Andreï Kolmogorov
Te above a priori estimates allow us to infer tat tere exists w and belonging to V LLrespectively, and passing to subsequences, if necessary, we find 2. Furthermore, the exdrcice of these two ensures that the mass does not escape to infinity, and the balance is optimal again see Lemma ee.
The negative contribution of f log f is controlled by the logarithmic momentum: Ten tese oscillations propagated along te bar see Figure 4.
The last interaction contribution is more tricky to handle with, and involves variables doubling, as it is known from granular media, see . Te approximate solution obtained for te redistribution of te contact node mass on te node preceding te contact see Mod 3 on Figure 3. Appendix A Te aim of tis section is to give te proofs of Teorem 1 and Lemma 3. From the logarithmic Hardy- Littlewood-Sobolev inequality  it comes that the functional. Te main difference comes from te coice of te fourirr problem.
Evolution As mentioned earlier, this situation corresponds to Dirichlet BC on c, and is qualitatively similar to the whole space model.
The Patlak, Keller and Segel PKS model [06, 53, 54] has been introduced in order to explain chemotactic cell aggregation by means of a coupled system of two equations: Ten te solution u to Problem 1. Te first step consists to introduce an auxiliary problem. Main results are summarized in section. In this case, the formation of plateaus occurs in a fast time scale, then these plateaus merge on a slow time scale .
Ten we may conclude tat tere exists a solution v to Problem 2.
The qualitative behaviour of this model results from the balance between two opposite tendencies. Of course continuity of T [S] in spaces L p loc is needed for passing to the limit in all cases.
Since the functional G[n] is not convex, we cannot conclude the uniqueness of minimisers for the discrete scheme, and thus, the scheme 8. Therefore, the following proof, in the line of , is more complete.
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Analyse des structures hyperstatiques: Let us write the differential inequality 3. Efficiency of a weigted mass redistribution metod for an elastodynamic problem wit unilateral constraints Te discrete evolution for te contact problem 3.
The outline of the paper is as follows. The analysis of these pictures provides morphometric data on the lesions and helps to establish space laws for the pattern.
Balo s concentric sclerosis is a rare and aggressive variant of MS where demyelination regions consists of concentric bands centered by a blood vessel. The de-localization induced by v or v in these formula is fundamental both for mathematical theory and biophysical interpretation. Multiplying the equation on n in 3.
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Moreover, if equi-integrability does not bib,ath on time T, i. First we prove existence of this steady state equilibrium, see Proposition 8. We establis now some a priori estimates, wic later will enable to infer te existence of a weak solution. We are convinced that a simple mathematical principle underlies such a robust process.
In particular, te courier still remains an open problem in oter frameworks. Chemotaxis models generally couple an equation for the movement of cells nt, x denotes the cell density together with the chemical constituent which is produced by the cells themselves ct, x denotes the concentration of the chemoattractant.
Finally we find 3. Te bar starts to fall down to te rigid obstacle wit an initial velocity equal to.
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Notice tat Teorem 1 is a straigtforward application of [14, p. Te convergence curves in dimension two illustrate tat te Lagrange multiplier does not converge in L2T ; H norm wen te standard mass matrix is used see Figure 4. However it comes out that our method of equilibrium factorization can be applied to a wider range of functionals of the type 7.
The secondary rearrangement of growing colloids at the back of the front inspired our model for BCS. Terefore, by using 1. As a summary, in the linear diffusion classical PKS model one has the following result: We combine the proof of lemma. More precisely, we prove below tat te energy associated to 1.
Namely it does not require any condition on the initial data, but on the domain s geometry. Here g t is a smoot, positive function see Figure 1.