1,680 research outputs found

    On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher

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    This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass McM_c such that the solutions exist globally in time if the mass is less than McM_c and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out. A number of open questions are also stated.Comment: 26 page

    Interaction of modulated pulses in the nonlinear Schroedinger equation with periodic potential

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    We consider a cubic nonlinear Schroedinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses

    Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

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    In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincar{\'e} inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work

    Global Well-Posedness Of A Non-Local Burgers Equation: The Periodic Case

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    This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads utuu+(u2)=0. u_t - u |\nabla| u + |\nabla|(u^2) = 0. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in LL^\infty. We show that any weak solution is instantaneously regularized into CC^\infty. We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.Comment: 27 pages, 11 figure

    Conditional regularity of solutions of the three dimensional Navier-Stokes equations and implications for intermittency

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    Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on L2mL^{2m}-norms of the vorticity, denoted by Ωm(t)\Omega_{m}(t), and particularly on Dm=ΩmαmD_{m} = \Omega_{m}^{\alpha_{m}}, where αm=2m/(4m3)\alpha_{m} = 2m/(4m-3) for m1m\geq 1. The first result, more appropriate for the unforced case, can be stated simply : if there exists an 1m<1\leq m < \infty for which the integral condition is satisfied (Zm=Dm+1/DmZ_{m}=D_{m+1}/D_{m}) 0tln(1+Zmc4,m)dτ0 \int_{0}^{t}\ln (\frac{1 + Z_{m}}{c_{4,m}}) d\tau \geq 0 then no singularity can occur on [0,t][0, t]. The constant c4,m2c_{4,m} \searrow 2 for large mm. Secondly, for the forced case, by imposing a critical \textit{lower} bound on 0tDmdτ\int_{0}^{t}D_{m} d\tau, no singularity can occur in Dm(t)D_{m}(t) for \textit{large} initial data. Movement across this critical lower bound shows how solutions can behave intermittently, in analogy with a relaxation oscillator. Potential singularities that drive 0tDmdτ\int_{0}^{t}D_{m} d\tau over this critical value can be ruled out whereas other types cannot.Comment: A frequency was missing in the definition of D_{m} in (I5) v3. 11 pages, 1 figur
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