In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is CΔt, where Δt is the time-step size. Secondly, we prove the convergence rate CΔt2 for the Strang's splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate CΔt.

In this paper we study the existence assertion of the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = \Delta e^{-\Delta u}$. This problem arises in the mathematical description of the evolution of crystal surfaces. Our investigations reveal that the exponent in the equation can have a singular part in the sense of the Lebesgue decomposition theorem, and the exponential nonlinearity somehow “cancels” it out. The net result is that we obtain a solution $u$ that satisfies the equation and the initial boundary conditions in the almost everywhere (a.e.) sense.

Abstract In this paper, we provide an alternative proof for the classical Sz. Nagy inequality in one dimension by a variational method and generalize it to higher dimensions J(h):=(∫Rd|h| dx)a611∫Rd|63h|2 dx(∫Rd|h|m+1 dx)a+1m+1≥β0, where m > 0 for d = 1, 2, for , and . The Euler–Lagrange equation for critical points of in the non-negative radial decreasing function space is given by a free boundary problem for a generalized Lane–Emden equation, which has a unique solution (denoted by h c ) and the solution determines the best constant for the above generalized Sz. Nagy inequality. The connection between the critical mass for the thin-film equation and the best constant of the Sz. Nagy inequality in one dimension was first noted by Witelski et al (2004 Eur. J. Appl. Math. 15 223–56). For the following critical thin film equation in multi-dimension ht+6366(h 63Δh)+6366(h 63hm)=0,x∈Rd, where m = 1 + 2/d, the critical mass is also given by . A finite time blow-up occurs for solutions with the initial mass larger than M c . On the other hand, if the initial mass is less than M c and a global non-negative entropy weak solution exists, then the second moment goes to infinity as or in for some subsequence . This shows that a part of the mass spreads to infinity.

This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0 < m < 2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m > 1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.

In this paper, we introduce a random particle blob method for the Keller-Segel equation (with dimension ) and establish a rigorous convergence analysis.