The tight transverse emittance budget for the LHC beam imposes severe constraints on the blow-up produced by injection errors. The modifications to the beam distributions and their possible parameterisation in terms of blow-up factors are discussed for betatron and dispersion mismatch as well as for injection offsets. The cases of initial uniform or gaussian distributions are considered in deta...

A degenerate Keller-Segel system with diffusion exponent m with 2n n+2 < m < 2 − 2 n in multi dimension is studied. An exact criterion for global existence and blow up of solution is obtained. The estimates on L 2n n+2 norm of the solution play important roles in our analysis. These estimates are closely related to the optimal constant in the HardyLittlewoodSobolev inequality. In the case of in...

* Correspondence: zhgs917@163. com Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China Full list of author information is available at the end of the article Abstract This article deals with the blow-up problems of the positive solutions to a nonlinear parabolic equation with nonlocal source and nonlocal boundary condition. The blow-up and globa...

We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for ut = u∆u + u ∫ Ω |∇u| in bounded domains Ω ⊂ R which arises in game theory. We prove that solutions converge to 0 if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with ...

We present results for finite time blow-up for filtration problems with nonlinear reaction under appropriate assumptions on the nonlinearities and the initial data. In particular, we prove first finite time blow up of solutions subject to sufficiently large initial data provided that the reaction term ”overpowers” the nonlinear diffusion in a certain sense. Secondly, under related assumptions o...

We consider the dyadic model, which is a toy model to test issues of wellposedness and blow-up for the Navier–Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier–Stokes. Likewise we prove wellposedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to ...

This paper is devoted to general balance laws (with a possibly non local source term) with a non-characteristic boundary. Basic well posedness results are obtained, trying to provide sharp estimates. In particular, bounds tend to blow up as the boundary tends to be characteristic. New uniqueness results for the solutions to conservation and/or balance laws with boundary are also provided. 2000 ...

We present evidence on the global existence of solutions of De Gregorio’s equation, based on numerical computation and a mathematical criterion analogous to the Beale–Kato–Majda theorem. Its meaning in the context of a generalized Constantin–Lax–Majda equation will be discussed. We then argue that a convection term, if set in a proper form and in a proper magnitude, can deplete solutions of blo...

We study in detail the blow-up procedure described in [BTW01]. We obtain a structure theorem for coreless polygroups as a double quotient space G/H, and a polygroup chunk theorem. Seeking to remove the arbitrary parameter needed for the blow-up, we find canonical ∅-invariant groupoids G > H analogous to G and H above, and show that H contains precisely all the arbitrary choices related to the b...

Stability of blow-up proole for equation of the type u t = u + juj p?1 u Abstract In this paper, we consider the following nonlinear equation u t = u + juj p?1 u u(:; 0) = u 0 ; (and various extensions of this equation, where the maximum principle do not apply). We rst describe precisely the behavior of a blow-up solution near blow-up time and point. We then show a stability result on this beha...