8 research outputs found
Development of singularities for the compressible Euler equations with external force in several dimensions
We consider solutions to the Euler equations in the whole space from a
certain class, which can be characterized, in particular, by finiteness of
mass, total energy and momentum. We prove that for a large class of right-hand
sides, including the viscous term, such solutions, no matter how smooth
initially, develop a singularity within a finite time. We find a sufficient
condition for the singularity formation, "the best sufficient condition", in
the sense that one can explicitly construct a global in time smooth solution
for which this condition is not satisfied "arbitrary little". Also compactly
supported perturbation of nontrivial constant state is considered. We
generalize the known theorem by Sideris on initial data resulting in
singularities. Finally, we investigate the influence of frictional damping and
rotation on the singularity formation.Comment: 23 page
Critical exponents for the absence of solutions for systems of quasilinear parabolic inequalities
Nonexistence of local solutions to semilinear partial differential inequalities
We investigate existence, nonexistence and asymptotical behaviour-both at the origin and at infinity-of radial self-similar solutions to a semilinear parabolic equation with inverse-square potential. These solutions are relevant to prove nonuniqueness of the Cauchy problem for the parabolic equation in certain Lebesgue spaces, generalizing the result proved by Haraux and Weissler [Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982) 167-189] for the case of vanishing potential
Positive solutions to a supercritical elliptic problem that concentrate along a thin spherical hole
We build solutions to a supercritical problem which blow-up along thin spherical holes
Positive solutions to a supercritical elliptic problem that concentrate along a thin spherical hole
A tribute to the 70th birthday of Professor Victor Burenkov
This editorial is the preface to the special volume dedicated to the 70th birthday of the outstanding mathematician Victor Ivanovich Burenkov, a leading authority on real analysis and the theory of function spaces, especially on Sobolev
spaces and spaces with fractional order of smoothness and their applications