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Register a new userPreventing blow up by convective terms in dissipative PDEs
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We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burgers type equations, convective Cahn-Hilliard equation, generalized Kuramoto-Sivashinsky equation and KdV type equations, we establish the following common scenario: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similarly to the case when the equation does not involve convective term. This kind of result has been previously known for the case of Burgers type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.
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We study initial boundary value problems for the convective Cahn-Hilliard equation $Dt u +px^4u +upx u+px^2(|u|^pu)=0$. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any $p>0$. In contrast to that, we show that the presence of the convective term $upx u$ in the Cahn-Hilliard equation prevents blow up at least for $0<p<frac49$. We also show that the blowing up solutions still exist if $p$ is large enough ($pge2$). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered.
We introduce and analyze the nonlocal variants of two Cahn-Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals. The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors.
We study finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and the critical regime. After general considerations on Palais-Smale sequences we determine precise blow up rates for subcritical solutions: in particular the possibility of tower bubbles is excluded in all dimensions. In subsequent papers we aim to establish the sharpness of this result, proving a converse existence statement, together with a one to one correspondence of blowing-up subcritical solutions and {em critical points at infinity}. This analysis will be then applied to deduce new existence results for the geometric problem.
Consider the energy critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.
These notes are devoted to the problem of finite-dimensional reduction for parabolic PDEs. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mane projection theorems. The recent counterexamples which show that the underlying dynamics may be in a sense infinite-dimensional if the spectral gap condition is violated as well as the discussion on the most important open problems are also included.
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