Short time existence and uniqueness in Hölder spaces for the 2D dynamics of dislocation densities

In this paper, we study the model of Groma and Balogh [I. Groma, P. Balogh, Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation, Acta Mater. 47 (1999) 3647–3654] describing the dynamics of dislocation densities. This is a two-dimensional model where the dislocation densities satisfy a system of two transport equations. The velocity vector field is the shear stress in the material solving the equations of elasticity. This shear stress can be related to Riesz transforms of the dislocation densities. Basing on some commutator estimates type, we show that this model has a unique local-in-time solution corresponding to any initial datum in the space Cr(R²)∩Lp(R²)$C^r({\mathbb{R}}^2)\cap L^p({\mathbb{R}}^2)$ for r>1$r>1$ and 1<p<+∞$1<p<+\infty$, where Cr(R²)$C^r({\mathbb{R}}^2)$ is the Hölder–Zygmund space.     

Benjamin–Ono periodic bifurcating water waves in presence of an essential spectrum

The mathematical study of travelling waves in the potential flow of two superposed layers of perfect fluid can be set as an ill-posed evolutionary problem, in which the horizontal unbounded space variable plays the role of “time”. In this paper we consider two problems for which the bottom layer of fluid is infinitely deep: for the first problem, the upper layer is bounded by a rigid top and there is no surface tension at the interface; for the second problem, there is a free surface with a large enough surface tension. In both problems, the linearized operator Lε$L_{\epsilon }$ (where ε is a combination of the physical parameters) around 0 possesses an essential spectrum filling the entire real line  , with in addition a simple eigenvalue in 0. Moreover, for ε<0$\epsilon <0$, there is a pair of imaginary eigenvalues which meet in 0 when ε=0$\epsilon =0$ and which disappear in the essential spectrum for ε>0$\epsilon >0$. For ε>0$\epsilon >0$ small enough, we prove in this paper the existence of a two parameter family of periodic travelling waves (corresponding to periodic solutions of the dynamical system). These solutions are obtained in showing that the full system can be seen as a perturbation of the Benjamin–Ono equation. The periods of these solutions run on an interval (T₀,∞)$(T_0,\infty )$ possibly except a discrete set of isolated points.     

Critical exponents for non-simultaneous blow-up in a localized parabolic system

The paper deals with the radially symmetric solutions of _ut=Δu+u^m(x,t)vⁿ(0,t)$u_t=\Delta u+u^m(x,t)v^n(0,t)$, _vt=Δv+u^p(0,t)v^q(x,t)$v_t=\Delta v+u^p(0,t)v^q(x,t)$, subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u$u$ (v$v$) blows up alone if and only if m>p+1$m>p+1$ (q>n+1$q>n+1$), which means that any blow-up is simultaneous if and only if m≤p+1$m\le p+1$, q≤n+1$q\le n+1$. (ii) Any blow-up is u$u$ (v$v$) blowing up with v$v$ (u$u$) remaining bounded if and only if m>p+1$m>p+1$, q≤n+1$q\le n+1$ (m≤p+1$m\le p+1$, q>n+1$q>n+1$). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1$m>p+1$, q>n+1$q>n+1$. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.     

Capturing two elements in unavoidable minors of 3-connected binary matroids

Let M be a 3-connected binary matroid and let n   be an integer exceeding 2. Ding, Oporowski, Oxley, and Vertigan proved that there is an integer f(n)$f(n)$ so that if |E(M)|>f(n)$|E(M)|>f(n)$, then M has a minor isomorphic to one of the rank-n wheel, the rank-n   tipless binary spike, or the cycle or bond matroid of _K3,n$K_{3,n}$. This result was recently extended by Chun, Oxley, and Whittle to show that there is an integer g(n)$g(n)$ so that if |E(M)|>g(n)$|E(M)|>g(n)$ and x∈E(M)$x\in E(M)$, then x is an element of a minor of M isomorphic to one of the rank-n wheel, the rank-n   binary spike with a tip and a cotip, or the cycle or bond matroid of _K1,1,1,n$K_{1,1,1,n}$. In this paper, we prove that, for each i   in {2,3}$\{2,3\}$, there is an integer _hi(n)$h_i(n)$ so that if |E(M)|>_hi(n)$|E(M)|>h_i(n)$ and Z is an i-element rank-2 subset of M, then M has a minor from the last list whose ground set contains Z.     

Gradient integrability and rigidity results for two-phase conductivities in two dimensions

This paper deals with higher gradient integrability for σ-harmonic functions u with discontinuous coefficients σ  , i.e. weak solutions of div(σ∇u)=0$\mathrm{div}(\sigma \mathrm{\nabla }u)=0$ in dimension two. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti and Nesi. When only the ellipticity is fixed and σ is otherwise unconstrained, the optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries.     

Asymptotic analysis of the p-Laplacian flow in an exterior domain

We consider the Dirichlet problem for the p  -Laplacian evolution equation, ut=Δpu$u_t={\mathrm{\Delta }}_{p}u$, where p>2$p>2$, posed in an exterior domain in RN${\mathbf{R}}^N$, with zero Dirichlet boundary condition and with integrable and nonnegative initial data. We are interested in describing the influence of the holes of the domain on the large time behaviour of the solutions. Such behaviour varies depending on the relative values of N and p  . We must distinguish between the behaviour near infinity of space (outer analysis), and near the holes (inner analysis). We obtain that the outer analysis is given in all cases by certain self-similar solutions and the inner analysis is given by quasi-stationary states. Logarithmic corrections to exact self-similarity appear in the critical case N=p$N=p$, which is mathematically more interesting. In this first paper we treat only the cases N>p$N>p$ and N=p$N=p$, the case N<p$N<p$ will be considered in a companion work.     

Best constants in a borderline case of second-order Moser type inequalities

We study optimal embeddings for the space of functions whose Laplacian Δu   belongs to L¹(Ω)$L^1(\Omega )$, where Ω⊂RN$\Omega \subset {\mathbb{R}}^N$ is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W2,1(Ω)$W^{2,1}(\Omega )$ in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when N=2$N=2$, we establish a sharp embedding inequality into the Zygmund space Lexp(Ω)$L_{\mathit{exp}}(\Omega )$. On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem Δu=f(x)∈L¹(Ω)$\mathrm{\Delta }u=f(x)\in L^1(\Omega )$, u=0$u=0$ on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger–Moser type inequalities to the case of higher-order derivatives. Extensions to dimension N?3$N?3$ are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions.