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Register a new userNon-oriented solutions of the eikonal equation
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We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Instead of a vector field grad u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with div P in L^2. We prove existence and uniqueness for solutions of the equation P div P=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve. The idea of the proof is to apply a generalized method of characteristics introduced in Jabin, Otto, Perthame, Line-energy Ginzburg-Landau models: zero-energy states, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), to a suitable vector field m satisfying P = m otimes m. This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals.
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The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose $Gamma$-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in the variational analysis of this problem. Establishing fine structures of entropy solutions of the Eikonal equation, e.g. concentration of entropy measures on $mathcal{H}^1$-rectifiable sets in $2$D, is arguably the key missing part for a proof of the full $Gamma$-convergence of the Aviles-Giga functional. In the first part of this work, for $pin left(1,frac{4}{3}right]$ we establish an $L^p$ version of the main theorem of Ghiraldin and Lamy [Comm. Pure Appl. Math. 73 (2020), no. 2, 317-349]. Specifically we show that if $m$ is a solution to the Eikonal equation, then $min B^{frac{1}{3}}_{3p,infty,loc}$ is equivalent to all entropy productions of $m$ being in $L^p_{loc}$. This result also shows that as a consequence of a weak form of the Aviles-Giga conjecture (namely the conjecture that all solutions to the Eikonal equation whose entropy productions are in $L^p_{loc}$ are rigid) - the rigidity/flexibility threshold of the Eikonal equation is exactly the space $ B^{frac{1}{3}}_{3,infty,loc}$. In the second part of this paper, under the assumption that all entropy productions are in $L^p_{loc}$, we establish a factorization formula for entropy productions of solutions of the Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this formula is control of all entropy productions by the Jin-Kohn entropies in the $L^p$ setting - this is a strong extension of an earlier result of the authors [Annales de lInstitut Henri Poincar{e}. Analyse Non Lin{e}aire 35 (2018), no. 2, 481-516].
It is known that in the parameters range $-2 leq gamma <-2s$ spectral gap does not exist for the linearized Boltzmann operator without cutoff but it does for the linearized Landau operator. This paper is devoted to the understanding of the formation of spectral gap in this range through the grazing limit. Precisely, we study the Cauchy problems of these two classical collisional kinetic equations around global Maxwellians in torus and establish the following results that are uniform in the vanishing grazing parameter $epsilon$: (i) spectral gap type estimates for the collision operators; (ii) global existence of small-amplitude solutions for initial data with low regularity; (iii) propagation of regularity in both space and velocity variables as well as velocity moments without smallness; (iv) global-in-time asymptotics of the Boltzmann solution toward the Landau solution at the rate $O(epsilon)$; (v) continuous transition of decay structure of the Boltzmann operator to the Landau operator. In particular, the result in part (v) captures the uniform-in-$epsilon$ transition of intrinsic optimal time decay structures of solutions that reveals how the spectrum of the linearized non-cutoff Boltzmann equation in the mentioned parameter range changes continuously under the grazing limit.
In the article a convergent numerical method for conservative solutions of the Hunter--Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction than the common CFL condition for conservation laws.
We show that for any $epsilonin ]0,1[$ there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball $BsubsetR^5$ which belongs to $C^{1,epsilon} (B)setminus C^{1,epsilon+} (B)$.
We consider a system of two coupled non-linear Klein-Gordon equations. We show the existence of standing waves solutions and the existence of a Lyapunov function for the ground state.
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