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Register a new userMeasure solutions to a system of continuity equations driven by Newtonian nonlocal interactions
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We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space $L^2(0,1)^2$ according to the classical theory by Brezis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the $L^m$-norms for all $min [1,+infty]$ and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.
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We prove existence and uniqueness of strong (pointwise) solutions to a linear nonlocal strongly coupled hyperbolic system of equations posed on all of Euclidean space. The system of equations comes from a linearization of a nonlocal model of elasticity in solid mechanics. It is a nonlocal analogue of the Navier-Lame system of classical elasticity. We use a well-known semigroup technique that hinges on the strong solvability of the corresponding steady-state elliptic system. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. For an operator possessing an asymmetric kernel comparable to that of the fractional Laplacian, we prove the $L^2$-solvability of the elliptic system in a Bessel potential space using the Fourier transform and textit{a priori} estimates. This $L^2$-solvability together with the Hille-Yosida theorem is used to prove the well posedness of the wave-type time dependent problem. For the fractional Laplacian kernel we extend the solvability to $L^p$ spaces using classical multiplier theorems.
We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic ``phases that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work ``Nonlocal self-improving properties Anal. PDE, 8(1):57--114 for the specific nonlinear setting under investigation in this manuscript.
In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in cite{Seis16a} and some tools from harmonic analysis introduced in cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~$L^1$, extending the proof for the $L^p$ case in cite{CrippaSpirito15}.
Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${mathbb L} u = u^p +lambda mu$ in a bounded domain $Omega$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $lambda>0$. The operator ${mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $mu$ is taken in the optimal weighted measure space. The interplay between the operator ${mathbb L}$, the source term $u^p$ and the datum $mu$ yields substantial difficulties and reveals the distinctive feature of the problem. We develop a new unifying technique based on a fine analysis on the Green kernel, which enables us to construct a theory for semilinear equation (E) in measure frameworks. A main thrust of the paper is to provide a fairly complete description of positive solutions to the Dirichlet problem for (E). In particular, we show that there exist a critical exponent $p^*$ and a threshold value $lambda^*$ such that the multiplicity holds for $1<p<p^*$ and $0<lambda<lambda^*$, the uniqueness holds for $1<p<p^*$ and $lambda=lambda^*$, and the nonexistence holds in other cases. Various types of nonlocal operator are discussed to exemplify the wide applicability of our theory.
Let $Omega subset mathbb{R}^{n+1}$ be an open set whose boundary may be composed of pieces of different dimensions. Assume that $Omega$ satisfies the quantitative openness and connectedness, and there exist doubling measures $m$ on $Omega$ and $mu$ on $partial Omega$ with appropriate size conditions. Let $Lu=-mathrm{div}(A abla u)$ be a real (not necessarily symmetric) degenerate elliptic operator in $Omega$. Write $omega_L$ for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) $omega_L in A_{infty}(mu)$, (ii) $L$ is $L^p(mu)$-solvable for some $p in (1, infty)$, (iii) every bounded null solution of $L$ satisfies a Carleson measure estimate with respect to $mu$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(mu)$ for some (or for all) $q in (0, infty)$ for any null solution of $L$, (v) $L$ is $mathrm{BMO}(mu)$-solvable, and (vi) every bounded null solution of $L$ is $varepsilon$-approximable for any $varepsilon>0$. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of $omega_L$ with respect to $mu$ in terms of local $L^2(mu)$ estimates of the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $mu$-almost everywhere of the truncated conical square function for any bounded null solution of $L$.
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