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Register a new userThe Solvability of a Strongly-Coupled Nonlocal System of Equations
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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.
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In this paper we study the asymptotic behavior of solutions to systems of strongly coupled integral equations with oscillatory coefficients. The system of equations is motivated by a peridynamic model of the deformation of heterogeneous media that additionally accounts for short-range forces. We consider the vanishing nonlocality limit on the same length scale as the heterogeneity and show that the systems effective behavior is characterized by a coupled system of local equations that are elliptic in the sense of Legendre-Hadamard. This effective system is characterized by a fourth-order tensor that shares properties with Cauchy elasticity tensors that appear in the classical equilibrium equations for linearized elasticity.
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.
This paper concerns inverse problems for strongly coupled Schrodinger equations. The purpose of this inverse problem is to retrieve a stationary potential in the strongly coupled Schrodinger equations from either boundary or internal measurements. Two stability results are derived from a new Carleman estimate for the strongly coupled Schrodinger equations.
We define the Ladyzhenskaya-Lions exponent $alpha_{rm {tiny sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-Delta)^{alpha}$ in ${Bbb R}^n$, for all $ngeq 2$. We review the proof of strong global solvability when $alphageq alpha_{rm {tiny sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1over 2} | abla u |^2_{L^2}$, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for $alpha<alpha_{rm {tiny sc l}} (n)$. The energy is critical under scale transformations only for $alpha=alpha_{rm {tiny sc l}} (n)$.
We introduce matrix coupled (local and nonlocal) dispersionless equations, construct wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and consider their asymptotics in some interesting cases. We consider the scalar cases of coupled, complex coupled and nonlocal dispersionless equations as well.
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