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Register a new userExistence and Convergence of Solutions of the Boundary Value Problem in Atomistic and Continuum Nonlinear Elasticity Theory
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We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy-Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail.
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We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the Gauss-Codazzi equations through examples for the Euler equations for fluids and the Euler-Lagrange equations for elastic solids. These results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics.
We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models on multilattices which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.
We consider a fluid-structure interaction problem with Navier-slip boundary conditions in which the fluid is considered as a non-Newtonian fluid and the structure is described by a nonlinear multi-layered model. The fluid domain is driven by a nonlinear elastic shell and thus is not fixed. To simplify the problem, we map the moving fluid domain into a fixed domain by applying an arbitrary Lagrange Euler mapping. Unlike the classical method by which we can consider the problem as its entirety, we utilize the time-discretization and split the problem into a fluid subproblem and a structure subproblem by an operator splitting scheme. Since the structure subproblem is nonlinear, Lax-Milgram lemma does not hold. Here we prove the existence and uniqueness by means of the traditional semigroup theory. Noticing that the Non-Newtonian fluid possesses a $ p- $Laplacian structure, we show the existence and uniqueness of solutions to the fluid subproblem by considering the Browder-Minty theorem. With the uniform energy estimates, we deduce the weak and weak* convergence respectively. By a generalized Aubin-Lions-Simon Lemma proposed by Muha and Canic [J. Differential Equations {bf 266} (2019), 8370--8418], we obtain the strong convergence. Finally, we construct the test functions and pass the approximate weak formulation to the limit as time step goes to zero with the convergence results.
This paper deals with the existence of positive solutions for the nonlinear system q(t)phi(p(t)u_{i}(t)))+f^{i}(t,textbf{u})=0,quad 0<t<1,quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here $textbf{u}=(u_{1},...,u_{n})$ and $f^{i}, i=1,2,...,n$ are continuous and nonnegative functions, $p(t), q(t)hbox{rm :} [0,1]to (0,oo)$ are continuous functions. Moreover, we characterize the eigenvalue intervals for (q(t)phi(p(t)u_{i}(t)))+lambda h_{i}(t)g^{i} (textbf{u})=0, quad 0<t<1,quad i=1,2,...,n. The proof is based on a well-known fixed point theorem in cones.
We exploit the existence and nonlinear stability of boundary spike/layer solutions of the Keller-Segel system with logarithmic singular sensitivity in the half space, where the physical zero-flux and Dirichlet boundary conditions are prescribed. We first prove that, under above boundary conditions, the Keller-Segel system admits a unique boundary spike-layer steady state where the first solution component (bacterial density) of the system concentrates at the boundary as a Dirac mass and the second solution component (chemical concentration) forms a boundary layer profile near the boundary as the chemical diffusion coefficient tends to zero. Then we show that this boundary spike-layer steady state is asymptotically nonlinearly stable under appropriate perturbations. As far as we know, this is the first result obtained on the global well-posedness of the singular Keller-Segel system with nonlinear consumption rate. We introduce a novel strategy of relegating the singularity, via a Cole-Hopf type transformation, to a nonlinear nonlocality which is resolved by the technique of taking antiderivatives, i.e. working at the level of the distribution function. Then, we carefully choose weight functions to prove our main results by suitable weighted energy estimates with Hardys inequality that fully captures the dissipative structure of the system.
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