Well-Posedness Theory for Aggregation Sheets
نویسندگان
چکیده
In this paper, we consider distribution solutions to the aggregation equation ρt + div(ρu) = 0, u = −∇V ∗ρ in R where the density ρ concentrates on a co-dimension one manifold. We show that an evolution equation for the manifold itself completely determines the dynamics of such solutions. We refer to such solutions aggregation sheets. When the equation for the sheet is linearly well-posed, we show that the fully non-linear evolution is also well-posed locally in time for the class of bi-Lipschitz surfaces. Moreover, we show that if the initial sheet is C then the solution itself remains C as long as it remains Lipschitz. Lastly, we provide conditions on the kernel g(s) = − ds that guarantee the solution remains a bi-Lipschitz surface globally in time, and construct explicit solutions that either collapse or blow up in finite time when these conditions fail.
منابع مشابه
Hadamard Well-posedness for a Family of Mixed Variational Inequalities and Inclusion Problems
In this paper, the concepts of well-posednesses and Hadamard well-posedness for a family of mixed variational inequalities are studied. Also, some metric characterizations of them are presented and some relations between well-posedness and Hadamard well-posedness of a family of mixed variational inequalities is studied. Finally, a relation between well-posedness for the family of mixed variatio...
متن کاملL Theory for the Multidimensional Aggregation Equation
We consider well-posedness of the aggregation equation ∂tu+ div(uv) = 0, v = −∇K ∗ u with initial data in P2(R)∩L(R), in dimensions two and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x|, α > 2−d, and prove local well-posedness in P2(R)∩L(R) for sufficiently large p > ps. In the special case of K(x) = |x|, the exponent ps = d/(d− 1) is sharp f...
متن کاملLp Theory for the Multidimensional Aggregation Equation
We consider well-posedness of the aggregation equation ∂tu + div(uv) = 0, v = −∇K ∗ u with initial data in P2(R)∩L(R), in dimensions two and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x|α , α > 2−d, and prove local well-posedness in P2(R)∩L(R) for sufficiently large p > ps. In the special case of K(x) = |x|, the exponent ps = d/(d−1) is sharp...
متن کاملBending Behavior of Sandwich Plates with Aggregated CNT-Reinforced Face Sheets
The main aim of this paper is to investigate bending behavior in sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets with considering the effects of carbon nanotube (CNT) aggregation. The sandwich plates are assumed resting on Winkler-Pasternak elastic foundation and a mesh-free method based on first order shear deformation theory (FSDT) is devel...
متن کامل