We study the Cauchy problem for chemotaxis Navier-Stokes equations and Keller-Segel-Navier-Stokes system. Local-in-time global-in-time solutions satisfying fundamental properties such as mass conservation nonnegativity preservation are constructed low regularity data in $2$ higher dimensions under suitable conditions. Our initial classes involve a new scale of function space, that is $\Y(\rn)$ ...

We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, functions being considered are defined on a metric measure space $(\mathcal{X}, d, \mu)$ satisfying doubling condition and supporting Poincar\'e inequality. For such parabolic minimizers that coincide with time-independent Cauchy-Dirichlet datum $u_0$ boundary of space-time-cylinder $\Omega \t...

A method of solving the eikonal equation, in either flat or curved space-times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and characteristic data for the same solution, namely they are related by a Legendre transformation. From the resulting solutions, we study and describe the wave-front si...

We study the matching of LRS spatially homogeneous collapsing dust space-times with non-stationary vacuum exteriors in cylindrical symmetry. Given an interior with diagonal metric we prove existence and uniqueness results for the exterior. The matched solutions contain trapped surfaces, singularities and Cauchy horizons. The solutions cannot be asymptotically flat and we present evidence that t...

We explore a function theory connected with the principal series representation of SL(2, R) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy space, the Cauchy-Riemann equation and the Taylor expansion. ∗Supported by grant 3GP03196 of the FWO-Vlaanderen (Fund of Scientific ResearchFlanders), Scientif...

and Applied Analysis 3 Let l∞ β denote the Banach space of all bounded sequences in Zβ with norm ‖x‖ sup n∈Zβ |xn| for x {xn}n∈Zβ ∈ l∞ β , B d,D { x {xn}n∈Zβ ∈ l∞ β : ‖x − d‖ ≤ D } for d {d}n∈Zβ ∈ l∞ β , D > 0 2.2 represent the closed ball centered at d and with radius D in l∞ β . By a solution of 1.1 , we mean a sequence {xn}n∈Zβ with a positive integer T ≥ n0 τ |β| such that 1.1 is satisfied ...