On questions of decay and existence for the viscous Camassa–Holm equations
On reduction of some classes of partial differential equations to equations with fewer variables and exact solutions.
On second-grade fluids with vanishing viscosity.
On shape optimization problems involving the fractional laplacian
Our concern is the computation of optimal shapes in problems involving (−Δ)1/2. We focus on the energy J(Ω) associated to the solution uΩ of the basic Dirichlet problem ( − Δ)1/2uΩ = 1 in Ω, u = 0 in Ωc. We show that regular minimizers Ω of this energy under a volume constraint are disks. Our proof goes through the explicit computation of the shape derivative (that seems to be completely new in the fractional context), and a refined adaptation of the moving plane method.
On similarity solution of a boundary layer problem for power-law fluids
The boundary layer equations for the non-Newtonian power law fluid are examined under the classical conditions of uniform flow past a semi infinite flat plate. We investigate the behavior of the similarity solution and employing the Crocco-like transformation we establish the power series representation of the solution near the plate.
On some Boussinesq systems in two space dimensions: theory and numerical analysis
A three-parameter family of Boussinesq type systems in two space dimensions is considered. These systems approximate the three-dimensional Euler equations, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. For a subset of these systems it is proved that their Cauchy problem is locally well-posed in suitable Sobolev classes. Further, a class of these systems is discretized...
On some inequalities for solutions of equations describing the motion of a viscous compressible heat-conducting capillary fluid bounded by a free surface
We derive inequalities for a local solution of a free boundary problem for a viscous compressible heat-conducting capillary fluid. The inequalities are crucial in proving the global existence of solutions belonging to certain anisotropic Sobolev-Slobodetskii space and close to an equilibrium state.
On some properties of solutions of transonic potential flow problems
The paper deals with solutions of transonic potential flow problems handled in the weak form or as variational inequalities. Using suitable generalized methods, which are well known for elliptic partial differential equations of the second order, some properties of these solutions are derived. A maximum principle, a comparison principle and some conclusions from both ones can be established.
On spatial decay estimates for derivatives of vorticities of the two dimensional Navier-Stokes flow
On step approximation for Roseau's analytical solution of water waves.
On Strongly Interacting Internal Solitary Waves.
On symmetric equilibrium of an isothermal gas with a free boundary and a body force.
On the behavior of the interface separating fresh and salt groundwater in a heterogeneous coastal aquifer.
On the blow up criterion for the 2-D compressible Navier-Stokes equations
Motivated by [10], we prove that the upper bound of the density function controls the finite time blow up of the classical solutions to the 2-D compressible isentropic Navier-Stokes equations. This result generalizes the corresponding result in [3] concerning the regularities to the weak solutions of the 2-D compressible Navier-Stokes equations in the periodic domain.
On the blow-up set for non-Newtonian equation with a nonlinear boundary condition.
On the boundary conditions at the contact interface between a porous medium and a free fluid
On the Cauchy problem for the equations of ideal compressible MHD fluids with radiation
We consider a system of balance laws describing the motion of an ionized compressible fluid interacting with magnetic fields and radiation effects. The local-in-time existence of a unique smooth solution for the Cauchy problem is proven. The proof follows from the method of successive approximations.
On the Charney Conjecture of Data Assimilation Employing Temperature Measurements Alone: The Paradigm of 3D Planetary Geostrophic Model
Analyzing the validity and success of a data assimilation algorithmwhen some state variable observations are not available is an important problem in meteorology and engineering. We present an improved data assimilation algorithm for recovering the exact full reference solution (i.e. the velocity and temperature) of the 3D Planetary Geostrophic model, at an exponential rate in time, by employing coarse spatial mesh observations of the temperature alone. This provides, in the case of this paradigm,...
On the conditional regularity of the Navier-Stokes and related equations
We present regularity conditions for a solution to the 3D Navier-Stokes equations, the 3D Euler equations and the 2D quasigeostrophic equations, considering the vorticity directions together with the vorticity magnitude. It is found that the regularity of the vorticity direction fields is most naturally measured in terms of norms of the Triebel-Lizorkin type.
On the Connection Between Stokes and Papkovich-Neuber Spherical Eigenfunctions in stokes flow