196 research outputs found
The linear hyperbolic initial and boundary value problems in a domain with corners
In this article, we consider linear hyperbolic Initial and Boundary Value
Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in
both the constant and variable coefficients cases. We use semigroup method
instead of Fourier analysis to achieve the well-posedness of the linear
hyperbolic system, and we find by diagonalization that there are only two
elementary modes in the system which we call hyperbolic and elliptic modes. The
hyperbolic system in consideration is either symmetric or
Friedrichs-symmetrizable.Comment: 41 page
A Result of Uniqueness of Solutions of the Shigesada-Kawasaki-Teramoto Equations
We derive the uniqueness of weak solutions to the Shigesada-Kawasaki-Teramoto
(SKT) systems using the adjoint problem argument. Combining with [PT17] we then
derive the well-posedness for the SKT systems in space dimension $d\le 4
The Linearized 2D Inviscid Shallow Water Equations in a Rectangle: Boundary Conditions and Well-Posedness
We consider the linearized 2D inviscid shallow water equations in a
rectangle. A set of boundary conditions is proposed which make these equations
well-posed. Several different cases occur depending on the relative values of
the reference velocities and reference height (sub- or
super-critical flow at each part of the boundary).Comment: 33 page
Global existence for fully nonlinear reaction-diffusion systems describing multicomponent reactive flows
We consider combustion problems in the presence of complex chemistry and
nonlinear diffusion laws leading to fully nonlinear multispecies
reaction-diffusion equations. We establish results of existence of solution and
maximum principle, i.e. positivity of the mass fractions, which rely on
specific properties of the models. The nonlinear diffusion coefficients are
obtained by resolution of the so-called Stefan-Maxwell equations
Pathwise Solutions of the 2D Stochastic Primitive Equations
In this work we consider a stochastic version of the Primitive Equations
(PEs) of the ocean and the atmosphere and establish the existence and
uniqueness of pathwise, strong solutions. The analysis employs novel techniques
in contrast to previous works in order to handle a general class of nonlinear
noise structures and to allow for physically relevant boundary conditions. The
proof relies on Cauchy estimates, stopping time arguments and anisotropic
estimates
Grisvard's shift theorem near L^infinity and Yudovich theory on polygonal domains
Let Omega be a bounded, simply connected domain with boundary of class
C^{1,1} except at finitely many points S_j where the boundary is locally a
corner of aperture alpha_j<=pi/2. Improving on results of Grisvard, we show
that the solution Gf to the Dirichlet problem on Omega with data f in
L^infinity(Omega) and homogeneous boundary conditions has exponentially
integrable second derivatives. The proof uses sharp L^p bounds for singular
integrals on power weighted spaces inspired by the work of Buckley. Our results
allow for the extension of the Yudovich theory of existence, uniqueness and
regularity of weak solutions to the Euler equations on Omega x (0,T) to
polygonal domains Omega as above.Comment: 20 pages; submitted. The authors are deeply grateful to Kabe Moen for
reading an early version of the manuscript and for allowing the inclusion of
his alternative proof of Proposition 5.2. The first named author wants to
thank Stefan Steinerberger for stimulating discussions on the subject of this
articl
Very weak solutions of the Stokes problem in a convex polygon
Motivated by the study of the corner singularities in the so-called cavity
flow, we establish in this article, the existence and uniqueness of solutions
in for the Stokes problem in a domain when
is a smooth domain or a convex polygon. We establish also a trace theorem and
show that the trace of can be arbitrary in The
results are also extended to the linear evolution Stokes problem
Conservative numerical schemes with optimal dispersive wave relations -- Part I. Derivations and analyses
An energy-conserving and an energy-and-enstrophy conserving numerical schemes
are derived, by approximating the Hamiltonian formulation, based on the Poisson
brackets and the vorticity-divergence variables, of the inviscid shallow water
flows. The conservation of the energy and/or enstrophy stems from skew-symmetry
of the Poisson brackets, which is retained in the discrete approximations.
These schemes operate on unstructured orthogonal dual meshes, over bounded or
unbounded domains, and they are also shown to possess the same optimal
dispersive wave relations as those of the Z-grid scheme
Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system
The motion of two contiguous incompressible and viscous fluids is described
within the diffuse interface theory by the so-called Model H. The system
consists of the Navier-Stokes equations, which are coupled with the
Cahn-Hilliard equation associated to the Ginzburg-Landau free energy with
physically relevant logarithmic potential. This model is studied in bounded
smooth domain in R^d, d=2 and d=3, and is supplemented with a no-slip condition
for the velocity, homogeneous Neumann boundary conditions for the order
parameter and the chemical potential, and suitable initial conditions. We study
uniqueness and regularity of weak and strong solutions. In a two-dimensional
domain, we show the uniqueness of weak solutions and the existence and
uniqueness of global strong solutions originating from an initial velocity u_0
in V, namely u_0 in H_0^1 such that div u_0=0. In addition, we prove further
regularity properties and the validity of the instantaneous separation
property. In a three-dimensional domain, we show the existence and uniqueness
of local strong solutions with initial velocity u_0 in V
Numerical Resolution near t = 0 of Nonlinear Evolution Equations in the Presence of Corner Singularities in Space Dimension 1
The incompatibilities between the initial and boundary data will cause
singularities at the time-space corners, which in turn adversely affect the
accuracy of the numerical schemes used to compute the solutions. We study the
corner singularity issue for nonlinear evolution equations in 1D, and propose
two remedy procedures that effectively recover much of the accuracy of the
numerical scheme in use. Applications of the remedy procedures to the 1D
viscous Burgers equation, and to the 1D nonlinear reaction-diffusion equation
are presented. The remedy procedures are applicable to other nonlinear
diffusion equations as well
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