Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFundamental Solution to 1D Degenerate Diffusion Equation with Locally Bounded Coefficients
76
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this work we study the degenerate diffusion equation $partial_{t}=x^{alpha}aleft(xright)partial_{x}^{2}+bleft(xright)partial_{x}$ for $left(x,tright)inleft(0,inftyright)^{2}$, equipped with a Cauchy initial data and the Dirichlet boundary condition at $0$. We assume that the order of degeneracy at 0 of the diffusion operator is $alphainleft(0,2right)$, and both $aleft(xright)$ and $bleft(xright)$ are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution $pleft(x,y,tright)$ and prove several properties for $pleft(x,y,tright)$; by conducting a localization procedure, we obtain an approximation for $pleft(x,y,tright)$ for $x,y$ in a neighborhood of 0 and $t$ sufficiently small, where the error estimates only rely on the local bounds of $aleft(xright)$ and $bleft(xright)$ (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of $alpha=1$. Our work extends part of the existing results to cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probability view (e.g., wellposedness of stochastic differential equations).
rate research
Read More
In this paper, we present strong numerical evidences that the $3$D incompressible axisymmetric Navier-Stokes equations with degenerate diffusion coefficients and smooth initial data of finite energy develop a potential finite time locally self-similar singularity at the origin. The spatial part of the degenerate diffusion coefficient is a smooth function of $r$ and $z$ independent of the solution and vanishes like $O(r^2)+O(z^2)$ near the origin. This potential singularity is induced by a potential singularity of the $3$D Euler equations. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin. The two-scale feature is characterized by the property that the center of the traveling wave approaches the origin at a slower rate than the rate of the collapse of the singularity. The driving mechanism for this potential singularity is due to two antisymmetric vortex dipoles that generate a strong shearing layer in both the radial and axial velocity fields. Without the viscous regularization, the $3$D Euler equations develop an additional small scale characterizing the thickness of the sharp front. On the other hand, the Navier-Stokes equations with a constant diffusion coefficient regularize the two-scale solution structure and do not develop a finite time singularity for the same initial data. The initial condition is designed in such a way that it generates a positive feedback loop that enforces a strong nonlinear alignment of vortex stretching, leading to a stable locally self-similar blowup at the origin. We perform careful resolution study and asymptotic scaling analysis to provide further support of the potential finite time locally self-similar blowup.
We consider properties of second-order operators $H = -sum^d_{i,j=1} partial_i , c_{ij} , partial_j$ on $Ri^d$ with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) geq 0$ almost everywhere, but allow for the possibility that $C$ is singular. We associate with $H$ a canonical self-adjoint viscosity operator $H_0$ and examine properties of the viscosity semigroup $S^{(0)}$ generated by $H_0$. The semigroup extends to a positive contraction semigroup on the $L_p$-spaces with $p in [1,infty]$. We establish that it conserves probability, satisfies $L_2$~off-diagonal bounds and that the wave equation associated with $H_0$ has finite speed of propagation. Nevertheless $S^{(0)}$ is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor Holder continuous. In particular one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in $C^{2-varepsilon}(Ri^d)$ with $varepsilon > 0$.
The goal of this work is to compute a boundary control of reaction-diffusion partial differential equation. The boundary control is subject to a constant delay, whereas the equation may be unstable without any control. For this system equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed. To do that we decompose the infinite-dimensional system into two parts: one finite-dimensional unstable part, and one stable infinite-dimensional part. An finite-dimensional delay controller is computed for the unstable part, and it is shown that this controller succeeds in stabilizing the whole partial differential equation. The proof is based on a an explicit form of the classical Artstein transformation, and an appropriate Lyapunov function. A numerical simulation illustrate the constructive design method.
We derive a diffusion approximation for the kinetic Vlasov-Fokker-Planck equation in bounded spatial domains with specular reflection type boundary conditions. The method of proof involves the construction of a particular class of test functions to be chosen in the weak formulation of the kinetic model. This involves the analysis of the underlying Hamiltonian dynamics of the kinetic equation coupled with the reflection laws at the boundary. This approach only demands the solution family to be weakly compact in some weighted Hilbert space rather than the much tricky $mathrm L^1$ setting.
In this paper, we study the consumption-chemotaxis-Stokes model with porous medium slow diffusion in a three dimensional bounded domain with zero-flux boundary conditions and no-slip boundary condition. In recent ten years, many efforts have been made to find the global bounded solutions of chemotaxis-Stokes system in three dimensional space. Although some important progress has been carried out in some papers, as mentioned by some authors, the question of identifying an optimal condition on m ensuring global boundedness in the three-dimensional framework remains an open challenge. In the present paper, we put forward a new estimation technique, completely proved the existence of global bounded solutions for arbitrary slow diffusion case, and partially answered the open problem proposed by Winkler.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
