Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equation
74
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find a equation to single out a unique characteristic curve through each initial point and prove the uniqueness of global admissible conservative weak solution without any additional assumptions.
rate research
Read More
The paper is concerned with the time-periodic (T-periodic) problem of the fractal Burgers equation with a T-periodic force on the real line. Based on the Galerkin approximates and Fourier series (transform) methods, we first prove the existence of T-periodic solution to a linearized version. Then, the existence and uniqueness of T-periodic solution to the nonlinear equation are established by the contraction mapping argument. Furthermore, we show that the unique T-periodic solution is asymptotically stable. This analysis, which is carried out in energy space $ H^{1}(0,T;H^{frac{alpha}{2}}(R))cap L^{2}(0,T;dot{H}^{alpha})$ with $1<alpha<frac{3}{2}$, extends the T-periodic viscid Burgers equation in cite{5} to the T-periodic fractional case.
We show that the Hunter-Saxton equation $u_t+uu_x=frac14big(int_{-infty}^x dmu(t,z)- int^{infty}_x dmu(t,z)big)$ and $mu_t+(umu)_x=0$ has a unique, global, weak, and conservative solution $(u,mu)$ of the Cauchy problem on the line.
For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical point for the functional on the $L^2$-normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous papers.
We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, the existence of the $W^{1,2}(0,1)$ weak steady-state solution is proved by the method of upper and lower solution, and the uniqueness by a generalized maximum principle. Also, the weak steady-state solution has $C^infty$ regularity, thereby being a classical solution. We further prove the global existence and uniqueness of the time-dependent solution in $C([0,1]times [0,+infty))cap C^{2,1}([0,1]times (0,+infty))$, which, for any continuous initial value, converges to the steady-state solution uniformly in space (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species.
In this paper, we consider unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in unsaturated porous media, modeled by a non-linear extension of Biots quasi-static consolidation model. The coupled, elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media obtained under similar assumptions as usually considered for Richards equation. In this work, the existence of a weak solution is established using regularization techniques, the Galerkin method, and compactness arguments. The final result holds under non-degeneracy conditions and natural continuity properties for the non-linearities. The assumptions are demonstrated to be reasonable in view of geotechnical applications.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
