THE CAUCHY PROBLEM FOR CONSERVATION LAWS WITH A MULTIPLICATIVE STOCHASTIC PERTURBATION

2012 ◽  
Vol 09 (04) ◽  
pp. 661-709 ◽  
Author(s):  
CAROLINE BAUZET ◽  
GUY VALLET ◽  
PETRA WITTBOLD
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Existence And Uniqueness ◽  
Entropy Solution ◽  
Stochastic Perturbation ◽  
Nonlinear Conservation Laws ◽  
The Cauchy Problem ◽  
Entropy Formulation

We study the Cauchy problem for multi-dimensional nonlinear conservation laws with multiplicative stochastic perturbation. Using the concept of measure-valued solutions and Kruzhkov's entropy formulation, the existence and uniqueness of an entropy solution is established.

scholarly journals Nonlinear conservation laws for the Schrödinger boundary value problems of second order

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ming Ren ◽  
Shiwei Yun ◽  
Zhenping Li
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Maximum Modulus ◽  
Second Order ◽  
Nonlinear Conservation Laws ◽  
The Cauchy Problem ◽  
Modulus Method ◽  
Semilinear Schrödinger Equation

AbstractIn this paper, we apply a reliable combination of maximum modulus method with respect to the Schrödinger operator and Phragmén–Lindelöf method to investigate nonlinear conservation laws for the Schrödinger boundary value problems of second order. As an application, we prove the global existence to the solution for the Cauchy problem of the semilinear Schrödinger equation. The results reveal that this method is effective and simple.

scholarly journals A degenerate parabolic–hyperbolic Cauchy problem with a stochastic force

2015 ◽  
Vol 12 (03) ◽  
pp. 501-533 ◽  
Author(s):  
Caroline Bauzet ◽  
Guy Vallet ◽  
Petra Wittbold
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Stochastic Forcing ◽  
Hyperbolic Problem ◽  
Stochastic Force ◽  
The Cauchy Problem ◽  
Degenerate Parabolic ◽  
Entropy Formulation ◽  
Uniqueness Of A Solution ◽  
Nonlinear Degenerate

In this paper, we are interested in the Cauchy problem for a nonlinear degenerate parabolic–hyperbolic problem with multiplicative stochastic forcing. Using an adapted entropy formulation a result of existence and uniqueness of a solution is proven.

ASYMPTOTIC STABILITY OF NON-PLANAR RIEMANN SOLUTIONS FOR MULTI-D SYSTEMS OF CONSERVATION LAWS WITH SYMMETRIC NONLINEARITIES

2004 ◽  
Vol 01 (03) ◽  
pp. 567-579 ◽  
Author(s):  
HERMANO FRID
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Conservation Laws ◽  
Entropy Solution ◽  
Smooth Functions ◽  
Riemann Solutions ◽  
The Cauchy Problem ◽  
Systems Of Conservation Laws ◽  
Self Similar

We study the asymptotic behavior of entropy solutions of the Cauchy problem for multi-dimensional systems of conservation laws of the form [Formula: see text], where the gα are real smooth functions defined in [0,+∞), and when the initial data are perturbations of two-state nonplanar Riemann data. Specifically, if R0(x) is such Riemann data and ψ∈L∞(ℝd)n satisfies ψ(Tx)→0 in [Formula: see text], as T→∞, then an entropy solution, u(x,t), of the Cauchy problem with u(x,0)=R0(x)+ψ(x) satisfies u(ξt,t)→R(ξ) in [Formula: see text], as t→∞, where R(x/t) turns out to be the unique self-similar entropy solution of the corresponding Riemann problem.

scholarly journals Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 66
Author(s):  
Azmat Ullah Khan Niazi ◽  
Jiawei He ◽  
Ramsha Shafqat ◽  
Bilal Ahmed
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Ulam Type Stability ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

scholarly journals ON A STOCHASTIC FIRST-ORDER HYPERBOLIC EQUATION IN A BOUNDED DOMAIN

2009 ◽  
Vol 12 (04) ◽  
pp. 613-651 ◽  
Author(s):  
GUY VALLET ◽  
PETRA WITTBOLD
Keyword(s):  
Hyperbolic Equation ◽  
Bounded Domain ◽  
Entropy Solution ◽  
Stochastic Perturbation ◽  
Dirichlet Condition ◽  
First Order ◽  
Scalar Conservation Law ◽  
Order Hyperbolic Equation ◽  
Scalar Conservation ◽  
Entropy Formulation

In this paper, we are interested in the stochastic perturbation of a first-order hyperbolic equation of nonlinear type. In order to illustrate our purposes, we have chosen a scalar conservation law in a bounded domain with homogeneous Dirichlet condition on the boundary. Using the concept of measure-valued solutions and Kruzhkov's entropy formulation, a result of existence and uniqueness of the entropy solution is given.

scholarly journals A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2843
Author(s):  
Ángel García ◽  
Mihaela Negreanu ◽  
Francisco Ureña ◽  
Antonio M. Vargas
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Meshless Method ◽  
Existence And Uniqueness ◽  
Fractional Laplacian ◽  
Porous Medium Equation ◽  
Medium Equation ◽  
The Cauchy Problem ◽  
Irregular Meshes ◽  
Complicated Geometry

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

scholarly journals Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1247-1257 ◽  
Author(s):  
Shijin Ding ◽  
Jinrui Huang ◽  
Fengguang Xia
Keyword(s):  
Cauchy Problem ◽  
Liquid Crystals ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Hydrodynamic Flow ◽  
Three Dimensions ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We consider the Cauchy problem for incompressible hydrodynamic flow of nematic liquid crystals in three dimensions. We prove the global existence and uniqueness of the strong solutions with nonnegative p0 and small initial data.

Sign in / Sign up

Export Citation Format

Share Document