Questions tagged [viscosity-solutions]

For questions on the definition, properties and applications of viscosity solutions.

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About viscosity solutions of elliptic equations

i am recently starting my studies on solutions of elliptic pde's of second order in the sense of viscosity. From this study some doubts result, they are: What is the motivation to introduce a new ...
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11 views

Unique Uniformily Viscosity Solution

Say we have the following pocess: $$dX^{\epsilon} = b(X^{\epsilon}(s))ds + \sqrt\epsilon \sigma(X^{\epsilon}(s))dW(s)$$ for $s \in [0, T]$ and $X^{\epsilon}(0) = x_0$ We want to estimate a quantity ...
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23 views

What is the equation for viscous heating?

I have been reading a paper which talks about viscosity in a plasma. They say that the viscous force is given by the divergence of the viscous stress tensor, $\boldsymbol{\nabla}\cdot\boldsymbol{\...
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22 views

Intuition on viscosity solutions [duplicate]

I would tou like to investigate two things: I) What is the motivation for the study of viscosity solutions of elliptic linear (fully non-linear) Pde's II) The relationship between viscosity ...
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1answer
49 views

Viscosity solution converging to conservation law solution

I am studying conservation laws, particularly the Burger's equation and I am dealing with the proposition that states that the solutions of the Burger's equation with viscosity converges to a solution ...
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51 views

Prove the comparison principle for viscosity solution of the Laplace equation [closed]

This is a basic question, but I cannot find where it is addressed. How does one pove a comparison principle for the Laplace equation $\Delta u = 0$ with Dirichlet boundary condition $u = 0$ on a ...
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1answer
131 views

Why should we give special attention to at most polynomially growing solutions of PDEs?

The equation \begin{gather} \frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\ u(0,x) = \...
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35 views

Touching points are dense (viscosity theory)

I am on a course about viscosity theory and we had this as an exercise: Assume $u: \Omega \to R$ is bounded and upper semicontinuous. Fix $x_0$. Prove that for every $r>0$ there is $\phi\in C^2(B(...
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1answer
50 views

Uniform convergence in $\sup$ and $\inf$ convolutions

In this paper, the authors introduce on p.645 ($11^{th}$ page of the paper) the $\sup$ and $\inf$ convolutions for a continuous, bounded function $w:\mathbb{R}^n \times [0, \infty) \to \mathbb{R}$ $$ ...
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1answer
86 views

Estimating a Lebesgue integral and Taylor's formula

Suppose $V:\mathbb{R}^n\to \mathbb{R}$ is a $C^1$ function and let $DV$ the gradient of $V$. Let $A\subset \mathbb{R}^m$ compact and $f:\mathbb{R}^n\times A\to \mathbb{R}^n$ be a function such that $f$...
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1answer
91 views

Viscosity solution for Hamilton-Jacobi equations and local extrema

Consider a general first order nonlinear partial differential equation of the form (the so called Hamilton-Jacobi equation) $$H(x,u,Du) = 0 \qquad x\in\Omega,\qquad[1]$$ where $\Omega$ is open in $\...
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1answer
78 views

Viscous Fluids at a Slope (Navier-Stokes)

An in-compressible viscous fluid flows down a flat slope of angle θ to the horizontal under the force of gravity, with g the acceleration due to gravity. What are the boundary conditions for the fluid ...
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1answer
79 views

Estimate the large-time behavior of the unique viscosity solution

I am looking for some help to determine the large-time behavior of the unique solution for the equation in $\mathbb R^+ \times \mathbb R$ $$u_t+\vert\nabla u\vert^\frac{2}{3}=0,\ \ \ \ u(0,x)=-\cos x$$...
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115 views

Determine solutions of the Jacobi-Hamilton problem $u_{t}+|u_x|^{2}=0$

How determine solutions of the initial value problem, $$u_{t}+|u_x|^{2}=0\qquad \mbox{in } \mathbb{R}\times(0,\infty)$$ With condition $u=0$ on $\mathbb{R}\times\{t=0\}$. Clearly one solution is $u(x,...
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1answer
81 views

Comparison principle for linear second order viscosity solutions

This may be an embarrassing question but could any one please tell me if we have a comparison principle for the viscosity solution of the following equation $$ \begin{cases} -\nabla\cdot A(x)\nabla u(...
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63 views

1D viscous flow upwards against gravity

Inviscid burgers equation for fluid flowing upwards against gravity: $$ u \frac{du}{dy} = -g $$ I can solve for the velocity profile by simple integration and applying Dirichlet b.c. $u(0)=u0$: $$...
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40 views

Why is this a viscosity super solution?

Consider the viscosity super solution to the equation $u_t - |u_x|=0$ given by $u(t,x) = \begin{cases} t+x \,\,\, \text{if } x \in (-t,0) \\ t-x \,\,\, \text{if } x \in (0,t) \\ 0 \,\,\, \text{else} ...
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1answer
60 views

Kruzkov's change of variable

Let $l:\Omega\to\mathbb{R}$ a sufficiently smooth function on an open set $\Omega$. Let the equations $$(I):\ \|\nabla u(x)\|=l(x)$$ $$(II):\ \|\nabla v(x)\|+l(x)v(x)=0$$ Prove that $u(x)$ is a ...
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66 views

Viscosity Solution for Hamilton Jacobi equation

I am trying to do a numerical approximation of the viscosity solution for the Hamilton-Jacobi problem $\hspace{5cm}\displaystyle \frac{du}{dt}+\frac{|u_x|^2}{2}=0 \hspace{0.5cm} x \in \mathbb{R},\...
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36 views

Problem with a result about Viscosity Solutions of the Euler-Lagrange PDE

I have a problem with the proof of the Proposition 5 in page 6 of the article https://arxiv.org/pdf/1308.5918.pdf. Until equation $(3.5)$ all is clear, but when the author says $"$By (3.1), strictness ...
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39 views

Where does the viscocity solution concept come from?

I understand that viscocity solutions are well suited for solving Optimal control problems involving Hamilton–Jacobi–Bellman equations. But how did the inventors get the idea for viscocity solutions? ...
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1answer
31 views

the 1st order jet is closed

Given continuous $u:\Omega\to\mathbb{R}$ and $x\in \Omega$, define $$J^{1,+}u(x)=\{p\in\mathbb{R}^n\mid u(x+z)\le u(x)+(p,z)+o(|z|)\},$$ as $z\to 0$. Here $f(z)\le o(|z|)$ means there exists a ...
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109 views

Continuity a viscosity solution: where am I wrong?

I am just starting to look into the theory of viscosity solutions. I can't see how the following simple argument fails. On the other hand the results seems too nice to be true. Let us consider an ...
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35 views

Conditions for existence of a viscoity or mild a solution

Suppose I have a PDE of the form $$ 0=-\partial_t f(t,z) + \phi(0,z)+ \sum_{i=1}^{\infty} b_i(t,z)\partial_{z_i}f(t,z) + \frac{1}{2}\sum_{i,j=1}^{\infty} a_{i,j}(t,z)(\partial_{z_j}\partial_{z_i}f(t,z)...
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1answer
150 views

Introduction to viscosity solutions theory

Can you recommend an introduction to viscosity solutions theory? More specifically, I'm looking for a modern treatment similar to Chapter 10 of Evans's Partial Differential Equations, but somewhat ...
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1answer
185 views

Crandall-Ishii lemma on unbounded domains

In the User's Guide to Viscosity Solutions, there is an example (Section 5.D; page 30) using the Crandall-Ishii lemma on unbounded domains. Equation (5.17) claims $$ -3\alpha\left(\begin{array}{cc} I\...
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51 views

Viscosity solution in for a non smooth value function

I am trying to find a solution for a free boundary problem, where the value function won't be smooth as it is in a multi period framework with fixed costs at each period. Resulting jumps in the ...
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1answer
877 views

Viscous Burgers' equation using Lax-Wendroff scheme

I was assigned to find a solution for the Burgers' equation $$u_t+uu_x=\upsilon u_{xx}$$ with initial condition $u(x,0) = \sin(x)$ and with boundary conditions $u(0,t)=u(\epsilon,t)$ with $\epsilon &...
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132 views

To show a given function is not the viscosity solution.

For the equation $ F(x,u,u',u'') = -au''-1 =0$ for $ x\in (0,2)$ with $ u(0) = 0 = u(2) $ and $a(x)$ is $1$ for $x\in (0,1)$ and $2$ for $x\in [1,2)$. Need to show that the function $$ u(x) = \begin{...
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93 views

continuous dependency estimate for viscosity solutions

Let $u^i$, $i=1,2$, be viscosity solutions of \begin{align*} u_t^i + H(Du^i,x) & = 0\quad\mathrm{in}\ \mathbb R^n\times (0,\infty)\\ u^i & = g^i\quad\mathrm{on}\ \mathbb R^n\times \{t=0\} \...
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1answer
94 views

User's guide to viscosity solutions basic question

Page 9 of the user's guide has a claim that if $u$ is twice differentiable at $\hat{x}$ and $$ u\left(x\right)\leq u\left(\hat{x}\right)+\left\langle p,x-\hat{x}\right\rangle +\frac{1}{2}\left\langle ...
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1answer
65 views

Determination of the modulus of continuity

I'm trying to prove the uniqueness of the viscosity solution of an Hamilton-Jacobi-Bellman equation. Thanks to a classical result, I'm left to check if it exist a modulus of continuity $\omega_1$ --- ...
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1answer
72 views

Is there a typo on this definition?

This is from Iishi's User's guide to viscosity solution. I don't understand the $\ni$ in the definition 2.6 at the end of its first line, is it a typo? User's guide page 11
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127 views

reference request: a graduate level textbook on viscosity solutions of IPDEs

I am particularly interested in IPDE which arises from optimal stopping/control problems so I would like some references on integral-partial differential equations, which are related this area. I have ...
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1answer
484 views

Viscosity Solution of Hamilton Jacobi Equation, result from Evans

I have a question about a proof in "Partial Differential Equation by Lawrence C. Evans". We look at the problem $$(1)\mbox{ }u_t+H(Du,x) = 0 \mbox{ in }\mathbb{R}^n\times (0,T] \mbox{ }$$ and $...
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2k views

viscosity solution vs. weak solution

viscosity solution vs. weak solution I am confused between the two. Is one a subset of the other or they are the same/completely different notions? Suppose I have an equation, $u_t=\mathcal{L}u$ for ...