Questions tagged [viscosity-solutions]

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

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44 views

If $w: [0, \infty) \to [0, \infty)$ is upper semicontinuous, increasing and $w(r)/r^2 = o(1)$, then $w$ is $C^2$ near $0$

If $w: [0, \infty) \to [0, \infty)$ is upper semicontinuous, increasing, $w(0) = 0$ and $w(r)/r^2 = o(1)$, then $w$ is $C^2$ near $0$. Why does the above hold? I understand that, by the definition of ...
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23 views

Prove that semiconvex function can be approximated by smooth functions with same semiconvexity constant

On page 60 of the user's guide to viscosity solutions, proof of Lemma A.3, it is stated that given $\varphi$, a semiconvex function, we can find a smooth approximation $\varphi_\epsilon$ with the ...
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27 views

About an application of maximum principle for viscosity solutions

I am reading by myself this paper and in the proof of the Theorem $6$ (it is on the page $7$), the authors stated Thus, by the maximum principle for viscosity solutions, the solution $u_t$ to the ...
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1answer
27 views

A comparison principle for Eikonal equation with viscosity solution

Given $u\in \mathrm{C}(\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$, then the followings are equivalent: $u$ is a viscosity solution of $|Du|\geq 1$ in $\Omega$. If $V$ is an open ...
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3answers
88 views

Given $p \in \mathbb R^N$ and $X$ a symmetric matrix, find $\varphi \in C^2$ such that $D\varphi = p, D^2\varphi = X$ at a given point

In their paper User's guide to viscosity solutions of second order differential partial equations, Crandall, Ishii and Lions define the superjet of an upper semicontinuous function $u$ at $\hat x \in \...
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1answer
14 views

Monotonicity of function with nonnegative super-differential (or viscosity solution-like)

I am trying to prove the following result: If $u\in \mathrm{C}([0,1])$ such that $u'\geq 0$ in the viscosity sense in $(0,1)$, i.e., for any $x\in (0,1)$, if $p\in D^-u(x)$ then $p\geq 0$, then $u$ is ...
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97 views

Limit operations with viscosity solutions

I'm reading the [user's guide to viscosity solutions][1]. In Lemma 4.2 we define $w(x) = \sup\{u(x):u\in\mathcal{F}\}$, where $\mathcal{F}$ is a family of subsolutions to a certain equation. Next we ...
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24 views

Solution to viscous Hamilton-Jacobi equation can be written as fixed points

Can anyone provide me some source to read more about the fact that, solution to the viscous (https://arxiv.org/abs/2002.06674) $$ -\Delta u + H(x,Du) = \alpha_0\quad \text{in}\;\mathbb{T}^n$$ can be ...
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0answers
25 views

Euristic and intuitive idea behind the theory of viscosity solutions

As the title suggests, I am kinda struggling to understand the basic idea behind viscosity solutions theory. The theorems are a lot different from what I saw in classical theory for PDEs (with Sobolev,...
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52 views

Construct solution of $u_t + (|u|)_x = \epsilon u_{xx}$ from solution of heat equation

I am trying to find the vanishing viscosity limit of $\begin{alignat}{3} \begin{cases} u^{\epsilon}_t + (|u^{\epsilon}|)_x &= \epsilon u^{\epsilon}_{xx} &\mathrm{in} &\mathbb{R} \times (0,...
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59 views

Any more explanations for proof $(Lu,v)_{L^{2}}=(U,L^*v)_{L^{2}}-(A_du|_{x=0},v|_{x=0})_{L^{2}} $?

I come accross the below identity which is montioned in the proof of Lemma2.2. 1 page 35 of this paper entitled Stability of Small Viscosity Noncharacteristic Boundary Layers by Guy Me ́tivier here, ...
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1answer
53 views

Developing a finite difference scheme for non-linear conservation law

When considering equations of the form $$u_t + f(u)_x = 0, $$ there are many finite difference schemes that will converge to smooth solutions, such as the one based on Lax-Friedrichs, $$\frac{u_{j}^{...
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33 views

A little push in a test function

Let $M$ be an $n\times n$ matrix of real entries, which has at least one positive eigenvalue. Now consider $f$ a function of class $C ^{2}$ in a domain of $\mathbb{R} ^ n$, and $g$ a continuous ...
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11 views

The little push in the graph “preserves” infimum

i have a limited domain $X \subset \mathbb{R}^{n}$, and two functions, $u$ continuous in $X$ and $ f \in C^{2}(X)$, such that $u-f$ has local minimum in a given $x_{0} \in X$, ie for some $p> 0$, ...
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40 views

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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34 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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1answer
86 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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1answer
182 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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43 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
117 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
97 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
117 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 $\...
2
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1answer
295 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 ...
3
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1answer
88 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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3answers
128 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,...
2
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1answer
130 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(...
2
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0answers
69 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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46 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
92 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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0answers
76 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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0answers
37 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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0answers
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
32 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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1answer
130 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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0answers
36 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
189 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
245 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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62 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 ...
2
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1answer
1k 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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137 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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1answer
111 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
107 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
73 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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131 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 ...
2
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1answer
528 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 $...
21
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2answers
3k 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 ...