Questions tagged [viscosity-solutions]
For questions on the definition, properties and applications of viscosity solutions.
47
questions
0
votes
1answer
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 ...
0
votes
1answer
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 ...
3
votes
0answers
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 ...
0
votes
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 ...
0
votes
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 \...
0
votes
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 ...
3
votes
2answers
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 ...
2
votes
0answers
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 ...
2
votes
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,...
2
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0answers
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,...
0
votes
0answers
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 Ģtivier here, ...
1
vote
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}^{...
1
vote
0answers
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 ...
0
votes
0answers
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$, ...
0
votes
0answers
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 ...
1
vote
0answers
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{\...
2
votes
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 ...
9
votes
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) = \...
2
votes
0answers
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(...
2
votes
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}$
$$ ...
1
vote
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 $f$...
2
votes
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
votes
1answer
295 views
Viscous Fluids at a Slope (Navier-Stokes)
An in-compressible viscous fluid flows down a flat slope of angle Īø to the
horizontal under the force of gravity, with g the acceleration due to gravity.
What are the boundary conditions for the fluid ...
3
votes
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 $\mathbb R^+ \times \mathbb R$ $$u_t+\vert\nabla u\vert^\frac{2}{3}=0,\ \ \ \ u(0,x)=-\cos x$$...
6
votes
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
votes
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
$$ \begin{cases}
-\nabla\cdot A(x)\nabla u(...
2
votes
0answers
69 views
1D viscous flow upwards against gravity
Inviscid burgers equation for fluid flowing upwards against gravity:
$$
u \frac{du}{dy} = -g
$$
I can solve for the velocity profile by simple integration and applying Dirichlet b.c. $u(0)=u0$:
$$...
1
vote
0answers
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} ...
3
votes
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 ...
1
vote
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},\...
1
vote
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 ...
1
vote
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 HamiltonāJacobiāBellman equations.
But how did the inventors get the idea for viscocity solutions? ...
1
vote
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 ...
0
votes
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 ...
1
vote
0answers
36 views
Conditions for existence of a viscoity or mild a solution
Suppose I have a PDE of the form
$$
0=-\partial_t f(t,z) + \phi(0,z)+ \sum_{i=1}^{\infty} b_i(t,z)\partial_{z_i}f(t,z)
+
\frac{1}{2}\sum_{i,j=1}^{\infty} a_{i,j}(t,z)(\partial_{z_j}\partial_{z_i}f(t,z)...
2
votes
1answer
189 views
Introduction to viscosity solutions theory
Can you recommend an introduction to viscosity solutions theory? More specifically, I'm looking for a modern treatment similar to Chapter 10 of Evans's Partial Differential Equations, but somewhat ...
0
votes
1answer
245 views
Crandall-Ishii lemma on unbounded domains
In the User's Guide to Viscosity Solutions, there is an example (Section 5.D; page 30) using the Crandall-Ishii lemma on unbounded domains. Equation (5.17) claims
$$
-3\alpha\left(\begin{array}{cc}
I\...
0
votes
0answers
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
votes
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 &...
4
votes
0answers
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 $$ u(x) = \begin{...
0
votes
1answer
111 views
continuous dependency estimate for viscosity solutions
Let $u^i$, $i=1,2$, be viscosity solutions of \begin{align*}
u_t^i + H(Du^i,x) & = 0\quad\mathrm{in}\Ā \mathbb R^n\times (0,\infty)\\
u^i & = g^i\quad\mathrm{on}\ \mathbb R^n\times \{t=0\}
\...
2
votes
1answer
107 views
User's guide to viscosity solutions basic question
Page 9 of the user's guide has a claim that if $u$ is twice differentiable at $\hat{x}$ and
$$
u\left(x\right)\leq u\left(\hat{x}\right)+\left\langle p,x-\hat{x}\right\rangle +\frac{1}{2}\left\langle ...
0
votes
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 $\omega_1$ --- ...
1
vote
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
2
votes
0answers
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
votes
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
votes
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 ...
