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Questions tagged [viscosity-solutions]

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

The viscosity solution is a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation . The viscosity solution is the natural solution concept to use in many applications of PDE's, including first order equations arising in optimal control (the Hamilton–Jacobi equation), differential games (the Isaacs equation) or front evolution problems.

72 questions
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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 an elliptic operator $\mathcal{L}$ with bad…
Medan
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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) = \varphi(x), \end{gather} is called Kolmogorov…
Joker123
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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 $u(x,t)=0$ (as in the answers), but how determine…
user570519
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Inequality $ (\nabla u)^2 \geq \left( \frac{1}{r} \partial_\theta u_r \right)^2 $ in Heywood paper

I am currently trying to understand the proof of Theorem 4 in John G. Heywood's paper On Uniqueness Questions in the Theory of Viscous Flow. Near the end of the proof, the inequality $ (\nabla u)^2 \geq \left( \frac{1}{r} \partial_\theta u_r…
ben_dvs
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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 $\mathbb R^+ \times \mathbb R$ $$u_t+\vert\nabla u\vert^\frac{2}{3}=0,\ \ \ \ u(0,x)=-\cos x$$ More specifically, I am thinking about how to…
Sam Wong
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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 viscosity solution of $(I)$ iff $v(x):=-e^{-u(x)}$ is a…
Veridian Dynamics
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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{cases} -x^2/2 + 5x/6 &\mbox{if } x \in (0,1]…
smiley06
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Desperate on Viscosity (Sub)solutions

The HJ equation is: $H(u'(x))+1=0, x\in(-1,1)$. And $H:\mathbb{R}\rightarrow\mathbb{R},H(p)=\min_{a\in[-1,1]}ap,p\in\mathbb{R}$. The question is show that $u(x)=1-|x|,x\in(-1,1)$ is a viscosity solution. I know that it's a supersolution since there…
Einherjar
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Viscosity solutions of (graph) mean curvature flow

Let $w=w(x,t)$, $x\in\mathbb{R},\ t\in[0,T)$ be a classical solution of the graph curve shortening flow (call (A)) $$ \begin{cases} w_t=\frac {w_{xx}}{1+w_x^2}\quad &\text{ in } \mathbb{R}\times(0,T),\\\\ w(x,0)=w_0(x) \quad &\text{ on }…
JJW
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Concepts of Solutions to Partial Differential Equations

I was wondering what the most used notions for solutions to partial differential equations is in current research. I'm aware, that the concept of solution strongly depends on the specific equation under consideration. But maybe there is a list of…
Peter Wacken
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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 geometric level set flow satisfies $u_t(x) \geq…
George
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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 consider the upper-semicontinuous envelope…
Jan Lynn
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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, Morrey, Campanato spaces, decay properties…
tommy1996q
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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 $w:\mathbb{R}^n \times [0, \infty) \to \mathbb{R}$ $$ w^{\varepsilon}(x,t) = \sup_{y,s} \left( w(y,s) -…
msm
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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(x) + b(x)\cdot \nabla u(x) = f(x), \quad x\in…
S.V.
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