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Results tagged with ap.analysis-of-pdes
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user 56892
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
0
answers
69
views
Weak-strong uniqueness for Hamilton-Jacobi equation?
Let $H \in C^{\infty}(\mathbb{R}^d;\mathbb{R})$ and $f \in W^{1,\infty}(\mathbb{R}_+\times \mathbb{R}^d ;\mathbb{R})$ be a Lipschitz function that satisfies
$$
\partial_t f - H(\nabla f) = 0 \qquad \t …
- 562
0
votes
Accepted
Uniqueness condition for Hamilton-Jacobi equation?
Proposition A.2 of https://arxiv.org/abs/2104.05360 shows that the answer is "yes". In fact, this is also valid for a general nonlinearity in the equation in place of the squared norm here.
- 562
4
votes
1
answer
225
views
Uniqueness condition for Hamilton-Jacobi equation?
Let $f= f(t,x) : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}$ be a Lipschitz function such that
$$
\partial_t f - |\nabla f|^2 = 0 \qquad \text{almost everywhere in } \mathbb{R}_+ \times \mathbb{R …
- 562
5
votes
0
answers
307
views
Reference for Hodge decomposition
Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the for …
- 562