Questions tagged [partial-differential-equations]
Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.
24,373 questions
2
votes
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views
Is this series expansion mathematically correct, and can we use it to derive the form of the series from the recurrence relation (13)?
I am working on deriving the scalar Green's function for the Helmholtz equation on a three-dimensional Riemannian manifold. I have established a master equation for the amplitude and expanded it into ...
- 31
1
vote
0
answers
15
views
Variational curve of closed geodesic of 2-sphere
$(S^2, g(t))$ is the smooth solution of normalized Ricci flow on 2-dimensional sphere. For fixed $t_0>0$, assume
$$
\gamma:[0,L] \rightarrow S^2
\tag{1}
$$
is a simple closed geodesic of $(S^2, g(...
- 7,264
0
votes
0
answers
67
views
Gradient estimate for a weak solution of a quasi-linear elliptic equation
Crossposted on MathOverflow
I consider a gradient estimate for a quasi-linear elliptic equation on a half-disc in the $2$-dimensional plane. Precisely, we set $B_{2r}^+=B_{2r}\cap \{y>0\}$ in $\...
- 970
1
vote
1
answer
167
views
Chemistry and Math
I remember throughout the course of university that professors would often comment "this math has applications in physics", "this comes from the bicycle model in physics", "...
- 562
3
votes
0
answers
77
views
Equivalence of Hardy norms
We define the real Hardy space as follows.
For any $\phi\in C_c^\infty(\mathbb{R}^n)$ and any $g\in L^1(\mathbb{R}^n)$, we define
\begin{equation*}
M_{\phi}g(x):=\sup_{r>0}|(g*\phi_r)(x)|,
\end{...
- 363
5
votes
1
answer
163
views
How should I integrate $\int Ce^{-3x}dy$ in this partial differential equation problem?
Solve the partial differential equation $3u_{y}+u_{xy}=0$. (Hint: Let $v=u_{y}.$)
Here's my work:
Consider the partial differential equation $3u_{y}+u_{xy}=0$.
Let $v=u_{y}$.
Then we have $v_{x}=u_{xy}...
- 53
6
votes
1
answer
75
views
On the heat equation with elliptic operator
I am studying the well-posedness of the following Cauchy problem on ℝⁿ:
$$\partial_t u(x,t) = Lu(x,t), \quad u(x,0) = \varphi(x)$$
where the operator $L$ is given in non-divergence form by
$$Lu(x) = \...
4
votes
0
answers
53
views
Variational curve of the shortest bisector of 2-dimensional sphere.
Assume $(S^2,g(t))$ is the smooth solution of normalized Ricci flow on 2-dimensional sphere. For fixed $t_0>0$,
$$
\gamma(l,t_0): [0,L]\rightarrow S^2
\tag{1}
$$
is the shortest bisector of $(S^2,...
- 7,264
1
vote
0
answers
120
views
Triviality of complexification of tangent bundle of the 2-sphere [duplicate]
Given a function $\mathbf{A}: \mathbb{R}^4 \to \mathbf{R}^3$, a solution to the Maxwell's equations $(\partial_t^2 - \nabla^2) \mathbf{A} = \mathbf{0}$,
in the Coulomb gauge $(\nabla \cdot \mathbf{A} =...
- 11
3
votes
0
answers
76
views
Short time existence of ancient solution of Ricci flow on $2$-dimensional sphere
Assume $S^2$ is a $2$-dimensional sphere, and $g_0$ is a smooth metric on $S^2$, and the curvature of $(S^2, g)$ is not constant. Consider the Ricci flow on $S^2$
$$
\frac{\partial}{\partial t}g(t) = ...
- 7,264
3
votes
2
answers
108
views
Coercivity of sesquilinear form
I have been trying to verify the following claim within a continuum-mechanical problem:
Define the Hilbert space
$$H^2_0=\left\{w\in H^2:w(1)=0\right\}$$
with inner product
$$\left<w,v\right>_{H^...
- 187
4
votes
2
answers
638
views
PDE textbooks and interpretations
I took grad level PDE in 1979! I decided to dig up my class notes and textbook,
(Fritz John, 3rd ed) and do some work by studying (again) and learning what I did over 45 years ago. I found the book to ...
- 75
2
votes
0
answers
79
views
Understanding of the Strichartz estimates [closed]
In the proof of the Strichartz estimates in Tao's book on dispersive equations, it is assumed that $F$ is a Schwartz function on spacetime. For a general $F$, is the transition made by density? Also, ...
- 21
0
votes
1
answer
128
views
How can I solve this half-line heat equation with a nonhomogeneous boundary condition? [closed]
I would like to ask for ideas on how to solve the following problem:
\begin{align*}
&\begin{cases}\partial_t u-\partial_{z z} u+\mu u=0, & z>0, t>0, \\ u(t, 0)=g(t), \\ u(t, z) \...
1
vote
0
answers
49
views
Proof that non-zero relative speed for moving solid boundary condition implies compressibility for Euler equations
I'm looking at the Euler equations in fluid mechanics
$$
\frac{\partial u}{\partial t} + u \cdot \nabla u = \frac{1}{\rho}\nabla p \\
\nabla \cdot u = 0
$$
The no-penetration boundary condition for a ...
- 31