# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

2,364 questions

**5**

votes

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55 views

### Laplace Beltrami eigenvalues on surface of polytopes

The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra
by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...

**0**

votes

**0**answers

43 views

### Energy estimates involving test functions for weak solutions of PDE problems

I was reading an article on Arxiv.org about Navier-Stokes system ([Breit]) and I stumble on this sentence on the second page:
"A weak (in the PDE sense) solution satisfying the energy inequality ...

**2**

votes

**1**answer

159 views

### Underdetermined system of linear PDEs

Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.
I ...

**1**

vote

**0**answers

58 views

### Conditions for the embeddig of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...

**1**

vote

**0**answers

27 views

### Local well-posedness of the quadratic NLS on the 1D torus

What is the proof of the local well-posedness of the quadratic nonlinear Schrödinger equation
$\mathrm{i} \,\partial_t u + \Delta u \pm \left|u\right| u = 0$
on the 1D torus in $H^s$ for $s > 1$ (a ...

**1**

vote

**1**answer

40 views

### Exact solution of two coupled transport equations

I want to solve the following system
$$\eqalign{
& y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr
& z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr
& y(0,x) = y_0,\,\,z(...

**0**

votes

**1**answer

61 views

### Reference request: Moving source to initial condition and vice versa in PDE problem

I am trying to find references in the literature that connect solutions of two problems given bellow. They deal with deterministic conservation laws.
Inhomogeneous Cauchy problem:
$$(1) \hspace{1cm} ...

**5**

votes

**0**answers

120 views

### Level Sets of Harmonic Maps

Can anybody point me in the direction of some references in which the level sets of harmonic maps between Riemannian manifolds are studied. (Sorry I am unfamiliar with the area and would like some ...

**10**

votes

**2**answers

292 views

### Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?

Good morning,
I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...

**5**

votes

**3**answers

298 views

### fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix

I am looking for the fundamental solution of the following PDE
$$\partial_i (a^{ij}\partial_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could ...

**3**

votes

**1**answer

89 views

### Interpretation of the Schouten bracket as an integrability condition

The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $\Lambda$ the vanishing of the Schouten bracket $[\Lambda,\Lambda]=0$ is referred to as a sort ...

**1**

vote

**0**answers

62 views

### $|\nabla \omega|^2= \alpha \omega^n+ \beta$ - References

I'm interested to know if anyone can give me some information (name, references, etc.) about this PDE
$$|\nabla \omega|^2= \alpha \omega^n+ \beta,$$
where $\omega$ is a scalar function of two or ...

**0**

votes

**0**answers

53 views

### Strichartz estimates for fractional Schrodinger equations

A pair $(q,r)$ is $\alpha-$fractional admissible if $q\geq 2, r\geq 2$ and
$$\frac{\alpha}{q} = d \left( \frac{1}{2} - \frac{1}{r} \right).$$
We take fractional Schrodinger propagator $U(t)=e^{it (...

**0**

votes

**0**answers

71 views

### Linearization around a traveling wave

In [1], Remark 2.1., the authors say the following: ".. in moving coordinates, the linearization of (37) around a traveling wave profile is given by
\begin{equation}
v_t = x \partial^2_x v + \frac{2}{...

**6**

votes

**1**answer

119 views

### “Overdetermined” Poisson equation

Consider the PDE $-\Delta u = f$ on a bounded domain $\Omega \subset \mathbb{R}^n$, where $f \in C^\infty(\bar{\Omega})$. I wish to consider both the boundary conditions $u = 0$ and $\frac{\partial u}{...

**4**

votes

**1**answer

225 views

### Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$

Consider the following PDE:
\begin{equation}
p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$}
\end{equation}
where $g$ is a flat function at the point (...

**1**

vote

**1**answer

55 views

### Is $H_0^1$ a redundant assumption in the 2D Agmon inequality?

The Wikipedia article on Agomon's inequality states the following:
Let $u\in H^2(\Omega)\cap H^1_0(\Omega)$ where $\Omega\subset\mathbb{R}^2$. Then Agmon's inequality in 2D states that there ...

**0**

votes

**0**answers

53 views

### Vector fields whose divergence is Gaussian

Let f be the pdf of a $n$ dimensional $N(0,C)$ distribution i.e up to a multiplicative constant, $f(x) = \exp(-\frac{1}{2} x'C^{-1}x)$.
Which vector fields $F$ are so that ${\rm div} (F)= f$ ?

**0**

votes

**0**answers

59 views

### A Generalized Bernstein's Problem

Suppose $\Sigma\subset \mathbb{R}^n$ be an area-minimizing hypersurfaces, the standard results in GMT tell that when $3\leq n\leq 7$, $\Sigma$ is a hyperplane; while for $n\geq 8$, there are examples ...

**3**

votes

**2**answers

201 views

### Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability

We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...

**2**

votes

**1**answer

92 views

### Optimal control theory of PDEs

This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.
Imagine one has two evolution equations
$$\partial_t u = p(x,\partial_x,f)u$$
$$\...

**2**

votes

**1**answer

130 views

### Quantitative finite speed of propagation property for ODE (cone of dependence)

Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...

**2**

votes

**0**answers

54 views

### Smooth dependence of convex functions on Monge-Ampère measure

Let $\Omega\subset\mathbb{R}^2$ be a bounded convex domain, $f$ be a positive smooth function on $\Omega$ and $\phi:\partial\Omega\rightarrow\mathbb{R}$ be a continuous function. It is known that as ...

**0**

votes

**2**answers

76 views

### Reference request for fractional Poincare inequality

Suppose we consider in $\mathbb R^n$, then how to show $\Vert f \Vert_{L^{p}} \leq C\Vert \nabla^{s}f \Vert_{L^{q}}^{\alpha}$, where $s>0$ is noninteger and $\alpha \in (0,1)$?

**3**

votes

**1**answer

68 views

### Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$

Is there any Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ in the literature?
More precisely I would like to know if there is an answer to the following
QUESTION: Let $f : \...

**0**

votes

**1**answer

85 views

### Separation of variables for PDE

Consider the PDE
$$ \partial_t f(t,x) = \Phi(x) f(t,x)+ \Psi(x)f(t,x-a)$$
$$f(0,x)=1$$
$$f(t,0)=1$$
where $a$ is a constant and $\Phi$ and $\Psi$ are some differentiable functions. To solve this I ...

**2**

votes

**1**answer

216 views

+50

### Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...

**2**

votes

**1**answer

104 views

### Liouville theorem for fractional Laplacian

Is there any Liouville type theorem for the half space problem
\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s v &= 0 &&\text{in } \mathbb R^N_+\\
v & =0 &&\text{...

**0**

votes

**0**answers

43 views

### linear fractional laplacian problem

If $U(x)$ is a classical solution of
\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s U+ U &=m(r)U &&\text{in } B \\
U &= 0 &&\text{in } \mathbb R^N \...

**1**

vote

**1**answer

74 views

### Extended Global approximation theorem

In Evans,
$\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there ...

**3**

votes

**1**answer

211 views

### Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...

**4**

votes

**1**answer

132 views

### Interpolation of some Lebesgue spaces

When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...

**1**

vote

**1**answer

97 views

### Exponential decay of resolvent kernel

For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...

**3**

votes

**1**answer

46 views

### Difference between linear and parabolic velocity profiles in Stokes flows of two fluids

For droplet interactions in low-Reynolds number flow, solutions are available when the underlying flow can be written as linear compositions of strain and rotation, see Batchelor & Green (1972a).
...

**4**

votes

**1**answer

99 views

### Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.
Consider the interval $[-\pi,\pi]$ let's say.
...

**1**

vote

**1**answer

173 views

### Boundary regularity for the Monge-Ampère equation $\det D^2u=1$

Let $\Delta$ be an open triangle in $\mathbb{R}^2$ and $u\in C^0(\overline{\Delta})\cap C^\infty(\Delta)$ be the convex function satisfying
$$
\det D^2u=1,\quad u|_{\partial\Delta}=0.
$$
Classical ...

**1**

vote

**0**answers

58 views

### Unique continuation from the boundary for inhomogeneous elliptic pde

Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real ...

**3**

votes

**0**answers

142 views

### Stationary Navier-Stokes solutions

Are there known nontrivial ($u\neq0$) stationary solutions to Navier-Stokes equations in $\mathbb R^3$ ? Not square integrable of course (that's impossible), but with self-similar amplitudes of ...

**1**

vote

**0**answers

70 views

### asymptotic behaviour of principal eigenfunctions and Large Deviations

Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...

**1**

vote

**0**answers

47 views

### On the non-existence of weak solutions to nonlinear evolutionary PDEs without time derivative

Imagine we have a nonlinear PDE, e.g. some velocity model
$$\begin{aligned} -\Delta v + (v \cdot \nabla) v + \nabla p = f \\ \text{div} \, v = 0 \end{aligned}$$
which has a weak solution for some ...

**2**

votes

**0**answers

53 views

### The space $H_\Delta(\Omega)$

I'm looking for good references for the study of the following space
$$H_\Delta(\Omega)=\{u\in L^2(\Omega) : \Delta u \in L^2(\Omega)\},$$
especially the trace theorems with proofs, where $\Omega \...

**1**

vote

**1**answer

83 views

### Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding:
\begin{equation}
C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T).
\end{equation}
what condition should be put ...

**1**

vote

**1**answer

109 views

### Existence and regularity for fractional Poisson-type equation

According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results.
Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...

**9**

votes

**2**answers

434 views

### Harmonic oscillator in spherical coordinates

It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.
More precisely, the operator
$$-\frac{d^2}{dx^2}+x^2$$
can be ...

**14**

votes

**3**answers

311 views

### What is known about sufficient conditions for the rigidity of a convex surface?

A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.
An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:...

**4**

votes

**2**answers

382 views

### Klein Gordon equation - references

The Klein Gordon equation of the form:
$\Delta u+ \lambda u^p=0$
is been studied for $p = 2$?
(i.e.$\Delta u+ \lambda u^2=0$)
If yes are there references?

**2**

votes

**0**answers

74 views

### geodesic balls in the conformal change

Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $...

**0**

votes

**0**answers

38 views

### What are tools we need to prove scattering result for NLS when propagator is unitary?

Consider nonlinear Schrödinger equation (NLS)
$$i\partial_t u + \Delta u =F(u) , u(x,0)=u_0$$
where $F$ is some nonlinearity.
Assume that there exists Banach space $X$ so that $\|e^{it\Delta} f\|_{X}...

**1**

vote

**0**answers

53 views

### Global interior estimate complex Monge-Ampere equation

Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ ...

**3**

votes

**1**answer

337 views

### Does current follow the path(s) of least (total) resistance?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (...