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Your equation is linear, first-order and can be written as $$ \frac{\partial f}{\partial t}-v(x)\frac{\partial f}{\partial x}=v'(x) f. \tag{$\ast$}$$ Using the characteristics of the vector field, yo …

Let me first recall what is the Bressan conjecture. Take a $BV\cap L^\infty$ vector field $X$ on some open subset of $\mathbb R^n$ such that there exists an $L^\infty$ function $\alpha\ge 1$ so that $ …

The answer to my query is in fact the following: Bressan's conjecture was proven in 2020 in a paper authored by Stefano Bianchini & Paolo Bonicatto, published by Invent. math. (2020) 220:255–393, "A u …

Let me change your notations slightly: you work in three dimensions and you want to compute $$ u_{jk}(x)=\int e^{2i\pi x\cdot \xi} \frac{\xi_j\xi_k}{\vert \xi\vert^4} d\xi, $$ where the integral shoul …

In the case of hypoelliptic operators with constant coefficients, you have a characterization found by L.Hörmander, which implies that you do have a pseudo-differential parametrix with a symbol in a c …

There are two a priori different points of view. The first one is the Lagrangean where you look at the ODE $$ \dot x(t)=X(x(t)), $$ where $X$ is your given vector field. Then, as you mention, a local …

Writing $$ \langle-\Delta u + au, u\rangle_{L^2(\Omega)}=\langle f, u\rangle_{L^2(\Omega)}, $$ using the Dirichlet boundary condition, you get $$ \Vert \nabla u\Vert_{L^2(\Omega)}^2+\langle au, u\rang …

Th fundamental solution of the heat equation in $\mathbb R^d$ is $$ E_d(x,t)=H(t) (4πt)^{-d/2} e^{-\frac{\vert x\vert^2}{4t}}, $$ so that $\Vert E_d(\cdot,t)\Vert_{L^1(\mathbb R^d)}=1,$ $ \nabla_x E_d …

Let me start with a constant coefficient operator $$ P(D)=\sum_{1\le j, k\le n} a_{jk} D_jD_k,\quad D_j=\frac{\partial }{i\partial x_j}. $$ Note that in two dimensions, you have elliptic operators wit …

Let me try to reformulate (too long for a comment) your question by specializing it to a more particular (and more stable) case. Let $u_0:\mathbb R^2\rightarrow \mathbb R$, be a (smooth) Morse functio …

Following your assumptions, it seems that the mapping $L^{-1}$ sends linearly and continuously the smooth compactly supported functions into distributions and thus, from the Schwartz (Laurent) kernel …

A semi-linear PDE reads $ \mathcal Lu=F(u), $ where $\mathcal L$ is a linear operator and $F$ is a function. A quasi-linear PDE with order $m$ reads $ \mathcal L\bigl((\partial_x^\alpha u)_{\vert \alp …

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form $$ u=\sum_{1\le j\le n} u_j dx_j,\quad …

You will find a study of self-adjoint operators of type $$ \sum_{j=1}^r(X_j^*-iY_j^*)(X_j+iY_j), $$ where $X_j, Y_j$ are real-valued vector fields for instance in the Helffer-Nier Lecture Note (Lectur …