# 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.

2,396 questions
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### Relative boundedness of the adjoint

Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$ ...
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Let $\Omega\subset\mathbb{R}^d$, $d>1$ be an open and bounded Lipschitz domain. It is well-known, that for a given function $f\in L^q(\Omega)$ with $1<q<\infty$ and and such that $\int_\Omega ... 0answers 79 views ### formula of spectral fractional laplacian In fact, if$Uis a solution of \ \ \left\{\begin{aligned} (-\Delta)^s U &=f &&\text{in } \Omega \\ U &= g &&\text{on } \partial \Omega \\ \end{... 0answers 39 views ### Existence (linear) parabolic second order system in non-divergence form I am looking for a hint to a reference: I am dealing with a system of parabolic equations of the form $$\partial_t u_i=\sum_{j=1}^m a_{ij}(x,t)\Delta u_j,\quad i=1,\ldots,m$$ set on a open and ... 1answer 107 views ### Intuition behind using energy estimate to prove existence and uniqueness of solution for Hyperbolic equation I am trying to understand the intution behind use of energy estimate to prove existence and uniqueness(which is clear the energy estimate) of solution to hyperbolic equations. What is the basic idea ... 0answers 69 views ### Backward heat equation [closed] I have seen many countre examples concerning the instability and the ill-posedness of the backward heat equation, but all these examples are done in the||.||_{\infty}$. My questions are: 1) Is the ... 0answers 68 views ### Connection between deterministic and stochastic problems in PDEs In the study of conservation or balance laws in partial differential equations relatively often we see this two problems (problem (1) more than problem (2)): Deterministic Cauchy problem: $$(1) \... 0answers 70 views ### Existence for -\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x) Let \Omega be a smooth bounded domain. Consider the equation$$-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)u|_{\partial\Omega} = 0$$where f,g are smooth functions on \Omega and \varphi... 1answer 381 views ### L^p-norm under the heat flow Let (M, g) be a compact Riemannian manifold. Assume that u_0 is a positive smooth function on M and let u_t = e^{t \Delta} u_0 be the solution to the heat equation on (M, g) with initial ... 2answers 274 views ### Functional decaying under the heat flow (?) Let (M, g) be a compact Riemannian manifold and let a, p be two real numbers greater than 1. For any positive function v, I set$$ J(v) = \int_M \left|\nabla(v^a)\right|^p d\mu^g. $$... 1answer 171 views +50 ### Is a Sobolev map with smooth minors smooth on the whole domain? Let d\ge 3 and 2 \le k \le d-1 be integers, where at least one of k,d is odd. Let \Omega \subseteq \mathbb{R}^d be open, and let f \in W^{1,p}(\Omega,\mathbb{R}^d), for some p \ge 1. ... 0answers 55 views ### Approximate by by a function h_{n}\in C^{\infty}(\Omega\times [0,T]\times\mathbb{R}) Let \Omega be a bounded smooth subset of \mathbb{R}^n and T>0 fixed. And let h be a function defined on \Omega\times [0,T]\times\mathbb{R} with values in \mathbb{R}. For almost ... 0answers 95 views ### trace inequality for Dirichlet Neumann operator Does there exists a Sobolev trace inequality of the form$$ \|U(x, 0)\|_{L^{q}((a, b))} \leq C\sqrt{q}\| \nabla U \|_{L^{2} (\mathcal C)} ; \forall U\in H^{1}_{0, L} (\mathcal C)$$and for any$q>...
Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property: For every $p \in M$ there exist ...